diff --git "a/MtAzT4oBgHgl3EQfV_zH/content/tmp_files/load_file.txt" "b/MtAzT4oBgHgl3EQfV_zH/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/MtAzT4oBgHgl3EQfV_zH/content/tmp_files/load_file.txt" @@ -0,0 +1,1365 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf,len=1364 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='01295v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='CV] 3 Jan 2023 NEVANLINNA THEORY ON COMPLETE K¨AHLER MANIFOLDS WITH NON-NEGATIVE RICCI CURVATURE XIANJING DONG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This paper considers the equiv-distribution of meromorphic mappings from a complete K¨ahler manifold with non-negative Ricci cur- vature into a complex projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' When the domain manifold is of maximal volume growth, one establishes a second main theorem in Nevanlinna theory with a refined error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' As an important result, we prove a sharp defect relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Furthermore, we apply our main theorems to the problems on the propagation of algebraic dependence, then finally we obtain some unicity results for dominant meromorphic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Motivation 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Main results 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Preliminaries 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Curvatures in differential geometry 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Comparison theorems on Riemannian manifolds 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Existence of positive global Green functions 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Positivity, ampleness and bigness for Q-line bundles 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Poincar´e-Lelong formula and Jensen-Dynkin formula 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Nevanlinna theory on complete K¨ahler manifolds 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Nevanlinna’s functions 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Calculus lemma and logarithmic derivative lemma 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Second main theorem and defect relation 25 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 32H30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 32H04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Nevanlinna theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' second main theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' defect relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' al- gebraic dependence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' unicity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The case for singular divisors 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Application to algebraic dependence problems 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Consequences of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Propagation of algebraic dependence 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Unicity theorems 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Targets are compact Riemann surfaces and Pn(C) 39 References 42 NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In 1972, Carlson-Griffiths [9] devised an equiv-distribution theory of holo- morphic mappings from Cm into a complex projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This theory is undoubtedly a great progress in the study of Nevanlinna theory [37], which was further generalized by Griffiths-King [22] to the affine algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Many famous scholars made efforts towards an extension of Carlson-Griffiths theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For example, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Stoll [45, 46] extended it to the parabolic manifolds, Lang-Cherry [33] promoted it onto a covering space of Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' More extensions and developments refer to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Noguchi [34, 35], M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Ru [38], B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Shiffman [39], F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Sakai [40, 41], Wong-Stoll [49], Wong-Wong [50] and Yang-Ru [51], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The study of Nevanlinna theory on K¨ahler manifolds via Brownian motion (initialized by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Carne [12]) can be traced back to the work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Atsuji [1] in 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Later, Atsuji wrote a series of papers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' [2, 3, 4, 5]) to study the value distribution of meromorphic functions on a complete K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' His excellent contribution to Nevanlinna theory seems to be the second main theorem of meromorphic functions on a complete K¨ahler manifold with non- positive sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Along the line of Atsuji, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Dong [17] further extended the notion for Nevanlinna’s functions and generalized the Carlson- Griffiths theory from Cm to the complete K¨ahler manifolds with non-positive sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' More details about the Nevanlinna theory via Brownian motion refer to Dong-He-Ru [16], Dong-Yang [18] and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Dong [19], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Although the Nevanlinna theory on K¨ahler manifolds has been studied for a long time, we know little about this theory if domain is not non-positively curved, particularly, if domain is of non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In fact, it is a long-standing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Refer to [42], we see that such class of manifolds are in abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Motivated by those, we shall focus on the Carlson-Griffith theory on complete K¨ahler manifolds with non-negative Ricci curvature, and we would like to derive a sharp defect relation in Nevanlinna theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Another motivation of this paper is the propagation problem of algebraic dependence of meromorphic mappings, which was first studied by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Smiley [43] in 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This problem has aroused widespread concern among scholars, referred to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Aihara [6, 7, 8], Dulock-Ru [14], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Drouilhet [15], H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Fujimote [20], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Ji [24, 25] and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Stoll [47], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' So far, the best result for propaga- tion theorems may be due to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Aihara [8] who obtained a beautiful criteria for the propagation of algebraic dependence of dominant meromorphic map- pings from an analytic finite covering space of Cm into a complex projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Aihara’s criteria improved the criteria of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Stoll [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In fact, he used the estimate for the ramification term obtained by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Noguchi [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In Aihara’s criteria, however, there still exist some unnatural restrictions, such as “fibers separated by mappings”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 4 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG In this paper, we shall apply our second main theorem to the propagation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The purpose is to prove some propagation theorems for algebraic dependence of dominant meromorphic mappings on a complete K¨ahler man- ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We not only generalize Aihara’s criteria to complete K¨ahler manifolds, but also remove those restrictions such as “ramification estimate” and “fibers separated by mappings” appeared in Aihara’s criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In Atsuji [5] and Dong [17], the Nevanlinna’s functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', characteristic function, proximity function and counting function) are defined on a geodesic ball of a complete K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, in order to establish a second main theorem, they must estimate the local Green functions for the geodesic balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A reasonable estimate was obtained under a non-positive sectional curvature condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' But, the estimate is not so accurate enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' That’s why a growth condition was assumed in their defect relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Not only that, their methods fail to the complete K¨ahler manifolds with non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It’s because that we don’t seem to obtain a suitable estimate (in terms of integral forms) on local Green functions for the geodesic balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is a great difficulty faced with in the study of Nevanlinna theory on a complete K¨ahler manifold with non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' To overcome this difficulty, our strategy is to use the global Green function instead of local Green functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Roughly speaking, we apply the global Green function to define the relatively compact r-domains which exhaust the manifold, and we then estimate the local Green functions for those domains and the harmonic measures for their boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Define the Nevanlinna’s functions on r-domains and use the approach in this paper, we establish the Carlson-Griffiths theory on such class of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete K¨ahler manifold with maximal volume growth and non-negative Ricci curvature, and let X be a complex projective manifold of complex dimension not greater than that of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A meromorphic mapping f : M → X is said to be differentiably non-degenerate, if the rank of differential df equals dimC X at some point in M \\I(f), where I(f) is the indeterminacy set of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In this case, we also call f a dominant meromorphic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Next, let us state the main results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Some notations will be introduced later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We obtain the following second main theorem: Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D ∈ |L| be a divisor of simple normal crossing type, where L is a positive line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a differentiably non- degenerate meromorphic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that Tf(r, L) + Tf(r, KX) + T(r, R) ≤ N f(r, D) + O � log+ Tf(r, L) + δ log r � holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 5 Later, we will note that Tf(r, L) ≥ O(log r) as r → ∞ for any nonconstant meromorphic mapping, which means that the error term δ log r in the above Theorem I is refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This yields a defect relation: ¯δf(D) ≤ �c1(K∗ X) c1(L) � − lim inf r→∞ T(r, R) Tf(r, L) ≤ �c1(K∗ X) c1(L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We give an application of Theorem I to the propagation problems of alge- braic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let S = S1 ∪ · · · ∪ Sq, where S1, · · · , Sq are hypersurfaces of M such that dimC Si ∩Sj ≤ dimC M −2 for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D = D1 +· · · +Dq, where D1, · · · , Dq ∈ |L| such that D has simple normal crossings, here L is an ample line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Now, given l dominant meromorphic mappings f1, · · · , fl : M → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that there are integers k1, · · · , kq (may be +∞) such that Sj = Suppkjf ∗ i Dj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 1 ≤ i ≤ l, 1 ≤ j ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set