diff --git "a/AtAyT4oBgHgl3EQf3_qJ/content/tmp_files/load_file.txt" "b/AtAyT4oBgHgl3EQf3_qJ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/AtAyT4oBgHgl3EQf3_qJ/content/tmp_files/load_file.txt" @@ -0,0 +1,1103 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf,len=1102 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='00779v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='AG] 2 Jan 2023 GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES, WITH APPLICATION TO BRILL-NOETHER THEORY GIUSEPPE PARESCHI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We introduce a variant of global generation for coherent sheaves on abelian varieties which, under certain circumstances, implies ampleness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This extends a criterion of Debarre asserting that a continuously globally generated coherent sheaf on an abelian variety is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We apply this to show the ampleness of certain sheaves, which we call naive Fourier-Mukai-Poincar´e transforms, and to study the structure of GV sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In turn, one of these applications allows to extend the classical existence and connectedness results of Brill-Noether theory to a wider context, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' singular curves equipped with a suitable morphism to an abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Another application is a general inequality of Brill-Noether type involving the Euler characteristic and the homological dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Motivation: continuous global generation, M-regularity, and ampleness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We work with projective varieties over an algebraically closed field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' On abelian (or, more generally, irreg- ular) varieties, the basic notion of global generation of a coherent sheaf (at a given point) admits a variant, introduced by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Popa and the author, called continuous global generation (CGG for short), see [PP1, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is as follows: given a coherent sheaf F on an abelian variety A, a closed point x ∈ A, and a Zariski open set U ⊂ Pic0A, one considers the sum of twisted evaluation maps evU(x) : � α∈U H0(A, F ⊗ Pα) ⊗ P −1 α → F|x , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) where: F|x := F ⊗ k(x) denotes the fiber of F at the point x, and Pα = P|A×{α} denotes the line bundle on A parametrized by a point α ∈ Pic0A via the (normalized) Poincar`e line bundle P on A × Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The sheaf F is said to be CGG at x if this map is surjective for all (non empty) Zariski open subsets of Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Of course, if this holds for every x ∈ A the sheaf F is said to be CGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 The condition of being CGG neither implies nor is implied by the usual global generaton (GG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' For example, the line bundle associated to a theta divisor on a p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' is CGG but not GG and the structure sheaf of an abelian variety is GG but not CGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' However the following three facts make the CGG property significant: Supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 1There is a more general version of these notions and their consequences holding for sheaves on any variety equipped with a morphism to an abelian variety, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this paper we mostly consider sheaves on abelian varieties 1 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI (i) Ampleness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let µN : A → A be the multiplication by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If F is a CGG coherent sheaf on A then there is a N >> 0 such that µ∗ N(F ⊗ Pα) is GG for all α ∈ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Consequently, a CGG sheaf on an abelian variety is ample ([D, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2 (ii) Criterion for global generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If F and L are respectively a CGG coherent sheaf on A, and a CGG line bundle on a subvariety of A, then F ⊗ L is GG (this follows by considering sections of the form sα · t−α with sα ∈ H0(F ⊗ Pα) and t−α ∈ H0(L ⊗ P −1 α ), [PP1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (iii) Cohomological criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' There is a natural vanishing condition on the higher cohomology, named M-regularity, implying CGG ([PP1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This package has found various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' To name a few: effective projective normality and syzygies of abelian varieties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' [PP3],[I]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' effective birationality of pluricanonical maps of irregular varieties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' [PP3], [JLT], [BLNP], [JS], [CCCJ]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' positivity of direct images of pluricanonical sheaves of varieties mapping to abelian varieties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' [D], [CJ], [LPS], [M1],[M2], [MP]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' birational geometry and volume of irregular varieties via eventual maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' [J]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The property of being CGG, as well as the M-regularity condition implying it, are quite strong and not often verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this paper we introduce the following weaker condition with the purpose of enlarging the range of applicability of this circle of ideas, especially Debarre’s criterion for ampleness mentioned in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Generation by a set of subvarieties of the dual abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Given a finite set of irreducible subvarieties of Pic0A, say Z = {Zi} (such that no subvariety is contained in another one), a coherent sheaf F on A is said to be generated by Z (at a given point x ∈ A) if the map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) is surjective for all open sets U of Pic0A such that U meets all subvarieties Zi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this language, to be CGG means to be generated by the trivial set Z = {Pic0A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' On the other hand, a GG sheaf is generated by the set formed by the origin alone: Z = {ˆ0} (but the converse does not hold in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We say that a coherent sheaf is generated if it is generated by some set of subvarieties (this notion of generation coincides with the notion of algebraic generation of [LY, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' There is a natural notion of irredundant generating set (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2), and it can be shown that, for any finite set of subvarieties Z = {Zi} as above, there are generated sheaves F having Z as irredundant generating set (Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Next, an irreducible subvariety Z of an abelian variety B is said to span B if N � �� � Z + · · · + Z = B for some N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A finite set of irreducible subvarieties Z = {Zi} is said to strongly span B if each subvariety Zi spans B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Items (i) and (ii) of the previous subsection generalize as follows: 2(a) The notion of ampleness have been extended to coherent sheaves by Kubota [K], and many properties holding for vector bundles extend to this setting, see [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (b) The quoted result is stated in loc cit over the complex numbers, but the proof works over any algebraically closed field 3In the same spirit, a related notion, namely weak CGG, was introduced by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Popa and the author in [PP2, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1], but the notion given here is better behaved GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 3 Theorem A (Nefness/ampleness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If F is generated then it is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If F is generated by a set of subvarieties Z strongly spanning Pic0A then F is ample (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Theorem B (Non-generation locus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If F is generated by a set Z = {Zi} and L is a CGG line bundle on a subvariety X of A, then the non-generation locus of F ⊗L (namely the locus B(F ⊗L) where F ⊗L is not globally generated) is explicitly controlled in function of the base loci of the line bundles L ⊗ Pα, with α ∈ Zi (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (As expected, it turns out that the bigger are the Zi, the smaller is B(F ⊗ L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Relationship with the Fourier-Mukai-Poincar´e transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We are left with the prob- lem of finding criteria or applicable methods for proving that a given coherent sheaf is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In view of Theorem A, it is especially relevant to understand whether a given sheaf F is generated by some set of subvarieties strongly spanning Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The author hopes that this method will be helpful in addressing the ampleness of vector bundles on subvarieties of abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The next result is a step in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Namely we provide a sufficient condition (close to be a characterization) ensuring that a given coherent sheaf F is generated, and we identify the irredundant set of subvarieties doing the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Roughly speaking, this is a subset of the set of irreducible components of the supports of the sheaves appearing in the torsion