k0 = max{k1, · · · , kq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Given an indecomposable hypersurface Σ of M1× · · × Ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Moreover, define a Q-line bundle L0 ∈ Pic(X) ⊗ Q by L0 = � q � j=1 kj kj + 1 � L ⊗ � − ˜γlk0 k0 + 1F0 � , where F0 is some big line bundle over X and ˜γ is a positive rational number depending only on Σ and F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Employing Theorem I, we obtain some propagation theorems of algebraic dependence in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For example, we show that Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, · · · , fl : M → X be dominant meromorphic mappings given as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that f1, · · · , fl are Σ-related on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If L0 ⊗ KX is big, then f1, · · · , fl are Σ-related on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The propagation theorems of algebraic dependence can apply to the unic- ity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For example, Theorem II derives a unicity theorem: Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → P1(C) be nonconstant holomorphic map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let a1, · · · , aq be distinct points in P1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have (a) Assume that Suppf ∗ 1 aj = Suppf ∗ 2 aj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If q ≥ 5, then f1 ≡ f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) Assume that Supp1f ∗ 1 aj = Supp1f ∗ 2 aj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If q ≥ 7, then f1 ≡ f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Curvatures in differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In order to simplify formulas, the Einstein’s convenience is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 6 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Curvatures of Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let (M, g) be a Riemannian manifold with Lapalce-Beltrami operator ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ∇ be the Levi-Civita connection of g on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Recall that the Riemannian curvature tensor is defined by R = Rijkldxi ⊗ dxj ⊗ dxk ⊗ dxl, where ∂j = ∂/∂xj and Rijkl = glpRp ijk with Γk ij = 1 2gkl (∂jgil + ∂igjl − ∂lgij) , Rl ijk = ∂kΓl ji − ∂jΓl ki + Γl kpΓp ji − Γl jpΓp ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For any point x ∈ M and every tangent 2-plane σ to M at x, the sectional curvature of σ is defined by K(X, Y ) = R(X, Y, Y, X) ∥X ∧ Y ∥2 , where X, Y ∈ TxM is a basis of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Define the Ricci curvature tensor by Ric(X, Y ) = R(X, ej, ej, Y ), where {e1, · · · , en} is an orthonormal basis of TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We can write Ric as Ric = Rijdxi ⊗ dxj, where Rij = Rp ipj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For 0 ̸= X ∈ TxM, the Ricci curvature at x in the direction X is defined by Ric(X, X)/∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Moreover, the scalar curvature of M is defined by s = gijRij, which is the trace of Ricci curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Curvatures of K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Now, we turn to Hermitian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let (M, h) be a Hermitian manifold with Hermitian connection ˜∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that M can be regarded as a Riemannian manifold with Riemannian metric g = ℜh, where g is extended linearly over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have the Levi-Civita connection ∇ on M as a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Also, we extend ∇ linearly over C to TM = TM ⊗C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In general, ˜∇ ̸= ∇ since the torsion tensor of ˜∇ may not vanish for the general Hermitian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Therefore, the Laplacian ˜∆ of ˜∇ does not coincide with the Laplace-Beltrami operator ∆ of ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' However, the case when ˜∇ = ∇ happens provided that M is a K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Consequently, ∆ = ˜∆ = 2hi¯j ∂2 ∂zi∂¯zj NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 7 acting on a C 2-class function when M is K¨ahlerian, where (hi¯j) is the inverse of metric matrix (hi¯j) of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Suppose that M is a K¨ahler manifold with K¨ahler metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The K¨ahlerness of h means that ⟨JX, JY ⟩ = ⟨X, Y ⟩ and ∇X(JY ) = J∇XY for X, Y ∈ TM, where J is the canonical complex structure of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Extending J and R linearly over C to TM = T 1,0 M ⊕ T 0,1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let X, Y ∈ T 1,0 M,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The holomorphic bisectional curvature H(X, Y ) of M at x in the holomorphic directions X, Y is defined by H(X, Y ) = R(X, X, Y, Y ) ∥X ∧ Y ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' When X = Y, we call H(X) = H(X, X) the holomorphic sectional curvature of M at x in the holomorphic direction X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The Ricci curvature tensor RicC of M is defined by RicC(X, Y ) = R(X, Y , ej, ej), where {e1, · · · , em} is an orthonormal basis of T 1,0 M,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Applying the K¨ahlerness of M, RicC can be computed in such way: RicC = Ri¯jdzi ⊗ d¯zj, where Ri¯j = − ∂2 ∂zi∂¯zj log det(hs¯t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Naturally, it defines the following Ricci form R = −ddc log det(hs¯t) = √−1 2π Ri¯jdzi ∧ d¯zj, where d = ∂ + ∂, dc = √−1 4π (∂ − ∂) so that ddc = √−1 2π ∂∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A well-known theorem by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Chern asserts that R defines a cohomology class in the de Rham cohomology group H2 DR(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We can define the Ricci curvature at x in the holomorphic direction X by RicC(X, X)/∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Indeed, we have the scalar curvature sC of M defined by (1) sC = hi¯jRi¯j = −1 2∆ log det(hs¯t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A comparison gives that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [52]) (2) s = 2sC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Comparison theorems on Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Fix a point o ∈ M, define ρ(x) = dist(o, x), ∀x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 8 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG It is well-known that ρ is Lipschitz-continuous and thus differentiable almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let Cut(o) be the cut locus of o, then Cut(o) has measure zero and ρ is smooth on M \\ Cut(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A space form is a simply-connected complete Riemannian manifold with constant sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ˜ M be a space form and ˜ρ(˜x) be the distance function of ˜x ∈ ˜ M from a fixed point ˜o ∈ ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ˜∆ be the Laplace-Beltrami operator on ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The Laplacian comparison theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [44]) states that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 (Laplacian comparison theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be an n-dimensional complete Riemannian manifold with Ric ≥ −(n − 1)K, where K is a non- negative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ˜ M be an n-dimensional space form with sectional curvature −K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that x ∈ M and ˜x ∈ ˜ M such that ρ(x) = ˜ρ(˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If x is not a cut point of ρ, then ∆ρ(x) ≤ ˜∆ρ(˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By finding the Jacobian field along a minimal geodesic, one can compute ˜∆˜ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 that Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be an n-dimensional complete Riemannian manifold with Ric ≥ −(n − 1)K, where K is a non-negative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then ∆ρ ≤ (n − 1) �√ K + 1 ρ � on M \\ Cut(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In particular, if Ric(M) ≥ 0, then ∆ρ ≤ n − 1 ρ in the sense of distributions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We also state the following Bishop’s volume comparison theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [44]): Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3 (Volume comparison theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be an n-dimensional complete Riemannian manifold with Ric ≥ (n−1)K, where K is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ˜ M be an n-dimensional space form with sectional curvature K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for r > 0, Vx(r)/V (K, r) is non-increasing in r and Vx(r) ≤ V (K, r), where Vx(r) is the volume of geodesic ball centered at x with radius r in M, and V (K, r) is the volume of geodesic ball with radius r in ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Existence of positive global Green functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be an n-dimensional non-compact complete Riemannian manifold, whose Laplace-Beltrami operator is ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set ρx(y) = dist(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In this paper, a positive global Green function (if it exists) for M means a Green function G(x, y) of ∆/2 on M ×M, which is a smooth function on M ×M \\diag(M × M) satisfying that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [44]) 1◦ G(x, y) ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2◦ G(x, y) = G(y, x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 3◦ ∆yG(x, y) = 0 for y ̸= x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 4◦ Fix any x, when y → x G(x, y) ∼ C2 log ρ−1 x (y), n = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' G(x, y) ∼ Cnρ2−n x (y), n > 2, where Cn (n ≥ 2) is a positive constant such that −1 2∆yG(x, y) = δx(y) in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that G(x, y) with n = 2 may change sign, thus the positivity of G(x, y) in this case should be understood to be outside a compact subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is known that there always exists a global Green function for any non- compact complete Riemannian manifold M, this fact was confirmed for the first time by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Malgrange [32], while a constructive proof, and best suited for applications, which was presented by Li-Tam [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Moreover, if M is non- parabolic (it admits a positive global Green function), Li-Tam’s construction produces a unique minimal positive global Green function for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In case that M has a nonconstant positive harmonic function, Schoen-Yau [44] confirmed that M admits a positive global Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In particular, they proved that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4 (Schoen-Yau).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete Riemann- ian manifold with lower bounded Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then M admits a positive global Green function, and thus there exists a unique minimal positive global Green function for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' There are necessary conditions for the existence of a positive global Green function for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Cheng-Yau [13] gave the first result in this direction, which involves only the volume growth of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The first major result for the suffi- ciency was due to Li-Yau [31] and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Varopoulos [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Utilizing the estimates of the heat kernel, Li-Yau [31] proved the following theorem: 10 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5 (Li-Yau).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete Riemannian manifold with non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If � ∞ 0 t Vx(t)dt < ∞, then there exists a positive global Green function for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Furthermore, the unique minimal positive global Green function G(x, y) for M satisfies that C−1 � ∞ ρx(y) t Vx(t)dt ≤ G(x, y) ≤ C � ∞ ρx(y) t Vx(t)dt for x ̸= y, where Vx(t) is the volume of geodesic ball centered at x with radius t, and C is a positive constant depending only on the dimension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Positivity, ampleness and bigness for Q-line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a compact complex manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We recall that a holomorphic line bundle L over M is said to be positive, if there exists a Hermitian metric h on L such that the Chern form c1(L, h) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' L is said to be very ample, if L provides a holomorphic embedding of X into a complex projective space, namely, there exist holomorphic sections e0, · · · , eN in H0(M, L) (the space of all holomorphic sections of L over M) such that the mapping [e0 : · · · : eN] : M ֒→ PN(C) is a holomorphic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is said that L is ample if L⊗ν (written as νL for short) is very ample whenever ν is a sufficiently large positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A known fact asserts that L is ample if and only if L is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A holomorphic line bundle F over M is said to be big, if dim H0(M, νL) ≥ Cνdim M for all sufficiently large integers ν > 0 and some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We introduce two well-known theorems by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Kodaira (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [21, 27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6 (Kodaira).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let F be a holomorphic line bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then F is big if and only if there exists a singular metric e−ψ on F such that ddc[ψ] ≥ δωL in the sense of currents for an arbitrary ample line bundle L over M, where ωL is a Hermitian metric on L and δ is a sufficiently small positive number depending on ωL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' As a matter of convenience, instead of ddc[ψ] ≥ δωL in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6, one writes F ≥ δL in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Indeed, we sometimes write F ⊗L = F +L in short for two Q-line bundles F, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let L be an ample line bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If F is a big line bundle over M, then F − δL is big for any sufficiently small number δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 11 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='8 (Kodaira).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let F be a big line bundle over M, and let L be a holomorphic line bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then there exists a positive integer µ such that µF ⊗ L is big, and H0(X, µF ⊗ L) ̸= 0 for any sufficiently large positive integer µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In general, a Q-line bunde is defined as an element belonging to Pic(M)⊗ Q, here Pic(M) is the Picard group over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let F ∈ Pic(M)⊗Q be a Q-line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then F is said to be positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' ample and big), if νF ∈ Pic(M) is positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' ample and big) for some integer ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10, it is trivial to see that Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let L be an ample Q-line bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If F is a big Q-line bundle over M, then F − δL is big for any sufficiently small positive number δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let F be a big Q-line bundle, and L be a Q-line bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then there exist positive integers µ, ν such that µF, νL ∈ Pic(M) and µF ⊗ νL is big, and H0(X, µF ⊗ νL) ̸= 0 for some sufficiently large positive integers µ, ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Poincar´e-Lelong formula and Jensen-Dynkin formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a m-dimensional K¨ahler manifold with Laplace-Beltrami oper- ator ∆ associated with the K¨ahler metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let us introduce two important formulas as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Poincar´e-Lelong formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let (L, hL) be a Hermitian holomorphic line over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We define the Chern form of L associated to hL by c1(L, hL) = −ddc log hL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If s is the canonical section of L over M with zero divisor D, then s defines a positive (1, 1)-current of integration [D] over D by [D](ϕ) = � D ϕ for any (m − 1, m − 1)-form ϕ with compact support on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let u be a plurisubharmonic function u on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If u is of C 2-class, then ddcu = ∂2u ∂zi∂¯zj √−1 2π dzi ∧ d¯zj ≥ 0, 12 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG by which we mean that the matrix of all the second order partial derivatives of u is semi-positive definite (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [36], Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', ∂2u ∂zi∂¯zj ξi ¯ξj ≥ 0 for every complex vector (ξ1, · · · , ξm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If u is not of C 2-class, one can use the notation ∂2[u]/∂zi∂¯zj in the sense of (Schwartz) distributions, which defines a positive Radon measure, so that ddc[u] = ∂2[u] ∂zi∂¯zj √−1 2π dzi ∧ d¯zj ≥ 0 which is a positive (1, 1)-current: ddc[u](ϕ) = � M ddc[u] ∧ ϕ for any (m−1, m−1)-form ϕ with compact support on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using the K¨ahler property of M, we see that ∆u = 2hi¯j ∂2[u] ∂zi∂¯zj ≥ 0 in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, u is subharmonic on M as a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In particular, if f is a nonzero meromorphic function on M, then log |f|2 is a plurisubharmonic function and it defines a positive (1, 1)-current ddc[log |f|2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We state the Poincar´e-Lelong formula (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [9]) as follows: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11 (Poincar´e-Lelong formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f be a nonzero meromorphic function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then ddc � log |f|2� = [(f = 0)] − [(f = ∞)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let (L, hL) be a Hermitian holomorphic line bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let s be the canonical section of L over M with zero divisor D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then ddc � log ∥s∥2 hL � = [D] − c1(L, hL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Remark that the K¨ahler condition for M is not necessary in both theorems above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In fact, we only need to require that M is a complex manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Jensen-Dynkin formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let Ω ⊊ M be a relatively compact domain with piecewise smooth bound- ary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Fix a point o ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let gΩ(o, x) denote the positive Green function of ∆/2 for Ω with a pole at o, satisfying Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that gΩ(o, x) defines a unique harmonic measure π∂Ω for ∂Ω with respect to o, say dπ∂Ω(x) = −1 2 ∂gΩ(o, x) ∂⃗ν dσ∂Ω(x), ∀x ∈ ∂Ω, NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 13 where ∂/∂⃗ν is the inward normal derivative on ∂Ω, dσ∂Ω is the Riemannian area element of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A δ-subharmonic function is defined as the difference of two subharmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We introduce the following Jensen-Dynkin formula that is viewed as a generalization of Green-Jensen formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The formula plays an important role in the study of Nevanlinna theory on complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The Jensen-Dynkin formula (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', [17, 18]) reads that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='13 (Jensen-Dynkin formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let u be a δ-subharmonic func- tion on M such that u(o) ̸= ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then � ∂Ω u(x)dπ∂Ω(x) − f(o) = 1 2 � Ω gΩ(o, x)∆u(x)dv(x), where dv is the Riemannian volume element of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We remark that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='13 remains true when M is just a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Recall that the K¨ahler metric form of M is defined by α = √−1 π hi¯jdzi ∧ d¯zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Wirtinger’s formula, dv = πmαm/m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='. Let u be a plurisubharmonic func- tion on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By the K¨ahler property of h, we see that u is subharmonic and ∆u = 4mddc[u] ∧ αm−1 αm in the sense of distributions or currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The following consequence follows from the above arguments and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let u be a plurisubharmonic function on M such that u(o) ̸= ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then � ∂Ω udπ∂Ω − f(o) = 2πm (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' � Ω gΩ(o, x)ddc[u] ∧ αm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Nevanlinna theory on complete K¨ahler manifolds 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Nevanlinna’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete K¨ahler manifold of complex dimension m, with Laplace-Beltrami operator ∆ associated to K¨ahler metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that M has non-negative Ricci curvature as a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Indeed, assume that M is of maximal volume growth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', (3) lim inf r→∞ r−2mV (r) > 0, where V (r) is the volume of geodesic ball B(r) centered at a reference point o with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 14 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG If m ≥ 2, using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4 or Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5, then we see that there exists the unique minimal positive global Green function G(x, y) of ∆/2 for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For r > 0, define the r-domain ∆(r) by ∆(r) = � x ∈ M : G−1(o, x) < r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that {∆(rn)}∞ 1 exhausts M compactly for a strictly increasing sequence {rn}∞ 1 with rn → ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set gr(o, x) = G(o, x) − r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Evidently, gr(o, x) defines the positive Green function of ∆/2 for ∆(r), with a pole at o satisfying Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Denote by πr the harmonic measure for ∂∆(r) with respect to o, which is defined by gr(o, x) as follows: (4) dπr(x) = −1 2 ∂gr(o, x) ∂⃗ν dσr(x), ∀x ∈ ∂∆(r), where ∂/∂⃗ν is the inward normal derivative on ∂∆(r), dσr is the Riemannian area element of ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If m = 1, the curvature assumption implies that M is a parabolic Riemann surface and each global Green function for M must change sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Again, since M is of maximal volume growth, we see that M is conformally equivalent to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Equip a conformal metric ds2 = λds2 0 on M, where λ is a positive smooth function and ds2 0 is the standard Euclidean metric on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is noted that the Laplacian ∆ in real dimension 2 is conformally invariant, thus it produces a global Green function G(x, y) of ∆/2 for M: (5) G(x, y) = 1 π log d(x, y), where d(x, y) is the Riemannian distance between x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In this case, we define the r-domain ∆(r) by ∆(r) = {x ∈ M : d(o, x) < r} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is clear that (6) gr(o, x) = 1 π log r d(o, x) gives the Green