filtration ([HuLe, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4]) of a certain coherent sheaf on Pic0A associated to F, referred to as the naive Fourier-Mukai-Poincar´e (FMP) transform of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' However, except for some cases, this is just a first step toward the solution of the above problem, since in general the torsion filtration of the naive FMP transform of F is not easy to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Passing to a more detailed description, we will consider the FMP functor in the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let P be the normalized Poincar`e line bundle on A × Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We consider the Fourier-Mukai equivalence ΦA P−1 : D(A) → D(Pic0A), ΦA P−1( · ) = Rq∗(p∗( · ) ⊗ P−1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) where p and q are the projections on A and Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We will also make use of the dualization functor in the following (unshifted) form: F∨ := RHom(F, OA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) The above mentioned naive FMP transform of a given coherent sheaf F on A is defined as follows T (F) := RgΦA P−1(F∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4) where g = dim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By Serre-Grothedieck duality, Hi(A, F∨ ⊗ P −1 α ) = 0 for all i > g and for all α ∈ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, by base change and duality, we have the canonical isomorphisms RgΦA P−1(F∨)|α ∼= Hg(A, F∨ ⊗ P −1 α ) ∼= H0(A, F ⊗ Pα)∨ for all α ∈ Pic0A (note that the above Hg is usually a hypercohomology group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Hence (up to duality) the coherent sheaf T (F) encodes the variation of the k-linear spaces H0(A, F ⊗ Pα) as α varies in Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore it is natural to expect that T (F) must be related somehow to the generation of the coherent sheaf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It turns out that it is the torsion of the sheaf T (F) what really matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Of course, in general it is not the case that the naive transform coincides (up to shift) with the transform in the derived category, namely that ΦA P−1(F∨) = RgΦA P−1(F∨)[−g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5) In fact the sheaf F is said to be a GV sheaf (generic vanishing sheaf) precisely when (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI The precise relation of the FMP transform with the generation problem is stated in Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It could be somewhat imprecisely summarized as follows Theorem C (Generation of sheaves and the FMP transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (a) The surjectivity of the map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) can be rephrased in terms of a condition involving both the FMP transform ΦA P−1(F∨) and the naive FMP transform RgΦA P−1(F∨);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (b) if F is generated, then a certain subset of the set of irreducible components of the supports of the sheaves appearing in the torsion filtration of the naive FMP transform of F form an (irredundant) generating set for F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' As mentioned above, the problem in applying condition (b) is that at present there is no general way of describing, in terms of F, the supports of the sheaves appearing in the torsion filtration of the naive FMP transform of F (only for GV sheaves there we have such a description, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Ampleness of naive FMP transforms (= generalized Picard bundles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The next result concerns a lucky class of coherent sheaves which turn out to be generated and ample, even when they are not GV sheaves: the naive FMP transforms themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' More generally, the same holds for the naive FMP transforms of coherent sheaves on certain reduced schemes mapping to abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' These are defined similarly, with the difference that the FMP functor is not an equivalence anymore (it turns out that, for the specific result described in the present subsection, this is unnecessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' given a equidimensional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' reduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Cohen-Macaulay projective scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' equipped with a morphism to an abelian variety f : X → A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' one defines PX = (f × id)∗P and considers the functor (not an equivalence anymore) ΦX P−1 X : D(X) → D(Pic0A) and the (unshifted) dualization functor ∆X : D(X) → D(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' ∆X(F) = RHom(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' ωX) The naive FMP transform of a coherent sheaf F on X is defined as T (F) = RdΦX P−1 X (∆X(F)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' where d = dim X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' and has the same meaning as discussed in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We will consider coherent sheaves F on a reduced scheme X as above with the property that all subsheaves of F have reduced scheme-theoretic support (equivalently, one can consider only the subsheaves appearing in the torsion filtration of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We will adopt the following notation: given a sheaf F on X, we denote Z(F) = {Zi} the set of irreducible components of the supports of the sheaves on X appearing in the torsion filtration of F, and f(Z(F)) := {f(Zi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Theorem D (Generation and ampleness of naive FMP transforms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the above setting, let F be a coherent sheaf on a scheme X as above, such that all subsheaves of F have reduced scheme-theoretic support (the simplest example are torsion free sheaves on X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then the naive FMP transform T (F) is generated by a subset of the collection f(Z(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In particular, if each subvariety of the set Z(F) maps, via the morphism f, to a subvariety spanning the abelian variety A, then T (F) is an ample sheaf on Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 5 The above theorem is in fact is a generalization of a result of Schnell, who proved that naive FM transforms of torsion free sheaves on abelian varieties are CGG ([Sch, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4 The argument used in the proof is essentially Schnell’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We remark that, although in a different language, a well known example of Theorem D is that of (dual) Picard bundles: here X is a smooth complex projective variety, equipped with its Albanese morphism a : X → Alb X, and F = L is a line bundle on X satisfying the following vanishing condition on the higher cohomology: Hi(X, L ⊗ Pα) = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6) for all α ∈ Pic0X and i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5 By base change this ensures that T (L) is a vector bundle on Pic0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Note that in this case the generating collection is given by a single subvariety, namely the Albanese image {a(X)}, which automatically spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By Theorem D, T (L) is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The dual of T (L), namely R0ΦX PX(F), is called the Picard bundle associated to the line bundle L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6 Picard bundles were classically known to be be negative for dim X = 1 ([ACGH]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This result was subsequently generalized by Lazarsfeld to all smooth projective varieties ([L, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='C]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Thus Theorem D can be seen as a vast generalization (with a completely different proof) of that result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It is worth to remark that in Theorem D we are not assuming condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6), nor any other vanishing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Interestingly, there are many examples of coherent sheaves not verifying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6) such that nevertheless their naive FMP transform is locally free (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Application to Brill-Noether theory of singular curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' One immediate application of Theorem D is to Brill-Noether theory of (complex) singular curves (even reducible and with non-planar singularities) equipped with a morphism to an abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this setting we provide analogues of the existence and connectedness theorems for special divisors, Ghione and Segre- Nagata theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We refer to Subsection 5 for these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A general inequality of Brill-Noether type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Another application of Theorem D is a general existence result of Brill-Noether type which, although somewhat weak, is optimal, and turns out to be useful in some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this subsection we will assume that the ground field is C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Given a coherent sheaf F on a complex abelian variety A,7 one defines the cohomological support loci V i(F) = {α ∈ Pic0A | hi(F ⊗ Pα) > 0} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7) and V >0(F) = � i>0 V i(F) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8) Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a coherent sheaf on a complex abelian variety A such that all its torsion subsheaves (if any) have reduced scheme-theoretic support, and each component of the support spans A (simplest example: a torsion free sheaf on A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Assume