function of ∆/2 for ∆(r), with a pole at o satisfying Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By (4), the harmonic measure πr for ∂∆(r) with respect to o is dπr(x) = 1 2πrdσr(x), ∀x ∈ ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let X be a complex projective manifold where we put an ample Hermitian line bunlde (L, hL) so that its Chern form c1(L, hL) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a meromorphic mapping, by which we mean that f is defined by a holomorphic mapping f0 : M \\ I → X such that the closure G(f0) of the graph G(f0) of f0 is an analytic subset of M ×X, and the natural projection π : G(f0) → M NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 15 is a proper mapping, where I is an analytic subset (called the indeterminacy set of f) of M satisfying dimC I ≤ m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that o ̸∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set ef = 2mf ∗c1(L, h) ∧ αm−1 αm = −1 2∆ log(h ◦ f), where α = √−1 π m � i,j=1 hi¯jdzi ∧ d¯zj is the K¨ahler form of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For any analytic hypersuface A of M, we put N(r, A) = πm (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' � A∩∆(r) gr(o, x)αm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The Nevanlinna’s functions (characteristic function, proximity function and counting function) are defined respectively by Tf(r, L) = 1 2 � ∆(r) gr(o, x)efdv, mf(r, D) = � ∂∆(r) log 1 ∥sD ◦ f∥dπr, Nf(r, D) = N(r, f ∗D), where sD is the canonical section of L with zero divisor D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Moreover, define the simple counting function by N f(r, D) = N(r, Suppf ∗D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using Jensen-Dynkin formula and Poincar´e-Lelong formula (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='13), we have the first main theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' [17] also): Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a meromorphic mapping such that f(o) ̸∈ SuppD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then mf(r, D) + Nf(r, D) = Tf(r, L) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Calculus lemma and logarithmic derivative lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Estimate on harmonic measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a m-dimensional complete Hermitian manifold with non-negative Ricci curvature, here m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Fix a point o ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For a convenience, we denote by ρ(x) the Riemannian distance function of x from o, and by V (r), A(r) the volume and area of B(r), ∂B(r), respectively, where B(r) is the geodesic ball centered at o with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that M is of maximal volume growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set θ(r) = (2m)−1r1−2mA(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 16 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG The Bishop’s volume comparison theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3) says that there exists a number θ > 0, independent of o, such that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' [29] also) (7) θ(r) ց θ, r−2mV (r) ց θ as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Laplacian comparison theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2), one obtains ∆ρ(x) ≤ 2m − 1 ρ , ∆ρ1−2m(x) ≥ 0 in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Recall that the notation G(o, x) is the minimal positive global Green function of ∆/2 for M with a pole at o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' According to the estimations of Li-Yau (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5) and (7), there exists a constant A(m, θ) > 0 depending only on m, θ such that A−1(m, θ)ρ2−2m(x) ≤ G(o, x) ≤ A(m, θ)ρ2−2m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In further, Colding-Minicozzi II [11] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Li-Tam-Wang [29] also) obtained the asymptotic behavior of G(o, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In fact, they showed that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a m-dimensional (m ≥ 2) Hermitian manifold with non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If M has maximal volume growth, then G(o, x) satisfies the asymptotic behavior (8) lim x→∞ 2m(m − 1)θG(o, x) ρ2−2m(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If m = 1, it was shown by Li-Tam [30] that there also has the asymptotic behavior lim x→∞ θG(o, x) log ρ(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Next, we give a gradient estimate on G(o, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have (a) If m = 1, then ∥∇G(o, x)∥ = π−1r−1, x ∈ ∂∆(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) If m ≥ 2, then there exists a constant Cm > 0 such that max x∈∂∆(r) ∥∇G(o, x)∥ ≤ Cmr− 2m−1 2m−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If m = 1, then ∆(r) is just the geodesic disc centered at o with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It means that ρ(x) = r for x ∈ ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, it follows from (5) that ∥∇G(o, x)∥ = ∂G(o, x) ∂⃗ν = π−1r−1, x ∈ ∂∆(r), where ∂/∂⃗ν is the inward normal derivative on ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 17 Next, we show that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since G(o, x) → 0, ρ2−2m(x) → 0 as x → ∞, utilizing L’Hˆospital’s rule, then it yields from (8) that for x ∈ ∂∆(r) (without loss of generality, we may assume that x is not a cut point of o): lim r→∞ 2m(m − 1)θ∂G(o, x)/∂⃗ν ∂ρ2−2m(x)/∂⃗ν = lim r→∞ −2m(m − 1)θ∥∇G(o, x)∥ (2 − 2m)ρ1−2m(x)∂ρ(x)/∂⃗ν = 1, where ∂/∂⃗ν stands for the inward normal derivative on ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On the other hand, (8) gives that lim r→∞ 2m(m − 1)θ rρ2−2m(x) = 1, x ∈ ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Or equivalently, lim r→∞ (2m(m − 1)θ) 2m−1 2m−2 r− 2m−1 2m−2 ρ1−2m(x) = 1, x ∈ ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, we see that ∆(r) turns increasingly to a geodesic ball centered at o, as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This implies that lim r→∞ ∂ρ(x) ∂⃗ν = 1, x ∈ ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining the above, we get lim r→∞ 2m(m − 1)θ∥∇G(o, x)∥ (2m − 2)ρ1−2m(x) = lim r→∞ 2m(m − 1)θ∥∇G(o, x)∥ (2m − 2)(2m(m − 1)θ) 2m−1 2m−2 r− 2m−1 2m−2 = 1 for x ∈ ∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Therefore, there exists a constant Cm > 0 such that max x∈∂∆(r) ∥∇G(o, x)∥ ≤ Cmr− 2m−1 2m−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have (a) If m = 1, then dπr = 1 2πrdσr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) If m ≥ 2, then dπr ≤ Cm2−1r− 2m−1 2m−2 dσr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In the above, dσr is the Riemannian area element of ∂∆(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 18 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The corollary follows immediately from the following relation dπr(x) = 1 2∥∇G(o, x)∥dσr(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Calculus lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' To establish the desired calculus lemma which plays a key role in deriving the second main theorem, we still need a lower bound of gr(o, x) in terms of integral forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set ρr = max x∈∂∆(r) ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have (a) If m = 1, then gr(o, x) = 1 π � r ρ(x) t−1dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) If m ≥ 2, then for any ǫ > 0, there exists ρ0 > 0 such that for all sufficiently large r > 0 and x ∈ ∆(r) with ρ(x) > ρ0, we have gr(o, x) ≥ � 1 mθ − ǫ � � ρr ρ(x) t1−2mdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' When m = 1, it follows immediately from (6) that (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In what follows, we show that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In view of (8), we obtain G(o, x) = ρ2−2m(x) 2m(m − 1)θ + o � ρ2−2m(x) 2m(m − 1)θ � as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that ∂∆(r) is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using the definition of ∂∆(r), the above equality implies that 1 r = ρ2−2m r 2m(m − 1)θ + o � ρ2−2m r 2m(m − 1)θ � as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, for any ǫ0 > 0, there exists ρ0 > 0 such that for sufficiently large r > 0 and x ∈ ∆(r) with ρ(x) > ρ0 gr(o, x) = G(o, x) − r−1 ≥ � 1 2m(m − 1)θ − ǫ0 � � ρ2−2m(x) − ρ2−2m r � ≥ � 1 mθ − (2m − 2)ǫ0 � � ρr ρ(x) t1−2mdt = � 1 mθ − ǫ � � ρr ρ(x) t1−2mdt, where ǫ = (2m − 2)ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 19 Now, we prove the following so-called calculus lemma: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let k ≥ 0 be a locally integrable function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that k is locally bounded at o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for any δ > 0, there exist a positive constant C and a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that (a) If m = 1, then � ∂∆(r) k(x)dπr(x) ≤ Crδ � � ∆(r) gr(o, x)k(x)dv(x) �(1+δ)2 holds for all r > 0 outside Eδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) If m ≥ 2, then � ∂∆(r) k(x)dπr(x) ≤ Cr 2m−1 2m−2 δ � � ∆(r) gr(o, x)k(x)dv(x) �(1+δ)2 holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If m = 1, then ∆(r) is the geodesic disc centered at o with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It yields from (6) that � ∆(r) gr(o, x)k(x)dv(x) = � r 0 dt � ∂∆(t) gr(o, x)k(x)dσt(x) = 1 π � r 0 � � r t s−1ds � ∂∆(t) k(x)dσt(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set Γ(r) = � r 0 � � r t s−1ds � ∂∆(t) k(x)dσt(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then dΓ(r) dr = r−1 � r 0 � � ∂∆(t) k(x)dσt(x) � dt, which yields from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4 that (9) d dr � rΓ′(r) � = � ∂∆(r) k(x)dσr(x) = 2πr � ∂∆(r) k(x)dπr(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On the other hand, applying Borel lemma to (rΓ′)′ twice, then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesque measure such that (10) d dr � rΓ′(r) � ≤ r1+δΓ(1+δ)2(r) holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining (9) with (10), we get � ∂∆(r) k(x)dπr(x) ≤ 1 2πrr1+δΓ(1+δ)2(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By the definition of Γ, we have (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 20 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Next, we show that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5 and (8), for any ǫ > 0, there is a sufficiently large number r0 > 0 such that gr(o, x) ≥ � 1 mθ − ǫ � � ρr ρt s1−2mds holds for all x ∈ ∂∆(t) with r0 < t ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, � ∆(r) gr(o, x)k(x)dv(x) (11) = � r r0 dt � ∂∆(t) gr(o, x)k(x)dσt(x) + O(1) ≥ C(m, ǫ) � r r0 � � ρr ρt s1−2mds � ∂∆(t) k(x)dσt(x) � dt, where C(m, ǫ) = 1 mθ − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set Λ(r) = � r r0 � � ρr ρt s1−2mds � ∂∆(t) k(x)dσt(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A direct computation leads to dΛ(r) dr = ρ1−2m r dρr dr � r r0 � � ∂∆(t) k(x)dσt(x) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In further, we have (12) d dr �ρ2m−1 r Λ′(r) ρ′r � = � ∂∆(r) k(x)dσr(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since dσr ≥ 2C−1 m r 2m−1 2m−2 dπr due to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4, then it follows from (12) that (13) � ∂∆(r) k(x)dπr(x) ≤ Cm2−1r− 2m−1 2m−2 d dr �ρ2m−1 r Λ′(r) ρ′r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using Borel lemma twice, for any δ > 0, we have (14) d dr �ρ2m−1 r Λ′(r) ρ′r � ≤ ρ(2m−1)(1+δ) r (ρ′r)1+δ Λ(1+δ)2(r) holds for all r > 0 outside a subset Fδ ⊆ (0, ∞) with finite Lebesque measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining (13) with (14), we get � ∂∆(r) k(x)dπr(x) ≤ Cm2−1r− 2m−1 2m−2 ρ(2m−1)(1+δ) r (ρ′r)1+δ Λ(1+δ)2(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 21 Since (8) implies that ρr ∼ (2m(m − 1)θ)− 1 2m−2 r 1 2m−2 , ρ′ r ∼ 1 as r → ∞, then for r0 > 0 sufficiently large, we have (0 < δ < 1) � ∂∆(r) k(x)dπr(x) ≤ Cm2−1r− 2m−1 2m−2 � 2 (2m(m − 1)θ)− 1 2m−2 r 1 2m−2 �(2m−1)(1+δ) (2−1)2 Λ(1+δ)2(r) ≤ Cm24m−1 (2m(m − 1)θ)− 2m−1 2m−2 r 2m−1 2m−2 δΛ(1+δ)2(r) holds for all r > r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, by this with (11) � ∂∆(r) k(x)dπr(x) ≤ Cm24m−1 (2m(m − 1)θ)− 2m−1 2m−2 r 2m−1 2m−2 δ C(1+δ)2(m, ǫ) �� ∆(r) gr(o, x)k(x)dv(x) �(1+δ)2 ≤ Cm24m−1 (2m(m − 1)θ)− 2m−1 2m−2 r 2m−1 2m−2 δ (2−1m−1θ−1)4 �� ∆(r) gr(o, x)k(x)dv(x) �(1+δ)2 = Cr 2m−1 2m−2 δ �� ∆(r) gr(o, x)k(x)dv(x) �(1+δ)2 holds for all r > 0 outside Eδ = Fδ ∪ (0, r0], where C = Cm24m+3m4θ4 (2m(m − 1)θ)− 2m−1 2m−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This shows that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Logarithmic derivative lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' To establish the second main theorem, we still need to prove a logarithmic derivative lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ∇ be the gradient operator on M associated with the K¨ahler metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ψ be a meromorphic function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The norm of the gradient of ψ is defined by ∥∇ψ∥2 = 2 m � i,j=1 hij ∂ψ ∂zi ∂ψ ∂zj , where (hij) is the inverse of (hij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Define T(r, ψ) = m(r, ψ) + N(r, ψ), 22 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG where m(r, ψ) = � ∂∆(r) log+ |ψ|dπr, N(r, ψ) = πm (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' � ψ∗∞∩∆(r) gr(o, x)αm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is trivial to show that T � r, 1 ψ − ζ � = T(r, ψ) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On P1(C), take a singular metric Ψ = 1 |ζ|2(1 + log2 |ζ|) √−1 4π2 dζ ∧ d¯ζ, which gives that � P1(C) Ψ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have 1 4π � ∆(r) gr(o, x) ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dv ≤ T(r, ψ) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|) = 4mπψ∗Ψ ∧ αm−1 αm , then it yields from Fubini’s theorem that 1 4π � ∆(r) gr(o, x) ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dv = m � ∆(r) gr(o, x)ψ∗Ψ ∧ αm−1 αm dv = πm (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' � P1(C) Ψ(ζ) � ψ−1(ζ)∩∆(r) gr(o, x)αm−1 = � P1(C) N � r, 1/(ψ − ζ) � Ψ(ζ) ≤ � P1(C) � T(r, ψ) + O(1) � Ψ = T(r, ψ) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 23 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that ψ ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that (a) If m = 1, then � ∂∆(r) log+ ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr ≤ (1 + δ)2 log+ T(r, ψ) + δ log r holds for all r > 0 outside Eδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) If m ≥ 2, then � ∂∆(r) log+ ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr ≤ (1 + δ)2 log+ T(r, ψ) + 2m − 1 2m − 2δ log r holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using the concavity of log, we have � ∂∆(r) log+ ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr ≤ log � ∂∆(r) � 1 + ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|) � dπr ≤ log+ � ∂∆(r) ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6, for m = 1 log+ � ∂∆(r) ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr ≤ (1 + δ)2 log+ � ∆(r) gr(o, x) ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dv + δ log r + O(1) ≤ (1 + δ)2 log+ T(r, ψ) + δ log r + O(1) for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Similarly, for m ≥ 2 log+ � ∂∆(r) ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr ≤ (1 + δ)2 log+ � ∆(r) gr(o, x) ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dv + 2m − 1 2m − 2δ log r + O(1) ≤ (1 + δ)2 log+ T(r, ψ) + 2m − 1 2m − 2δ log r + O(1) for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Adjust Eδ so that O(1) is absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Define m � r, ∥∇ψ∥ |ψ| � = � ∂∆(r) log+ ∥∇ψ∥ |ψ| dπr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 24 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ψ be a nonconstant meromorphic function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for any δ > 0, there exist a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that (a) If m = 1, then m � r, ∥∇ψ∥ |ψ| � ≤ 2 + (1 + δ)2 2 log+ T(r, ψ) + δ 2 log r holds for all r > 0 outside Eδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) If m ≥ 2, then m � r, ∥∇ψ∥ |ψ| � ≤ 2 + (1 + δ)2 2 log+ T(r, ψ) + 2m − 1 4m − 4δ log r holds for all r > 0 outside Eδ, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that m � r, ∥∇ψ∥ |ψ| � ≤ 1 2 � ∂∆(r) log+ ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr +1 2 � ∂∆(r) log � 1 + log2 |ψ| � dπr = 1 2 � ∂∆(r) log+ ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr +1 2 � ∂∆(r) log � 1 + � log+ |ψ| + log+ 1 |ψ| �2� dπr ≤ 1 2 � ∂∆(r) log+ ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr + log � ∂∆(r) � log+ |ψ| + log+ 1 |ψ| � dπr + O(1) ≤ 1 2 � ∂∆(r) log+ ∥∇ψ∥2 |ψ|2(1 + log2 |ψ|)dπr + log+ T(r, ψ) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='8, for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that, for m = 1 m � r, ∥∇ψ∥ |ψ| � ≤ 2 + (1 + δ)2 2 log+ T(r, ψ) + δ 2 log r + O(1) holds for all r > 0 outside Eδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' and for m ≥ 2 m � r, ∥∇ψ∥ |ψ| � ≤ 2 + (1 + δ)2 2 log+ T(r, ψ) + 2m − 1 4m − 4δ log r + O(1) holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Adjust Eδ so that O(1) is absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Second main theorem and defect relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D = D1+· · ·+Dq be a reduced divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' D is said to be of simple normal crossing type if every Dj is smooth and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' at every point x ∈ X, there exist a local holomorphic coordinate neighborhood U(z1, · · · , zm) and a non- negative integer k with 0 ≤ k ≤ m such that U ∩ D = � z1 · · · zk = 0 � for U ∩ D ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We establish the following second main theorem: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete K¨ahler manifold with maximal volume growth and non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let X be a com- plex projective manifold of complex dimension not greater than that of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D ∈ |L| be a divisor of simple normal crossing type, where L is a positive line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a differentiably non-degenerate mero- morphic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that Tf(r, L) + Tf(r, KX) + T(r, R) ≤ N f(r, D) + O � log+ Tf(r, L) + δ log r � holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Equip every holomorphic line bundle O(Dj) with a Hermitian metric hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It can induce a Hermitian metric hL = h1⊗· · ·⊗hq on L with c1(L, hL) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We see that Ω = cn 1(L, hL) is a volume form on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Pick sj ∈ H0(X, O(Dj) such that (sj) = Dj and ∥sj∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On X, define a singular volume form Φ = Ω �q j=1 ∥sj∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set f ∗Φ ∧ αm−n = ξαm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that αm = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' det(hi¯j) m � j=1 √−1 π dzj ∧ d¯zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is clear that ddc[log ξ] ≥ f ∗c1(L, hL) − f ∗Ric(Ω) + R − [Suppf ∗D] in the sense of currents, where R = −ddc log det(hi¯j) is the Ricci form of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It yields that 1 4 � ∆(r) gr(o, x)∆ log ξdv (15) ≥ Tf(r, L) + Tf(r, KX) + T(r, R) − N f(r, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 26 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Next, we need to estimate the upper bound of the first term in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since D is of simple normal crossing type, then there exist a finite open covering {Uλ} of X and finitely many rational functions wλ1, · · · , wλn on X such that wλ1, · · · , wλn are holomorphic on Uλ satisfying that dwλ1 ∧ · · · ∧ dwλn(x) ̸= 0, ∀x ∈ Uλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' D ∩ Uλ = � wλ1 · · · wλhλ = 0 � , ∃hλ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Moreover, we can require that O(Dj)|Uλ ∼= Uλ × C for λ, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On Uλ, write Φ = eλ |wλ1|2 · · · |wλhλ|2 n� k=1 √−1 π dwλk ∧ d ¯wλk, where eλ is a positive smooth function on Uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set Φλ = φλeλ |wλ1|2 · · · |wλhλ|2 n� k=1 √−1 π dwλk ∧ d ¯wλk, where {φλ} is a partition of unity subordinate to {Uλ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set fλk = wλk ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On f −1(Uλ), we have f ∗Φλ = φλ ◦ f · eλ ◦ f |fλ1|2 · · · |fλhλ|2 n� k=1 √−1 π dfλk ∧ d ¯fλk = φλ ◦ f · eλ ◦ f � 1≤i1̸=···̸=in≤m ���∂fλ1 ∂zi1 ��� 2 |fλ1|2 · · · ��� ∂fλhλ ∂z ihλ ��� 2 |fλhλ|2 ���� ∂fλ(hλ+1) ∂zihλ+1 ���� 2 · · ���� ∂fλn ∂zin ���� 2 �√−1 π �n dzi1 ∧ d¯zi1 ∧ · · · ∧ dzin ∧ d¯zin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Fix any x0 ∈ M, we take local holomorphic coordinates z1, · · · , zm near x0 and local holomorphic coordinates ζ1, · · · , ζn near f(x0) such that α|x0 = √−1 π m � j=1 dzj ∧ d¯zj, c1(L, hL)|f(x0) = √−1 π n � j=1 dζj ∧ d¯ζj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Put f ∗Φλ ∧ αm−n = ξλαm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Clearly, we have ξ = � ξλ and ξλ = φλ ◦ f · eλ ◦ f � 1≤i1̸=···̸=in≤m ��� ∂fλ1 ∂zi1 ��� 2 |fλ1|2 · · · ��� ∂fλhλ ∂z ihλ ��� 2 |fλhλ|2 ���� ∂fλ(hλ+1) ∂zihλ+1 ���� 2 · · ���� ∂fλn ∂zin ���� 2 ≤ φλ ◦ f · eλ ◦ f � 1≤i1̸=···̸=in≤m ��∇fλ1 ��2 |fλ1|2 · · ��∇fλhλ ��2 |fλhλ|2 ��∇fλ(hλ+1) ��2 · · · ��∇fλn ��2 NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 27 at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Define a non-negative function ̺ on M by (16) f ∗c1(L, hL) ∧ αm−1 = ̺αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set fj = ζj ◦ f for 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, at x0 f ∗c1(L, hL) ∧ αm−1 = (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2 m � j=1 ��∇fj ��2αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' That is, ̺ = (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' n � i=1 m � j=1 ��� ∂fi ∂zj ��� 2 = (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 2 n � j=1 ��∇fj ��2 at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining the above, we are led to ξλ ≤ φλ ◦ f · eλ ◦ f · (2̺)n−hλ (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='n−hλ � 1≤i1̸=···̸=in≤m ��∇fλ1 ��2 |fλ1|2 · · ��∇fλhλ ��2 |fλhλ|2 on f −1(Uλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that φλ ◦f ·eλ ◦f is bounded on M, whence it yields from log+ ξ ≤ � λ log+ ξλ + O(1) that (17) log+ ξ ≤ O � log+ ̺ + � k,λ log+ ∥∇fλk∥ |fλk| � + O(1) on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Jensen-Dynkin formula (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='13) (18) 1 4 � ∆(r) gr(o, x)∆ log ξdv = 1 2 � ∂∆(r) log ξdπr + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining (17) with (18) and using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9, 1 4 � ∆(r) gr(o, x)∆ log ξdv ≤ O � � k,λ m � r, ∥∇fλk∥ |fλk| � + log+ � ∂∆(r) ̺dπr � + O(1) ≤ O � � k,λ log+ T(r, fλk) + log+ � ∂∆(r) ̺dπr � + O(1) ≤ O � log+ Tf(r, L) + log+ � ∂∆(r) ̺dπr � + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6 and (16), for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that log+ � ∂∆(r) ̺dπr ≤ O � log+ Tf(r, L) + δ log r � 28 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, we conclude that (19) 1 4 � ∆(r) gr(o, x)∆ log ξdv ≤ O � log+ Tf(r, L) + δ log r � + O(1) for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining (15) with (19), we prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ In view of (1) and (2), we see that T(r, R) has the alternative expressions T(r, R) = 1 2 � ∆(r) gr(o, x)sCdv = 1 4 � ∆(r) gr(o, x)sdv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The curvature condition implies that T(r, R) ≥ 0, thus we deduce that Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume the same conditions as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that Tf(r, L) + Tf(r, KX) ≤ N f(r, D) + O � log+ Tf(r, L) + δ log r � holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We continue to consider a defect relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Define the defect and the simple defect ¯δf(D) of f with respect to D, respectively by δf(D) = 1 − lim sup r→∞ Nf(r, D) Tf(r, L) , ¯δf(D) = 1 − lim sup r→∞ N f(r, D) Tf(r, L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By the first main theorem, we have 0 ≤ δf(D) ≤ ¯δf(D) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For any two holomorphic line bundles L1, L2 over X, define �c1(L2) c1(L1) � = inf � t ∈ R : ω2 < tω1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' ∃ω1 ∈ c1(L1), ∃ω2 ∈ c1(L2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Invoking the volume growth condition (3) and the Green function estimate (8) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5 also) for M, one can deduce that Tf(r, L) ≥ O(log r) for a nonconstant meromorphic mapping f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, we obtain a defect relation: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume the same conditions