that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' V >0(F) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 4Of course Schnell does not use this terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Moreover his result is stated in the language of the symmetric Fourier transform, introduced in the same paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 5When dealing with sheaves on abelian varieties this condition is usually referred to as the IT(0) condition (namely F satisfied the index theorem with index 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 6It is easily seen that this definition is equivalent to the one of [L, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='C], where the Picard bundle is defined as a vector bundle on PicλX, where λ is the algebraic equivalence class of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 7also in this case one could extend the discussion of this topic to varieties equipped of a finite morphism to an abelian varieties, but for sake of simplicity we will stick to abelian varieties 6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI Then χ(F) ≥ hd(F) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Here hd(F) denotes the homological dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Easy examples showing that (at least the way it is stated), Theorem E is optimal, are: (a) locally free sheaves F on abelian varieties (of arbitrarily high rank) such that V >0(F) = ∅ and χ(F) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (It is known that such sheaves are of the form F = ΦP−1(L∨), where L is any ample line bundle on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then χ(F) = 1 and rk(F) = χ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=') (b) Let i : C → J(C) a Abel-Jacobi embedding of a curve in its Jacobian and let F = i∗L, where L is a line bundle on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this case V >0(F) = V 1(F), which is non-empty if and only if deg(L) ≤ 2g − 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' χ(F) ≤ g − 1 = hd(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' An example of application of Theorem E is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let J (D) be the multiplier ideal sheaf of an effective Q-divisor D on an abelian variety A, and let L be an ample line bundle on A such that L − D is nef and big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By Nadel’s vanishing the sheaf J (D) ⊗ L satisfies the assumptions of Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Letting Z the scheme of zeroes of J (D), it follows that χ(J (D) ⊗ L) ≥ codim Z (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='9) (meaning the maximal codimension of a component of Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is instrumental in the proof of a result of the author on singularities of divisors on complex simple abelian varieties ([P2], see also Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The case of GV sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' GV sheaves are those sheaves such that their naive FMP transform T (F) coincides, up to shift, with the FMP transform in the derived category (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8 They are very special cases of naive FMP transforms since it follows from the inversion formula for the FMP equivalence that F = T (T (F)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10) (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore part (a) of the next result, on the generation and ampleness of such sheaves, follows from Theorems C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' As M regular sheaves form a subclass of GV sheaves, this recovers as a particular case the M-regularity criterion (iii) of Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Item (b) is a structure result for such GV sheaves, following from the analysis of the torsion filtration of the FMP transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Loosely speaking, the content is that the building blocks for GV sheaves are either CGG (hence ample) sheaves, or sheaves of the form (p∗G) ⊗ Pα, where p : A → B is a surjective homomorphism with connected kernel onto a lower dimensional abelian variety and G is a CGG (hence ample) sheaf on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Theorem F (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a GV sheaf on a g-dimensional abelian variety A, and assume that all subsheaves of the FMP transform T (F) have reduced support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then: (a) F is generated and an irredundant generating set can be explicitly described;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (b) F has a cofiltration whose kernels have in turn a cofiltration whose kernels are dominated either by ample sheaves or by sheaves of the form (p∗G) ⊗ Pα as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Concerning (a), let us recall that not all GV sheaves are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is shown by the well known example of non-trivial unipotent vector bundles on abelian varieties (see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is caused by the fact that, unless F is trivial, the FMP transform T (F) is a torsion sheaf supported at a non-reduced zero-dimensional scheme, set-theoretically supported at the origin of 8If this is the case the sheaf T (F) is often denoted � F∨ in the literature GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 7 Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It is worth to recall that a generation criterion for GV sheaves was already given by Popa and the author (WIT criterion, [PP2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Item (a) is a more precise version of that result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In turn, item (b) can be seen as a weak analog of the Chen-Jiang decomposition ([CJ]), a quite useful property satisfied by direct images of pluricanonical sheaves under morphisms to abelian varieties (see Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Generated and ample coherent sheaves on abelian varieties In this section we provide some generalities on the notion of generation of coherent sheaves on abelian varieties introduced in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2, and prove Theorems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Generalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' To begin with, we remark that, although the focus of this paper is on coherent sheaves on abelian varieties, one can extend the definition given in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2 to the following setting: let X be a projective variety, equipped with a morphism to an abelian variety f : X → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let Z = {Zi}i∈I be a finite set of irreducible subvarieties of Pic0A, such that no subvariety belonging to Z is contained in another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A coherent sheaf F on X is said to be generated by Z at a given point x ∈ X (with respect to te morphism f) if the map evU(x) : � α∈U H0(F ⊗ f ∗Pα) ⊗ f ∗P −1 α → F|x is surjective for all open sets U such that U meets Zi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If this happens for all x ∈ X, namely the map evU : � α∈U H0(F ⊗ f ∗Pα) ⊗ f ∗P −1 α → F is surjective, the sheaf F is said to be generated by the set Z (with respect to the morphism f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If Z = {Pic0A} then F is said to be continuously globally generated (CGG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Moreover F is said to be generated (with respect to the morphism f) if there is some set of subvarieties Z generating F (with respect to the morphism f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is equivalent to the surjectivity of the map evPic0A : � α∈Pic0A H0(F ⊗ f ∗Pα) ⊗ f ∗P −1 α → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A finite set of subvarieties Y = {Yj} as above is said to be covered by another Z = {Zi} if for all i there is a j such that Yj is contained in Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If a sheaf F is generated by a set Z then, trivially, it is generated by any set covered by Z, and, as it will be clear in a moment, it turns out that is useful to identify a maximal (with respect to the relation of being covered) set doing the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Such a set of subvariety will be called an irredundant generating set for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In practice in some arguments it may happen to find generating sets Z = {Zi} such that Zj is contained in Zk for some j and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' However this does not cause any problem because, if this is the case, to be generated by the set Z is the same thing of being generated by the set Z ∖ {Zk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Hence, by taking the subset of minimal subvarieties belonging to set Z, one can always reduce to the assumption of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By noetherianity and quasi-compactness, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 can be equivalently for- mulated replacing the map evU of the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 with the sum of finite number of evaluation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The required condition is the existence of positive integer N0 and a collection of positive integers {Ni}i∈I such that the sum of twisted evaluation maps � i∈I∪{0},1≤j≤Ni H0(F ⊗ f ∗Pαi,j) ⊗ f ∗P −1 αi,j → F is surjective for all sufficiently general (α0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' α0,N) ∈ (Pic0A)N and (αi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' , αi,Ni) ∈ (Zi)Ni for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore also in the sum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) one can take a finite number of summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In particular, the notion of generated coherent sheaf introduced here coincides with the the notion of algebraically generated coherent sheaf introduced in the recent paper [LY, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It follows that a coherent generated (with respect to any morphism) sheaf is nef, as it is the quotient of the direct sum of numerically trivial