as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then ¯δf(D) ≤ �c1(K∗ X) c1(L) � − lim inf r→∞ T(r, R) Tf(r, L) ≤ �c1(K∗ X) c1(L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that the above defect relation in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='12 is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' There exist no differentiably non-degenerate meromorphic mappings f : M → X satisfying the growth condition lim inf r→∞ T(r, R) Tf(r, L) > �c1(K∗ X) c1(L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The case for singular divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We consider a general hypersurface D of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let S be the set of the points of D at which D has a non-normal-crossing singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hironaka’s resolution of singularities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' [23]) says that there exists a proper modification τ : ˜X → X such that ˜X \\ ˜S is biholomorphic to X \\S through τ, and ˜D has only normal crossing singularities, where ˜S = τ −1(S) and ˜D = τ −1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Write ˆD = ˜D \\ ˜S and denote by ˜Sj the irreducible components of ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Put (20) τ ∗D = ˆD + � pj ˜Sj = ˜D + � (pj − 1) ˜Sj, Rτ = � qj ˜Sj, where Rτ is ramification divisor of τ, and pj, qj are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set (21) S∗ = � ςj ˜Sj, ςj = max � pj − qj − 1, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Endow O(S∗) with a Hermitian metric ∥ · ∥ and take a holomorphic section σ of O(S∗) such that the zero divisor (σ) = S∗ and ∥σ∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a meromorphic mapping such that f(M) ̸⊆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Define the proximity function of f with respect to the singularities of D by mf(r, Sing(D)) = � ∂∆(r) log 1 ∥σ ◦ τ −1 ◦ f∥dπr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Consider the lift ˜f : M → ˜X defined via τ ◦ ˜f = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, ˜f is a holomorphic mapping on M \\ ˜I, where ˜I = I ∪ f −1(S) with the indeterminacy set I of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' One can similarly define the Nevanlinna’s functions of ˜f through the lift of f via τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is not difficult to verify that (22) mf(r, Sing(D)) = m ˜f(r, S∗) = � ςjm ˜f(r, ˜Sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The following second main theorem is an extension of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete K¨ahler manifold with maximal volume growth and non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let X be a com- plex projective manifold of complex dimension not greater that of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D ∈ |L| be a hypersurface of X, where L is a positive line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a differentiably non-degenerate meromorphic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that Tf(r, L) + Tf(r, KX) + T(r, R) ≤ N f(r, D) + mf(r, Sing(D)) + O � log+ Tf(r, L) + δ log r � holds for all r > 0 outside Eδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 30 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We first show the case where D is the union of smooth hypersurfaces, namely, no irreducible component of ˜D crosses itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let E be the union of generic hyperplane sections of X so that the set A = ˜D ∪E has only normal crossing singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By (20) and K ˜ X = τ ∗KX + O(Rτ) K ˜ X + O( ˜D) = τ ∗KX + τ ∗O(D) + (1 − pj + qj)O( ˜Sj) (23) = τ ∗KX + τ ∗L + (1 − pj + qj)O( ˜Sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10 to ˜f, we obtain T ˜f(r, O(A)) + T ˜f(r, K ˜ X) + T(r, R) ≤ N ˜f(r, A) + O � log+ T ˜f(r, τ ∗O(A)) + δ log r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The First Main Theorem gives that T ˜f(r, O(A)) = m ˜f(r, A) + N ˜f(r, A) + O(1) = m ˜f(r, ˜D) + m ˜f(r, E) + N ˜f(r, A) + O(1) ≥ m ˜f(r, ˜D) + N ˜f(r, A) + O(1) = T ˜f(r, O( ˜D)) − N ˜f(r, ˜D) + N ˜f(r, A) + O(1), which yields T ˜f(r, O(A)) − N ˜f(r, A) ≥ T ˜f(r, O( ˜D)) − N ˜f(r, ˜D) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since T ˜f(r, τ ∗L) = Tf(r, L) and N ˜f(r, ˜D) = N f(r, D), then T ˜f(r, O( ˜D)) + T ˜f(r, K ˜ X) + T(r, R) (24) ≤ N ˜f(r, ˜D) + O � log Tf(r, L) + δ log r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It follows from (23) that T ˜f(r, O( ˜D)) + T ˜f(r, K ˜ X) (25) = T ˜f(r, τ ∗L) + T ˜f(r, τ ∗KX) + � (1 − pj + qj)T ˜f(r, O( ˜Sj)) = Tf(r, L) + Tf(r, KX) + � (1 − pj + qj)T ˜f(r, O( ˜Sj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that N ˜f(r, ˜S) = 0, thus it yields from (21) and (22) that � (1 − pj + qj)T ˜f(r, O( ˜Sj)) = � (1 − pj + qj)m ˜f(r, ˜Sj) + O(1) (26) ≤ � ςjm ˜f(r, ˜Sj) + O(1) = mf(r, Sing(D)) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining (24) and (25) with (26), we have the theorem proved in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For the general case, according to those arguments stated above, it suffices to prove the case that a hypersurface D is of normal crossing type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using the arguments from [[39], pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 175], there exists a proper modification τ : ˜X → X NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 31 such that ˜D = τ −1(D) is the union of a collection of hypersurfaces of simple normal crossing type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It means that mf(r, Sing(D)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Invoking Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10, we can prove the theorem in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume the same conditions as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then ¯δf(D) ≤ �c1(K∗ X) c1(L) � + lim sup r→∞ mf(r, Sing(D)) − T(r, R) Tf(r, L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Application to algebraic dependence problems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Consequences of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The purpose of this section is to apply the Nevanlinna theory established in Section 3 to the study of algebraic dependence problems on the dominant meromorphic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We shall point out that some arguments essentially follow Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Aihara [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let X be a complex projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let L be an ample line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For any holomorphic line bundle F over X, define �F L � = inf � γ ∈ Q : γL ⊗ F −1 is big � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' According to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9, we see that [F/L] < 0 if and only if F −1 is big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a meromorphic mapping, where M is a K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For F ∈ Pic(X) ⊗ Q, we define Tf(r, F) = 1 ν Tf(r, νF), where ν is a positive integer such that νF ∈ Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Evidently, this is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Next, we would like to consider another defect relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The second main theorem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10) yields that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete K¨ahler manifold with maximal volume growth and non-negative Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let X be a com- plex projective manifold of complex dimension not greater than that of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D ∈ |L| be a divisor of simple normal crossing type, where L is an ample line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a dominant meromorphic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then ¯δf(D) ≤ � K−1 X L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It follows from the definition of [K−1 X /L] that ([K−1 X /L] + ǫ)L ⊗ KX is big for any rational number ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9, we obtain �� K−1 X /L � + ǫ � L ⊗ KX ≥ δL 32 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG for a sufficiently small rational number δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This implies that Tf(r, K−1 X ) ≤ �� K−1 X /L � − δ + ǫ � Tf(r, L) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Invoking Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10, we conclude that ¯δf(D) ≤ � K−1 X L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ The first main theorem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1) also yields that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a K¨ahler manifold and X be a complex projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → X be a dominant meromorphic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that µF ⊗L−1 is big for some positive integer µ, where F is a big line bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then Tf(r, L) ≤ µTf(r, F) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='8, the bigness of µF ⊗ L−1 implies that there exists a nonzero holomorphic section s ∈ H0(X, ν(µF ⊗L−1)) for a sufficiently large positive integer ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 remains true for a general K¨ahler manifold M, whence Nf(r, (s)) ≤ Tf(r, ν(µF ⊗ L−1)) + O(1) = µνTf(r, F) − νTf(r, L) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This leads to the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Propagation of algebraic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let M be a non-compact complete K¨ahler manifold with maximal volume growth and non-negative Ricci curvature, and let X be a complex projective manifold with complex dimension not greater than that of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Fix an integer l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A proper algebraic subset Σ of Xl is said to be decomposible, if there exist positive integers l1, · · · , ls with l = l1 + · · · + ls for some integer s ≤ l and algebraic subsets Σj ⊆ Xlj for 1 ≤ j ≤ s, such that Σ = Σ1×· · ·×Σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If Σ is not decomposable, we say that Σ is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For l meromorphic mappings f1, · · · , fl : M → X, there is a meromorphic mapping f1×· · ·×fl : M → Xl, defined by (f1 × · · · × fl)(x) = (f1(x), · · · , fl(x)), ∀x ∈ M \\ l� j=1 I(fj), where I(fj) denotes the indeterminacy set of fj for 1 ≤ j ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' As a matter of convenience, set ˜f = f1 × · · · × fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 33 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let S be an analytic subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The nonconstant mero- morphic mappings f1, · · · , fl : M → X are said to be algebraically dependent on S, if there exists a proper indecomposable algebraic subset Σ of Xl such that ˜f(S) ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In this case, we say that f1, · · · , fl are Σ-related on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let L be an ample line bundle over X, and let D1, · · · , Dq ∈ |L| such that D1 + · · · + Dq has only simple normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set G = � f : M → X is a dominant meromorphic mapping � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let S1, · · · , Sq be hypersurfaces of M such that dimC Si ∩ Sj ≤ m − 2 for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Fix q positive integers k1, · · · , kq which may be +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Denote by (27) F = F � f ∈ G ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' {kj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (M, {Sj});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (X, {Dj}) � the set of all f ∈ G such that Sj = Suppkjf ∗Dj, 1 ≤ j ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ˜L be a big line bundle over Xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In general, we have ˜L ̸∈ π∗ 1Pic(X) ⊕ · · · ⊕ π∗ l Pic(X), where πk : Xl → X is the natural projection on the k-th factor for 1 ≤ k ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let F1, · · · , Fl be big line bundles over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, it defines a line bundle over Xl by ˜F = π∗ 1F1 ⊗ · · · ⊗ π∗ l Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If ˜L ̸= ˜F, we assume that there is a rational number ˜γ > 0 such that ˜γ ˜F ⊗ ˜L−1 is big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If ˜L = ˜F, we shall take ˜γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In further, assume that there is a line bundle F0 ∈ {F1, · · · , Fl} such that F0 ⊗ F −1 j is either big or trivial for 1 ≤ j ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let H be the set of all indecomposable hypersurfaces Σ in Xl satisfying Σ = Supp ˜D for some ˜D ∈ |˜L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set S = S1 ∪ · · · ∪ Sq, k0 = max{k1, · · · , kq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, · · · , fl be meromorphic mappings in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that ˜f(S) ⊆ Σ and ˜f(M) ̸⊆ Σ for some Σ ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then N(r, S) ≤ ˜γ l � j=1 Tfj(r, Fj) + O(1) ≤ ˜γ l � j=1 Tfj(r, F0) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Take ˜D ∈ |˜L| such that Σ = Supp ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' As mentioned earlier, ˜γ ˜F ⊗ ˜L−1 is big for ˜γ ̸= 1 and trivial for ˜γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, by conditions with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 34 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2, we conclude that N(r, S) ≤ T ˜f(r, ˜L) + O(1) ≤ ˜γT ˜f(r, ˜F ) + O(1) ≤ ˜γ l � j=1 Tfj(r, Fj) + O(1) ≤ ˜γ l � j=1 Tfj(r, F0) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f be a meromorphic mapping in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then qTf(r, L) + Tf(r, KX) ≤ k0 k0 + 1N(r, S) + q � j=1 1 kj + 1Nf(r, Dj) + o � Tf(r, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Sj = Suppkjf ∗Dj, we have N f(r, Dj) ≤ N(r, Sj) ≤ kj kj + 1N(r, Sj) + 1 kj + 1Nf(r, Dj) ≤ k0 k0 + 1N(r, Sj) + 1 kj + 1Nf(r, Dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining this with the second main theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10), then one can easily prove the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Define a Q-line bundle L0 ∈ Pic(X) ⊗ Q by (28) L0 = � q � j=1 kj kj + 1 � L ⊗ � − ˜γlk0 k0 + 1F0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Again, for an arbitrary Q-line bundle H ∈ Pic(X) ⊗ Q, define T(r, H) = l � j=1 Tfj(r, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In what follows, we are to show a theorem for the propagation of algebraic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, · · · , fl be meromorphic mappings in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that f1, · · · , fl are Σ-related on S for some Σ ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If L0 ⊗ KX is big, then f1, · · · , fl are Σ-related on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 35 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It suffices to prove ˜f(M) ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Otherwise, we assume that ˜f(M) ̸⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5, for i = 1, · · · , l qTfi(r, L) + Tfi(r, KX) ≤ k0 k0 + 1N(r, S) + q � j=1 1 kj + 1Tfi(r, L) + o � Tfi(r, L) � , which yields from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4 that q � j=1 kj kj + 1Tfi(r, L) + Tfi(r, KX) ≤ k0 k0 + 1N(r, S) + o � Tfi(r, L) � ≤ ˜γk0 k0 + 1 l � j=1 Tfj(r, F0) + o � Tfi(r, L) � = ˜γk0 k0 + 1T(r, F0) + o � Tfi(r, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, we get q � j=1 kj kj + 1T(r, L) + T(r, KX) ≤ ˜γlk0 k0 + 1T(r, F0) + o � T(r, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It follows that (29) T(r, L0) + T(r, KX) ≤ o � T(r, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On the other hand, the bigness of L0⊗KX implies that there exists a positive integer µ such that µ(L0 ⊗ KX) ⊗ L−1 is a big line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2 T(r, L) ≤ µ � T(r, L0) + T(r, KX) � + O(1), which contradicts with (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We conclude that ˜f(M) ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Set γ0 = � L−1 0 ⊗ K−1 X L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, L0 ⊗ KX is big if and only if γ0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Whence, we obtain Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, · · · , fl be meromorphic mappings in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that f1, · · · , fl are Σ-related on S for some Σ ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If γ0 < 0, then f1, · · · , fl are Σ-related on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In the case where γ0 ≥ 0, we cannot conclude the propagation of algebraic dependence in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Particularly, for γ0 > 0, the propagation of algebraic dependence does not occur even if one assumes the existence of Picard’s deficient divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For instance, consider the situation that M = C, X = P1(C) and l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set L = F1 = F2 = O(1), where O(1) stands for the 36 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG hyperplane line bundle over P1(C), which means that F0 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, one can take ˜L = ˜F = π∗ 1O(1) ⊗ π∗ 2O(1) which implies that ˜γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1(z) = ez, f2(z) = e−z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' D1 = 0, D2 = ∞, D3 = −1, D4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, D1, D2 are Picard’s deficient values of f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Put Sj = Supp1f ∗ 1 Dj, which means that kj = 1 for 1 ≤ j ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let S = S1 ∪ · · · ∪ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is trivial to check that f2 ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A direct computation leads to L0 = � 4 � j=1 1 1 + 1 � O(1) ⊗ � − 2 1 + 1O(1) � = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It yields that γ0 = �L−1 0 ⊗ K−1 P1(C) L � = �−O(1) ⊗ (2O(1)) O(1) � = 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Taking Σ as the diagonal of P1(C) × P1(C), then we have Σ ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is clear that (f1 ×f2)(S) ⊆ Σ and (f1 ×f2)(C) ̸⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Therefore, one cannot conclude the propagation of algebraic dependence if γ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' However, for γ0 = 0, the propagation of algebraic dependence may occur under a defect condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In fact, we have the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, · · · , fl be meromorphic mappings in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that f1, · · · , fl are Σ-related on S for some Σ ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If γ0 = 0 and δfi(Dj) > 0 for some i ∈ {1, · · · , l} and some j ∈ {1, · · · , q}, then f1, · · · , fl are Σ-related on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9, we need a lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, · · · , fl be meromorphic mappings in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that f1, · · · , fl are Σ-related on S for some Σ ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If γ0 = 0 and ˜f(M) ̸⊆ Σ, then there exist positive constants C1, C2 such that for j = 1, · · · , l C1 ≤ Tfj(r, L) Tf1(r, L) ≤ C2 holds for all sufficiently large r ̸∈ I, where I = ∪l j=1I(fj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ν be a rational number with 0 < ν < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We claim that (30) ˜γνT(r, F0) ≤ N(r, S) + O(1) for all sufficiently large r ̸∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Otherwise, there exists a monotone increasing sequence {rn}∞ n=1 outside I with rn → ∞ as n → ∞, such that ˜γνT(rn, F0) > N(rn, S) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 37 Using Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5, q � j=1 kj kj + 1T(rn, L) + T(rn, KX) ≤ ˜γν0lk0 k0 + 1 T(rn, F0) + o � T(rn, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This yields that (31) T(rn, L0 ⊗ KX) + λT(rn, F0) ≤ o � T(rn, L) � , where λ = ˜γlk0(1 − ν) k0 + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since γ0 = 0, the bigness of F0 implies the bigness of L0 ⊗KX ⊗λF0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, L0 ⊗ KX ⊗ λF0 ≥ δL for a sufficiently small rational number δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, δT(rn, L) ≤ T(rn, L0) + T(rn, KX) + λT(rn, F0) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By this with (31) δT(rn, L) ≤ o(T(rn, L)), which is a contradiction and it thus confirms (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1 N(r, S) = q � j=1 N(r, Sj) ≤ q � j=1 Nf1(r, Dj) ≤ qTf1(r, L) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Combining this with (30), we get ˜γνT(r, F0) ≤ qTf1(r, L) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since F0 is big, for a sufficiently small rational number ǫ > 0 ǫ˜γνTfj(r, L) ≤ ǫ˜γνT(r, L) ≤ q˜γνT(r, F0) ≤ Tf1(r, L) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9 It suffices to show that ˜f(M) ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Otherwise, we assume that ˜f(M) ̸⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' From the definition of the defect, for any ǫ > 0, there is a subset Eǫ ⊆ (0, ∞) with finite Lebesgue measure such that Nfs(r, Dj) < (1 − δfs(Dj) + ǫ)Tfs(r, L) for all r > 0 outside Eǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By (28) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='5 T(r, L0) + T(r, KX) ≤ − l � s=1 q � j=1 1 kj + 1(δfs(Dj) − ǫ)Tfs(r, L) + o � T(r, L) � ≤ qǫT(r, L) − 1 kj + 1δfi(Dj)Tfi(r, L) + o � T(r, L) � , which yields that 1 kj + 1δfi(Dj)Tfi(r, L) + T(r, L0) + T(r, KX) − qǫT(r, L) ≤ o � T(r, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 38 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Since [L−1 0 ⊗K−1 X /L] = 0, we have L0 +KX ≥ −γL for an arbitrary rational number γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It implies that −γT(r, L) ≤ T(r, L0) + T(r, KX) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Whence, 1 kj + 1δfi(Dj)Tfi(r, L) − (γ + qǫ)T(r, L) ≤ o � T(r, L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that δfi(Dj) > 0 and ǫ, γ can be be small arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' However, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='10 implies that the above inequality does not hold if ǫ, γ are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, we have ˜f(M) ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Unicity theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We apply the propagation theorems of algebraic dependence to the unicity problems for meromorphic mappings from a complete K¨ahler manifold into a complex projective manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since X is projective, there is a holomorphic embedding Φ : X ֒→ PN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let O(1) be the hyperplane line bundle over PN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Now, take l = 2 and F1 = F2 = Φ∗O(1) which gives that F0 = Φ∗O(1) and ˜F = π∗ 1 (Φ∗O(1)) ⊗ π∗ 2 (Φ∗O(1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Again, take ˜L = ˜F, then ˜γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In view of (28), we have (32) L0 = � q � j=1 kj kj + 1 � L ⊗ � − 2k0 k0 + 1Φ∗O(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Suppose that D1, · · · , Dq ∈ |L| satisfy that D1 + · · · + Dq has only simple normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For f0 ∈ F, one says that the set {Dj}q j=1 is generic with respect to f0 and φ, if ˆ Ms = f0(M − I(f0)) ∩ SuppDs ∩ {x ∈ X : rank(dφ(x)) = dimC X} ̸= ∅ for at least one s ∈ {1, · · · , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Next, assume that the set {Dj}q j=1 is generic with respect to f0 and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Denote by F0 the set of all meromorphic mappings f ∈ F such that f = f0 on the hypersuface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If L0 ⊗ KX is big, then F0 contains exactly one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f ∈ F0 be an arbitrary meromorphic mapping, it suffices to show that f ≡ f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Recall that Φ : X ֒→ PN(C) is a holomorphic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since f = f0 on S, we have Φ◦f = Φ◦f0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Now, we assert that Φ◦f ≡ Φ◦f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Otherwise, we may assume that Φ◦f ̸≡ Φ◦f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let ∆ denote the diagonal of PN(C)×PN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Put ˜Φ = Φ×Φ and ˜f = f ×f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Then, it gives a meromorphic mapping φ = ˜Φ ◦ ˜f : M → PN(C) × PN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 