line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The main point is in the following easy lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the setting of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, let Z = {Zi}n i=1 and Y = {Yj}m j=1 be two finite sets of subvarieties of Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let moreover F and G be coherent sheaves on X respectively generated by Z and Y (with respect to the morphism f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then F ⊗ G is generated by the set of subvarieties (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3): Z + Y := {Zi + Yj}(n,m) (i,j)=(1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Here Zi + Yj denotes, as usual, the image of Zi × Yj via the group law of Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' For open subsets of Pic0A, say U and V , respectively meeting all the Zi’s and all the Yi’s we have the surjective map � (α,β)∈U×V H0(F ⊗ f ∗Pα) ⊗ H0(G ⊗ f ∗Pβ) ⊗ f ∗P −1 α ⊗ f ∗P −1 β ∥ �� α∈U H0(A, F ⊗ f ∗Pα � ⊗ f ∗P −1 α ) ⊗ �� β∈V H0(A, G ⊗ f ∗Pβ) ⊗ f ∗P −1 β � −→ F ⊗ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This map factors through the map � γ∈U+V H0(F ⊗ G ⊗ f ∗Pγ) ⊗ f ∗P −1 γ → F ⊗ G which is, therefore, surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Finally, any open subset of Pic0A meeting all the subvarieties Xi+Yj, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' , n and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' , m, contains some open subset of the form U + V , with U and V as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ The following result is Theorem A of the Introduction, in a slightly more general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It is an extension of the above quoted result of Debarre asserting that a CGG sheaf with respect to a finite onto its image morphism to an abelian variety is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the setting of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 let us assume that the morphism f : X → A is finite onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a coherent sheaf on X such that F is generated by a finite set of subvarieties Z = {Zi}i∈I strongly spanning Pic0A (see Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then F is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' To begin with, we note that the fact that the set Z = {Zi}i∈I strongly spans Pic0A implies that there is a positive integer M such that Zi1 + · · · + ZiM = Pic0A for all (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' , iM) ∈ IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 9 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 we know that the sheaf F⊗M is generated by the set {Zi1 + · · · + ZiM }(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=',iM)∈IM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Hence F⊗M is CGG and therefore the same holds true for SMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, by Debarre’s theorem, SMF is ample, hence F is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ As a particular case, a coherent sheaf on an abelian variety which is generated by a single irreducible subvariety Z spanning Pic0A is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Obviously this is in general false if F is generated by the set of components of a reducible subvariety spanning Pic0A, but some individual component do not (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' on a product of abelian varieties A = A1 × A2, let F = p∗ 1F1 ⊕ p∗ 2F2, with Fi CGG (hence ample) sheaves on Ai for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The sheaf F is not ample, but is generated by the collection {Pic0A1 × {ˆ0}, {ˆ0} × Pic0A2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' As mentioned in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 (item (ii)), the CGG condition provides a useful criterion for global generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The natural extension of this is the following9 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the setting of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, let F be a coherent sheaf on X, generated (with respect to the morphism f) by a set Z = {Zi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let Y be a subvariety of X and let L be a CGG (with respect to f) line bundle on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then B(F ⊗ L) ⊂ � i � � α∈Zi B(L ⊗ P −1 α ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Note that for a line bundle L, to be CGG means that the intersection of the base loci � α∈V B(L ⊗ Pα) is empty for all (non-empty) open subsets V ⊆ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Since L is CGG the subset {α ∈ Pic0A | y ̸∈ B(L ⊗ P −1 α )} contains an open subset Uy(L) ⊆ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Assume that y ̸∈ � i(� α∈Zi B(L ⊗ P −1 α )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then the open set Uy(L) meets all irreducible subvarieties Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, given another open set U ⊂ Pic0A meeting all irreducible subvarieties Zi, also the open subset U ∩ Uy(L) meets all subvarieties Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, in the commutative diagram � α∈U∩Uy(L) H0(F ⊗ Pα) ⊗ H0(L ⊗ P −1 α ) � � H0(F ⊗ L) � � α∈U∩Uy(L) H0(F ⊗ Pα) ⊗ (L ⊗ P −1 α )|y � (F ⊗ L)|y both the left arrow and the bottom arrow are surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Hence the right arrow is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Relationship with the FMP transform 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The basic relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The relevance of the FMP functor in the study of the generation of coherent sheaves on abelian varieties stems from the following relation between the evaluation maps of a coherent sheaf on A and of its naive FMP transform on Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is known to the experts (see Schnell’s paper [Sch], proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In what follows we will keep the setting and notation of the Introduction, Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Given an open subset U of Pic0A, the continuous evaluation map at x ∈ A evU(x) : � α∈U H0(F ⊗ Pα) ⊗ P −1 α → F|x (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) 9also this result is an extension of a result of Popa and the author in the context of the above mentioned notion of weak global generation ([PP2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4(c)]) 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI factors through the map � α∈U H0(F ⊗ Pα) ⊗ (P −1 α )|x → F|x (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) is surjective if and only if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let us consider the individual maps of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' H0(F ⊗ Pα) ⊗ P −1 α ⊗ k(x) → F ⊗ k(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) The next Proposition describes the image of the dual map of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) via the (contravariant) equiva- lence F : D(A) → D(Pic0A), F( · ) = ΦA P−1(( · )∨) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The functor F identifies the Serre-Grothendieck dual of the linear map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) to the linear map Hg(ΦA P−1(F∨) ⊗ Px) → (RgΦA P−1(F∨) ⊗ Px) ⊗ k(α) factoring as follows Hg(ΦP−1(F∨) ⊗ Px) edx � �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ H0(RgΦP−1(F∨) ⊗ Px) evx(α) � (RgΦP−1(F∨) ⊗ Px)|α (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4) where evx(α) is the evaluation at the point α ∈ �A of the coherent sheaf RgΦP−1(F∨) ⊗ Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The proof is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We identify the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) to H0(F ⊗ Pα) ⊗ H0(P −1 α ⊗ k(x)) → H0(F ⊗ k(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5) Applying Serre-Grothedieck duality, we write the dual map of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5) as HomD(A)(k(x), F∨[g]) = Extg(k(x), F∨) → Hom(H0(P −1 α ⊗ k(x)), Hg(F∨ ⊗ P −1 α )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6) Applying the functor ΦA P−1 to the the source of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6), we have the chain of isomorphisms HomD(A)(k(x), F∨[g]) ∼= HomD(Pic0A)(P −1 x , ΦA P−1(F∨)[g]) ∼= ∼= Extg(P −1 x , ΦA P−1(F∨)) ∼= Hg(ΦA P−1(F∨) ⊗ Px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7) Concerning the target of the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6), there are the canonical identifications H0(P −1 α ⊗ k(x)) ∼= R0ΦA P−1 � k(x) � ⊗ k(α) = ΦA P−1(k(x)) ⊗ k(α) = P −1 x ⊗ k(α) and Hg(F∨ ⊗ P −1 α ) ∼= � RgΦA P−1(F∨) � ⊗ k(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We conclude that the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6) is identified, via the functor ΦA P−1, to a linear map Hg(ΦA P−1(F∨) ⊗ Px) → � RgΦA P−1(F∨) ⊗ Px � ⊗ k(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' factorizing trough the evaluation map of the sheaf RgΦA P−1(F∨) ⊗ Px at the point α, as stated in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 11 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The map edx : Hg(ΦA P−1(F∨) ⊗ Px) → H0(RgΦA P−1(F∨) ⊗ Px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8) appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4) is in fact the edge map in the hypercohomology spectral sequence Hp(RqΦA P−1(F∨) ⊗ Px) =⇒ Hp+q(ΦA P−1(F∨) ⊗ Px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' From this point we will adopt the notation of the Introduction (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4), namely T (F) := RgΦA P−1(F∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='9) This sheaf will be referred to as the naive FMP transform of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 allows to express the surjectivity of the map evU(x) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) as follows (this is Theorem C(a) of the Introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let U be an open subset of Pic0A and let x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The map evU(x) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) is surjective if and only if the following two conditions hold: (1) the map edx of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8) is injective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (2) the restriction of the simultaneous evaluation map evx(U) : H0(T (F) ⊗ Px) −→ � α∈U (T (F) ⊗ Px)|α (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10) to the image of the map edx is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Notice that the map edx depends only on x and not on the open subset U ⊂ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore if F is generated at x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' the map evU(x) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1)) is surjective for some U, then edx is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In view of the previous corollary, it is useful to understand the kernel of the evaluation maps evx(U) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10) for a non empty open subset U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A non-zero section s ∈ ker evx(U) must be a torsion section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' the image of the map s : OPic0A → T (F) ⊗ Px must be a torsion sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If the support of such image is reduced then it is the closure of the subset of α ∈ Pic0A such that s|α ∈ (T (F) ⊗ Px)|α is non zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If this is the case then U must be contained in the complement of such support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Of course for all x ∈ A the torsion subsheaves of T (F) ⊗ Px coincide, after tensorization with P −1 x , with the torsion subsheaves of T (F), hence they do not depend on x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Moreover all the irreducible components of supports of torsion subsheaves of T (F) appear in the support of sheaves appearing in the torsion filtration of T (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Summarizing, we have the following Corollary, which is Theorem C(b) of the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the statement we denote �V 0(τ) = {x ∈ A | H0(τ ⊗ Px) ∩ Im(edx) ̸= 0} Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a coherent sheaf on an abelian variety A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Assume that: (a) the maps edx of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8) are injective for all x ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (b) all sheaves τ appearing in the torsion filtration of the naive FMP transform T (F), and such that �V 0(τ) is non empty, have reduced scheme-theoretic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then F is generated by the set of minimal irreducible components of supports of all such sheaves τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In general, it is not known how to identify in function of F the various sheaves τ appearing in the torsion filtration of the naive FMP transform T (F), let alone those such that �V0(τ) in non empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Not surprisingly, in the case of GV sheaves this can be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Recalling the 12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI definition and notation for cohomological support loci given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7), it is well known that F is a GV sheaf if and only if, for all i ≥ 0 codimPic0A V i(F) ≥ i ([PP6, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10] or [PP5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If this is the case, the components of supports of the torsion sheaves appearing in the torsion filtration of the sheaf T (F) are components W of the loci V i(F) of the minimalcodimension, namely codimPic0A W = i (this follows from a result of Popa and the author, see [P1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We refer also to [CLP, §9] where an even more precise description of the torsion filtration is given in the case of coherent sheaves admitting a Chen-Jiang decomposition, a special class of GV-sheaves which includes direct images of pluricanonical bundles with respect to morphism to an abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' See Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5 below for more on these sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Generation and ampleness of naive FMP transforms and generalized Picard bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A known class of examples of ample vector bundles on abelian varieties is the one of (dual) Picard bundles (see Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4 of the Introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the case where X is a smooth curve naturally embedded in its Jacobian, it was classically known that the projectivization of the dual of the Picard bundle of a line bundle of degree d ≥ 2g − 1 is the d-symmetric product of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Based on this observation, it follows that the dual of Picard bundles of curves are ample, and, as a consequence, one gets the existence and connectedness theoremsn of Brill-Noether theory ([ACGH, Chapter VII]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The ampleness of dual Picard bundles and some of its applications were subsequently generalized to all smooth complex projective varieties by Lazarsfeld ([L, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='C and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='C]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this section we show that the FMP methods of the previous section quickly provide (with a different argument) a vast generalization of such results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Naive FMP transforms of sheaves on abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In order to give a quick idea of the result, we start from a particular case already met in the previous section, namely naive FMP transforms of coherent sheaves on abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We keep the notation of Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4, especially on the set of subvarieties Z(F) (irreducible components of supports of coherent sheaves appearing in the torsion filtration of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a coherent sheaf on an abelian variety A such that all of its subsheaves have reduced scheme-theoretic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then the naive FMP transform T (F) = RgΦA P−1(F∨) is generated by a subset of the set Z(F) (see Theorem D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In particular, T (F) is an ample sheaf on Pic0A as soon as all subvarieties appearing in the set Z(F) span A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' As mentioned in the Introduction, this result, for torsion free coherent sheaves, is already present in Schnell’s paper [Sch, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The present argument is borrowed from his.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We use the notation on evaluation maps of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The arguments consists in considering diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4) and let x (rather than α, vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Specifically, the generation of T (F) means the surjectivity of the map evA(α) : � x∈A : H0(RgΦA P−1(F∨) ⊗ Px) ⊗ (P −1 x )|α −→ RgΦA P−1(F∨)|α for all α ∈ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 (and its proof) the map evα(A) : H0(F ⊗ Pα) −→ � x∈A (F ⊗ Pα)|x GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 13 is the dual of a map factorizing trough evA(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore asserted surjectivity follows from the injectivity of the map evα(A), which follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4 because we are assuming there are no subsheaves with non reduced scheme-theoretic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In fact, also the map evα(V ) : H0(F ⊗ Pα) −→ � x∈V (F ⊗ Pα)|x is injective, for all open subsets V of A meeting all irreducible components of the support of all subsheaves of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In fact it is enough to consider the subsheaves appearing in the torsion filtration of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Generalization to transforms from reduced schemes mapping to abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this subsection we prove Theorem D of the Introduction, namely the generalization of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 to naive transforms from certain reduced schemes mapping to abelian varieties, rather than from abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Again, we keep the notation and setting introduced in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let f : X → A be a reduced Cohen-Macaulay equidimensional (of dimension d) projective scheme equipped with a morphism to an abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a coherent sheaf on X such that all of its subsheaves have reduced scheme theoretic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then its naive FMP transform T (F) = RdΦX P−1 X (∆X(F)) is generated by a subset of the set f(Z(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In particular, T (F) is ample sheaf on Pic0A as soon as, for each i, the image f(Zi) of the subvariety Zi ∈ Z(F) spans A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' this happens when f(X) spans A and F is a torsion free sheaf on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The argument is similar to the one of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 but needs to be adjusted because the functor ΦX P−1 X is not anymore an equivalence, hence Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 does not hold anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We still consider the linear map H0(F ⊗ f ∗Pα) ⊗ H0(f ∗P −1 α ⊗ k(x)) → H0(F ⊗ k(x)) Dualizing we get HomD(X)(k(x), ∆X(F)[d]) = Extd(k(x), ∆X(F)) → Hom(H0(f ∗P −1 α ⊗k(x)), Hd(∆X(F)⊗f ∗P −1 α )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) We still have that ΦX P−1 X (k(x)) = R0ΦX P−1 X (k(x)) = P −1 f(x) and H0(f ∗P −1 α ⊗ k(x)) ∼= R0ΦP−1 X (k(x)) ⊗ k(α) = ΦP−1 X (k(x)) ⊗ k(α) = P −1 f(x) ⊗ k(α) Therefore the target of the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) is still identified by the functor ΦP−1 X to the fibre (RdΦP−1 X (∆X(F)) ⊗ f ∗Pf(x))|α Even if the