39 Again, set ˜ O(1) = π∗ 1O(1)⊗π∗ 2O(1), which is a holomorphic line bundle over PN(C) × PN(C), where O(1) is the hyperplane line bundle over PN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It is easy to see that ˜L = ˜Φ∗ ˜ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since Φ◦f ̸≡ Φ◦f0, there exists a holomorphic global section ˜σ of ˜ O(1) which satisfies that φ∗˜σ ̸= 0 and ∆ ⊆ Supp(˜σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Take Σ = Supp˜Φ∗(˜σ), it yields that ˜f(S) ⊆ Σ and ˜f(M) ̸⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' On the other hand, with the aid of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6, the bigness of L0 ⊗KX implies that ˜f(M) ⊆ Σ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, we obtain Φ◦f ≡ Φ◦f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By the assumption, ˆ Ms ̸= ∅ for some s ∈ {1, · · · , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Take a point P ∈ ˆ Ms, then there is an open neighborhood UP of P such that Φ|UP : UP → Φ(UP) is biholomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let V = UP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since Φ ◦ f = Φ ◦ f0 and f = f0 on S, we deduce that f = f0 on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By the uniqueness theorem on analytic functions, we see that f ≡ f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Using the similar arguments as in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9, we can also prove the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that [L−1 0 ⊗ K−1 X /L] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If δf0(Ds) > 0 for some s ∈ {1, · · · , q}, then F0 contains exactly one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Targets are compact Riemann surfaces and Pn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In this section, we consider some consequences of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='12 (and Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9 too) in the case that X is a compact Riemann surface or a complex projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' X is a compact Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let R be a compact Riemann surface with genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that R is viewed as P1(C) for g = 0, and a torus for g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f : M → R be a nonconstant holomorphic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let a1, · · · , aq be distinct points in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='12 gives a defect relation: (33) q � j=1 ¯δf(aj) ≤ 2 − 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A1 The cases g = 0 and g ≥ 2 In the case of g = 0, we have Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → P1(C) be nonconstant holomorphic map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let a1, · · · , aq be distinct points in P1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have (a) Assume that Suppf ∗ 1 aj = Suppf ∗ 2 aj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If q ≥ 5, then f1 ≡ f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) Assume that Supp1f ∗ 1 aj = Supp1f ∗ 2 aj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If q ≥ 7, then f1 ≡ f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11, we put X = P1(C) and L = O(1) as well as kj = ∞ for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that KP1(C) = −2O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' It yields from (32) that L0 ⊗ KP1(C) = qO(1) ⊗ (−2O(1)) ⊗ (−2O(1)) = (q − 4)O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 40 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Hence, L0 ⊗KP1(C) is big provided that q ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='11, we have (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For (b), we just need to put kj = 1 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In this case, L0 ⊗ KP1(C) = q 2O(1) ⊗ (−O(1)) ⊗ (−2O(1)) = q − 6 2 O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Clearly, L0⊗KP1(C) is big provided that q ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ In the case that g ≥ 2, it yields from (33) that each holomorphic mapping f : M → R is a constant mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Therefore, this case is actually trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A2 The case g = 1 By (33), each nonconstant holomorphic mapping f : M → R is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Here, we only consider the case where the torus R is a smooth elliptic curve, denoted by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' When M is a finite analytic ramified covering of Cm, Aihara proved a unicity theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' [8], Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In particular, when M = Cm, he showed that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' [8], Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='14 (Aihara).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : Cm → E be nonconstant holomorphic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D1 = {a1, · · · , aq} be a set of points in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set D2 = φ(D1) with #(D2) = p, here φ ∈ End(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that Supp1f ∗ 1D1 = Supp1f ∗ 2D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If pq > (p + q)(deg ψ + 1), then f2 = φ(f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using the theorems on the propagation of algebraic dependence obtained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2, we see that those arguments in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='14 can be carried to our situation where the domain is M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, there is no need to state the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We give a generalization of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='14 as follows: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → E be nonconstant holomorphic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D1 = {a1, · · · , aq} be a set of points in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set D2 = φ(D1) with #(D2) = p, here φ ∈ End(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that Supp1f ∗ 1 D1 = Supp1f ∗ 2 D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If pq > (p + q)(deg ψ + 1), then f2 = φ(f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In particular, when φ = Id, we have deg φ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, it yields that Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → E be nonconstant holomorphic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let a1, · · · , aq be distinct points in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that Supp1f ∗ 1 aj = Supp1f ∗ 2 aj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If q ≥ 5, then f1 ≡ f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' X is a complex projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We consider the case X = Pn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' For a Q-line bundle F ∈ Pic(Pn(C))⊗Q, note that F is big if and only if F is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let F1, F2 be ample line bundles over Pn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since Pic(Pn(C)) ∼= Z, then there are positive integers d1, d2 such that F1 = d1O(1) and F2 = d2O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Again, define a holomorphic line bundle ˜F over Pn(C) × Pn(C) by ˜F = π∗ 1F1 ⊗ π∗ 2F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' A well-known fact says that Pic (Pn(C) × Pn(C)) = π∗ 1Pic(Pn(C)) ⊕ π∗ 2Pic(Pn(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE 41 Thus, we may assume that ˜L = ˜F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let σ be a holomorphic section of ˜L over Pn(C)×Pn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Note that σ can be identified with a homogeneous polynomial P(ξ, ζ) of degree d1 in ξ = [ξ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' ξn] and degree d2 in ζ = [ζ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' ζn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Set d0 = max{d1, d2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let L be an ample line bundle over Pn(C) with L = dO(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let S = S1 ∪ · · · ∪ Sq, where S1, · · · , Sq are q hypersufaces of M such that dimC Si ∩Sj ≤ m−2 for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let D1, · · · , Dq ∈ |L| such that D1 +· · ·+Dq has simple normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Based on the above notations, we give the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → Pn(C) be dominant meromorphic map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that Suppkf ∗ 1 Dj = Suppkf ∗ 2Dj = Sj with 1 ≤ k ≤ +∞ for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Suppose, in addition, that P(f1, f2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If q > 2d0 + (1 + n)(1 + k−1) d , then P(f1, f2) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Recall that F0 is defined as an element in {F1, F2} such that F0⊗F −1 j is either big or trivial for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, we have F0 = d0O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Since ˜L = ˜F, then ˜γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We also note that l = 2, k0 = kj = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Thus, it follows from (28) that L0 = � q � j=1 k k + 1 � dO(1) ⊗ � − 2kd0 k + 1O(1) � = k(dq − 2d0) k + 1 O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' By KPn(C) = −(n + 1)O(1), we get L0 ⊗ KPn(C) = k(dq − 2d0) − (k + 1)(n + 1) (k + 1) O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Hence, L0 ⊗ KPn(C) is big if k(dq − 2d0) − (k + 1)(n + 1) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=', q > 2d0 + (1 + n)(1 + k−1) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Invoking Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='6, we have the theorem proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Take L = F1 = F2 = O(1), then d = d1 = d2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' In this case, D1, · �� · , Dq reduce to q hyperplanes H1, · · · , Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='17, it yields that Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → Pn(C) be dominant meromorphic map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that Suppkf ∗ 1 Hj = Suppkf ∗ 2Hj = Sj with 1 ≤ k ≤ +∞ for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Suppose, in addition, that P(f1, f2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If q > 2 + (1 + n)(1 + k−1) then P(f1, f2) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Keep the same notations as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='17, we get that 42 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' DONG Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → Pn(C) be dominant meromorphic map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Assume that Suppkf ∗ 1 Dj = Suppkf ∗ 2Dj = Sj with 1 ≤ k ≤ +∞ for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Suppose, in addition, that P(f1, f2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If δf1(Dj) > 0 for some j ∈ {1, · · · , q} and q = 2d0 + (1 + n)(1 + k−1) d ∈ Z, then P(f1, f2) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' The given relation between q, d, d0 implies that L0 ⊗KPn(C) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Invoking Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='9, the theorem can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' □ Similarly, take L = F1 = F2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='19 yields that Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Let f1, f2 : M → Pn(C) be dominant meromorphic map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' We have (a) Assume that Suppf ∗ 1 Dj = Suppf ∗ 2 Dj = Sj for all j, and assume that P(f1, f2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If δf1(Dj) > 0 for some j ∈ {1, · · · , q} with q = n + 3, then P(f1, f2) ≡ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' (b) Assume that Supp1f ∗ 1Dj = Supp1f ∗ 2 Dj = Sj for all j, and assume that P(f1, f2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' If δf1(Dj) > 0 for some j ∈ {1, · · · , q} with q = 2n + 4, then P(f1, f2) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Atsuji, Nevanlinna theory via stochastic calculus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfV_zH/content/2301.01295v1.pdf'} +page_content=' 132 (1995), 473- 510.' 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