analog, in the present setting, of the first map in the chain of isomorphisms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7) is not an isomorphism anymore, still it follows that the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) factorizes, via the functor ΦX P−1 X , trough the evaluation map of global sections at the point α ∈ Pic0A of the naive FMP transform, twisted by the line bundle Pf(x): Extd(k(x), ∆X(F)) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ H0(RdΦP−1 X (∆X(F)) ⊗ f ∗Pf(x)) evx(α) � (RdΦP−1 X (∆X(F)) ⊗ f ∗Pf(x))|α (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) At this point the argument goes exactly as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ 14 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Note that in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 we are not assuming the vanishing condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='6), namely that Hi(X, F ⊗ f ∗Pα) = 0 for all α ∈ Pic0A and i > 0, which implies that the naive FMP transform (or Picard sheaf) T (F) is a locally free sheaf on Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It is worth to note that in fact there are many examples where such condition does not hold but still T (F) is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, even if one is interested only in ample vector bundles the above results produce a wider class of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let us outline a simple way to produce such sheaves, similar to Raynaud’s construction of stable vector bundles ”without theta divisor” on curves ([R]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The simplest example is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let C be a smooth curve embedded in its Jacobian J(C) := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' For odd coprime positive integers a, b let Wa,b be the semhomogenous vector bundles on a p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=', as J(C), considered by Mukai and Oprea ([Mu1], Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 and Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3, [O]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Denoting µb : A → A the multiplication by b, and L = OA(Θ), these are vector bundles on A such that µ∗ aWa,b = (Lab)⊕ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) We claim that if a and b are such that a b ∈ (0, 1) then hi(F ⊗ Pα) is non-zero and constant for α ∈ Pic0A for both for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This can be shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3), we have that hi((Wa,b)|C ⊗ Pα) = ag a2g hi(a∗ A(OC ⊗ Pα) ⊗ OA(abΘ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' for a general α ∈ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This shows that, for i = 0, 1 and α general in Pic0A, by definition, the rational number 1 ag hi((Wa,b)|C ⊗Pα) equals the value of the cohomological rank functions hi OC(xθ), x = b a (see [JP]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' These functions have been computed in loc cit, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 837, proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Specifically, in the interval [0, 1], they are h0 OC(xθ) = xg and h1 OC(xθ) = xg − g + 1 − xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4) If there was a non empty jump locus for the function α �→ hi((Wa,b)|C ⊗ Pα) (α ∈ Pic0A) then as it is easy to see, this would cause a critical point of the functions hi OC(xθ) (see §5, loc cit) in the interval (0, 1) but (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4) shows that there is none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore the functions α �→ hi((Wa,b)|C ⊗ Pα) are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It follows, in particular, that T ((Wa,b)|C) is a locally free sheaf on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Similar examples can be obtained starting from any subvariety X of a ppav and any coherent sheaf G on X, considering the sheaves F = G ⊗Wa,b for b a in a suitable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Constructions of this type can be performed on abelian varieties with arbitrary polarizations (not necessarily principal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Applications to Brill-Noether theory 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Singular curves equipped with a morphism to an abelian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The interest of results as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, besides providing examples of ample vector bundles, is in their appli- cation to Brill-Noether theory via the nowdays classical results of Kempf, Kleiman-Laksov, and Fulton-Lazarsfeld on non-emptyness and connectedness of degeneracy loci [ACGH, Chapter VII], [L, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Such applications mainly concern locally free sheaves on smooth curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Via Theorem D they can be generalized e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' to the following setting: torsion free sheaves F on singular curves C (connected reduced 1-dimensional Cohen-Macaulay projective schemes), equipped with a morphism f : C → A to an abelian variety, such that the image of each component of spans the abelian variety A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this sort of application we will mainly concerned with Brill-Noether loci V 0,r f (X, F) = {α ∈ Pic0A | h0(X, F ⊗ f ∗Pα) ≥ r + 1} GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 15 According to the previous notation, the locus V 0,0 f (X, F) is simply denoted V 0 f (X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The loci V 0,r f (X, F) can be realized as degeneracy loci of a map of locally free sheaves in the usual way, namely considering the exact sequence 0 → F → F(D) → F(D)|D → 0 where D is a Cartier divisor avoiding the singular points of X, such that D is of degree high enough on all components of X so that Hi(X, F(D) ⊗ f ∗Pα) = 0 for all α ∈ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It turns out that ΦX PX(F(D)) = R0ΦX PX(F(D)) is a locally free sheaf on Pic0A (dual to the naive FMP transform T (F)) whose fibre at α is identified to H0(X, F(D) ⊗ f ∗Pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore the locus V 0,r f (X, F) is the k’th degeneracy locus of the map of vector bundles ϕ : ΦX P−1 X (F(D)) → ΦX P−1 X (F(D)|D) where k = χ(F(D))−(r+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The target is a locally free sheaf on Pic0A of rank equal to � νi deg Di, where (ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='., νh) is the multirank of F (the tuple of the ranks of F at the various components of X) and deg Di is the degree of D at the various components of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The expected dimension of the locus V 0,r f (X, F) is (r + 1)( � νi deg Di) − (χ(F(D)) − (r + 1))) = (r + 1)(r + 1 − χ(F)) By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 the source of the map ϕ is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' On the other hand it is well known that the target is a homogeneous vector bundle, namely admitting a filtration whose quotients are line bundles parametrized by Pic0(Pic0A)) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, by the Theorem of Fulton-Lazarsfeld, we have Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the above setting, the locus V 0,r f (X, F) is nonempty (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' connected) as soon as (r + 1)(r + 1 − χ(F)) ≤ dim A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (r + 1)(r + 1 − χ(F)) < dim A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In particular V 0 f (F) is non empty as soon as χ(F) ≥ − dim A + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This statement recovers (for smooth curves embedded in their Jacobians) the classical exis- tence and connectedness theorems of Brill-Noether theory, as well as Ghione’s theorem ([L, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' As a consequence, we have the following version of the theorem of Segre-Nagata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' To sim- plfy, assume furthermore that, in the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, the torsion free sheaf F has uniform rank ν (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' ν1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' = νh = ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let us define deg F := χ(F) − ν(1 − pa(X)) Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The sheaf F has an invertible subsheaf B of degree deg B ≥ 1 − pa(X) + deg F + dim A − 1 ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By the last inequality of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 it follows that the locus V 0 f (F ⊗ B−1) is non-empty as soon as χ(F) − ν deg B ≥ − dim A + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ Similarly, one can extend to the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 the result of [L, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 16 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A Brill-Noether inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a coherent sheaf on an abelian variety A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Recalling the notation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8) about the cohomological support loci of F, assume that the loci V >0(F) is strictly contained in Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then χ(F) ≥ 0 (since it is equal to the generic value of h0(F ⊗Pα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It turns out that this easy inequality can be improved if the cohomological loci V i(F), i > 0, are suitably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' For example, the ”Castelnuovo - De Franchis inequality” of Popa and the author ([PP4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3]) states that, if F is a GV sheaf and V >0(F) is non empty, then χ(F) ≥ gvα(F) := mini>0codimα{V i(F) − i | i > 0} (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) for all α ∈ Pic0A, where codimαV i(F) denotes the codimension of V i(F) in Pic0A, in the neigh- borhood of a given point α ∈ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The result stated as Theorem E in the Introduction complements the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1), dealing with the case where the locus V >0(F) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The statement asserts that if this is the case, and the coherent sheaf F is torsion free, or, more generally, all of its subsheaves have all reduced support, and each irreducible component of the support spans A, then χ(F) ≥ hd(F) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (of Theorem E) By Theorem D the naive FMP transform of F, namely T (F) = RgΦA P−1(F∨) is an ample sheaf on Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Moreover the hypothesis that V >0(F) = ∅ yields, by base change, that T (F) is a locally free sheaf on Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We claim that Supp � Exti(F, OA) � ⊆ V i(T (F)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) The assertion of Theorem E follows from the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Indeed, by Le Potier vanishing, V j(T (F)) = ∅ as soon as j ≥ rkT (F) = χ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) yields that χ(F) > max {j | Extj(F, OA) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' To prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3), note that the hypothesis V >0(F) = ∅ yields trivially that RiΦA P−1(F∨) = 0 for i ̸= g, and therefore T (F) = ΦA P−1(F∨)[g] (in other words F is a GV-sheaf (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, since the inverse of the FMP equivalence ΦA P−1 : D(A) → D(Pic0A) is the equivalence ΦPic0A P [g] : D(Pic0A) → D(A), it follows that ΦP(T (F)) = F∨ = RHom(F, OA) hence RiΦP(T (F)) = Exti(F, OA) (see also Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2 below for a related statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This yields (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) since, by base change, SuppRiΦP(T (F)) ⊆ V i(T (F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Interestingly, the main point of the proof of Theorem E is Le Potier vanishing theorem, while the essential ingredient of the proof of inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) is the syzygy theorem of Evans-Griffith ([EG, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7]), which is in turn quite related to the theorem of Le Potier (as shown by Ein, [E]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore it seems possible that the inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) are both manifestations of some more general phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Theorem E was already implicitly used by the author (in a special case) in the proof of the following result (conjectured in [DH, §6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 17 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2 ([P2] Theorem B(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let A be a complex simple abelian variety and D an effective Q-divisor on A such that (A, D) is a non klt pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If L is a line bundle on A such that L − D is nef and big then χ(L) ≥ dim A + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The hypothesis means that J (D), the multiplier ideal sheaf of D, is non trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By Nadel’s vanishing V >0(J (D) ⊗ L) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let Z be the zero-scheme of J(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By Theorem E, χ(J (D) ⊗ L) ≥ codim Z, where codim Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' dim Z) denotes the maximal codimension (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' maximal dimension) of a component of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' On the other hand, an inequality of Debarre-Hacon ([DH, Lemma 5(e)]) states that, if A is simple, χ(J (D) ⊗ L) ≤ χ(L) − dim Z − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Combining the two inequalities it follows that χ(L) ≥ codim Z + dim Z + 1 ≥ dim A + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' GV sheaves 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Recap on GV and M-regular sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We recall from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5) that a sheaf F on abelian variety A is said to be GV if ΦP−1(F∨) is a sheaf in cohomological degree g, so that ΦP−1(F∨)[g] coincides with the naive FMP transform T (F) = ΦP−1(F∨)[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore for a GV sheaf F the naive FMP transform will be simply referred to as the FMP transform of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10 Moreover a M-regular sheaf is a GV sheaf such that T (F) is torsion free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It is also worth to recall that to be GV (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' M-regular) is equivalent to the follow- ing condition on the cohomological support loci of F (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='7)): codimPic0A V i(F) ≥ i (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' codimPic0A V i(F) > i) for all i > 0 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' [PP5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A result of Popa and the author states that a M-regular sheaf is CGG ((iii) of Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is a particular case of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Indeed if F is GV then the source and target of the map edx of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8) simply coincide, and the map is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Moreover, if F is, in addition, M-regular, the FMP transform T (F) is torsion free and therefore all maps evx(U) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='10) are injective for all (non-empty) open subsets U of Pic0A and for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It also follows that a M-regular sheaf is ample by the mentioned result of Debarre ((i) of the Introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The purpose of this section is to extend this analysis from M-regular to GV sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Generation of GV sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The generation of GV sheaves follows in the same way (under the usual reducedness assumption) from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore we have the following result, partly known by work of Popa and the author in [PP2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a GV sheaf on an abelian variety A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Assume that all subsheaves ap- pearing in the torsion filtration of the FMP transform T (F) have reduced scheme-theoretic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then F is generated by the set of minimal irreducible components of the supports of such sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Alternatively, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 follows from Theorem D (or Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) combined with the following inversion result 10In the literature of GV sheaves, including papers of the author, this is usually denoted � F∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this paper we changed notation because we have been considering also non-GV sheaves, where the naive FMP transform does not coincide with T (F) = ΦP−1(F∨)[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 18 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' If F is a GV sheaf then also TA(F) is a GV sheaf (on Pic0A) and TPic0A(TA(F)) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Note that, in the the statement above, we have denoted TA(F) = RgΦA P−1(F∨) and, for a coherent sheaf G on Pic0A, TPic0A(G) = RgΦPic0A P−1 (G∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' However in the sequel the ambient abelian variety will be omitted from the notation, unless this will be cause of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By definition a sheaf F is a GV sheaf if TA(F) = ΦA P−1(F∨)[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, since the inverse of the FMP equivalence ΦA P−1 : D(A) → D(Pic0A) is the equivalence ΦPic0A P [g] : D(Pic0A) → D(A), it follows that ΦPic0A P (TA(F)) = F∨ = RHom(F, OA) Therefore, by Grothendieck duality ([Mu2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8)]), ΦPic0A P−1 (TA(F)∨)[g] = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3 (Unipotent vector bundles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Concerning the assumption of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, its necessity is shown by the well known example of unipotent vector bundles on abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' These are, by definition, vector bundles F admitting a filtration whose successive quotients are trivial line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Since a trivial bundle on an abelian variety is evidently a GV sheaf, any unipotent bundle is a GV sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' For such vector bundles H0(F ⊗ Pα) ̸= 0 if and only if α = ˆ0 (the identity point of Pic0A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore the only possible set of subvarieties generating F is {ˆ0}, and this happens if and only if F is trivial, because otherwise H0(F) < rk F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In conclusion, any non-trivial unipotent vector bundle is GV but not generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This is explained by the fact that the FMP transform T (F) is supported at zero, but the scheme theoretic support is non-reduced, unless the sheaf F is trivial ([Mu1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Other examples obtained from this one are e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' homogeneous vector bundles (direct sums of unipotent vector bundles twisted by line bundles parametrized by Pic0A), and, on a product of abelian varieties A = B × C, coherent sheaves of the form p∗ BG ⊠ p∗ CU, where G is GV on B and U is homogeneous on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4 (Any subset of subvarieties can be realized as a irredundant generating subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, combined with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2, can be used to construct examples showing that any subset of subvarieties is the irredundant generating set of some (locally free) sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Indeed let A be an abelian variety and let G be any coherent sheaf on Pic0A, such that all of its subsheaves have reduced scheme-theoretic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Twisting with a sufficiently high power of an ample line bundle L on Pic0A we have that V >0(G⊗L) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore, ΦPic0A P−1 ((G⊗L)∨) = RgΦPic0A P−1 ((G⊗L)∨)[−g] = TA(G ⊗ L)[−g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F := TA(G ⊗ L) From Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2 it follows that F is a (locally free) GV sheaf and TPic0A(F) = G ⊗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore the assertion follows from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, because it can be easily shown, with the help of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='4, that the generating set of minimal irreducible components of support of the sheaves appearing in the torsion filtration of G ⊗ L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' of G, is actually irredundant, and G is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' For the next example illustrating Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, it is useful to recall the compatibility property of the FMP functor with respect to homomorphisms of abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let pB : A → B be a surjective homomorphism and iB := b∗ : Pic0B → Pic0A the dual inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then ΦA P−1 A ◦ p∗ B = iB∗ ◦ p∗ΦB P−1 B ΦPic0A P−1 A iB∗ = p∗ B ◦ p∗ΦPic0B P−1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) (This can be deduced e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' from [Sch, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' GENERATION AND AMPLENESS OF COHERENT SHEAVES ON ABELIAN VARIETIES 19 Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5 (Sheaves admitting a Chen-Jiang decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' A significant class of GV sheaves where the assumption of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1(a) is always verified is given by coherent sheaves having the Chen-Jiang decomposition property ([LPS, §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' They were already mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By definition, these sheaves admit a (essentially canonical) decompo- sition F = � i (p∗ BiGi) ⊗ Pαi (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) where pBi : A → Bi are surjective homomorphisms, with connected kernel, of abelian varieties, Gi are M-regular sheaves (hence ample) on Bi, and αi ∈ Pic0A are torsion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It follows that such sheaves are generated and semiample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This class is important because it contains higher direct images of canonical sheaves of smooth complex projective varieties mapping to abelian varieties ([PPS, Theorem A]), as well as direct images of pluricanonical sheaves ([LPS, Theorem C]), and direct images of log-pluricanonical sheaves for klt pairs ([M1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1) that each summand in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) is a GV sheaf, and its FMP transform is T ((p∗ BiGi) ⊗ Pαi) = t∗ −αi(iB∗TB(Gi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In particular the various FMP transforms of the summands are scheme-theoretically supported on the translated abelian subvarieties Pic0Bi − αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore such sheaves, as well as F, satisfy the assumption of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (In fact, in the proof of the above quoted results about direct images of pluricanonical sheaves, this is an essential point, whose proof revolves around the minimal extension property of a positively curves metrics on such direct images, whose existence was shown by Cao-Paun [CP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' See also the survey [HPS] on this matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=') Therefore, as noted in [CLP, §9], the torsion filtration of the FMP transform: T0 ⊂ · · · ⊂ Td ⊂ T (F), where d = dim T (F), is given by Tk = � dim Bi≤k t∗ −αi(iB∗TB(Gi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (Note that if d = dim A, among the summands of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) there is one with Bi = A, hence Gi is already M-regular in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Its transform is the torsion free sheaf T (F)/Tg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=') 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Structure of GV sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In the previous example, applying the inverse FMP transform ΦPic0A P : D(Pic0A) → D(A) to the torsion filtration of T (F) one gets back the cofiltration F = Fd ։ · · · ։ F0 → 0 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) where Fk = � dim Bi≤k(p∗ BiGi) ⊗ Pαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' (In particular, if d = dim A(= g) the kernel of F ։ Fg−1 is M-regular hence ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=') This is a sort of weak version of the decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2) The following results says that, under the reducedness assumption of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, the structure of GV sheaves can be described by a weak analog of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let F be a GV sheaf on an abelian variety A, satisfying the assumption of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then F has a cofiltration F ϕ0 ։ F1 ϕ1 ։ · · · ϕs−1 ։ Fs ϕs → 0 such that: (a) The kernel of the surjection ϕ0 : F ϕ0 ։ F1 is either zero or ample, the latter case holding if and only if dim T (F) = dim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' 20 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='PARESCHI (b) For each i ≥ 1, the sheaf ker ϕi has, in turn, a cofiltration ker ϕi ϕ0i ։ F1i ϕ1i ։ · · · ϕs−1 i ։ Fsi i ϕsi i → 0 such that for all (j, i), there is a surjection fj i : p∗ BiFj i ⊗ Pαj i ։ ker ϕj i, where: pBj i : A → Bj i is a surjective homomorphism of abelian varieties with connected kernel, Fj i is an ample coherent sheaf on Bj i and αj i ∈ Pic0A Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We begin with the general, and well known to the experts (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' [Sch, Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2]), Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let p : A → B be a surjective homomorphism of abelian varieties, with connected kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' i : Pic0B ֒→ A the dual inclusion, τ a coherent sheaf on Pic0B and α ∈ Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Then TPic0A(t∗ −αi∗τ) = p∗TPic0B(τ) ⊗ Pα here tα denotes the translation by α on Pic0A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Recalling that, on an abelian variety C, the functor TC((·)) is defined as Rdim CΦP−1((·)∨) (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3) for the dualizing functor), the assertion of the Lemma follows from: (i) the second identity in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1), (ii) the fact that RHomPic0A(i∗τ, OPic0A) = i∗RHomPic0B(τ, OB)[dim A − dim B], and, (iii) the well known identity ΦPic0A P−1 (t∗ −α(·)) = ΦPic0A P−1 (·) ⊗ Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This proves Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Going back to the Theorem, the initial step of the cofiltration is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let τ be the torsion part of TA(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We apply the right exact contravariant functor TPic0A(·) to the exact sequence 0 → τ → TA(F) → TA(F)/τ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2 we get the exact sequence TPic0A(TA(F)/τ) −→ F ϕ0 −→ TPic0A(τ) → 0 We define F1 := TPic0A(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Since the sheaf TA(F)/τ is either zero or torsion free, the sheaf TPic0A(TA(F)/τ) is either zero or ample (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore the same is true for ker ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Hence we have accomplished the first step of the cofiltration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Next, let d = dim τ and 0 → τd−1 → τ → τ/τd−1 → 0 the first step of the torsion filtration of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Applying the functor TPic0A(·) we get the exact sequence TPic0A(τ/τd−1) → F1 ϕ1 → F2 := TPic0A(τd−1) → 0 We need to prove condition (b) on ker ϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' The sheaf σ := τ/τd−1 has pure dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Let us pick an irreducible component V of the support of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We apply the functor TPic0A(·) to the exact sequence 0 → K → σ → σ|V → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We get TPic0A(σ|V ) → TPic0A(σ) → TPic0A(K) → 0 By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1 if V spans Pic0A then the sheaf TPic0A(σ|V ) is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Otherwise V is a subvariety of a translate of an abelian subvariety C of Pic0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this case, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='2, the sheaf TPic0A(σ|V ) is of the form p∗TPic0B(G)⊗Pα, where B is the dual of C, and the sheaf TPic0B(G) is ample on B again by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' This settles the first step of the cofiltration on ker ϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Next, the sheaf K is either zero or of pure dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' Therefore we can apply the same procedure to the sheaf K, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' In this way, after a finite number of steps the cofiltration on ker ϕ1 is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' We now proceed inductively applying the same procedure to the sheaves τk for k ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=' □ References [ACGH] Arbarello, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQf3_qJ/content/2301.00779v1.pdf'} +page_content=', 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