* [119] D. Dwivedi, P. K. Yemula, and M. Pal, “Dynamopmu: A physics informed anomaly detection, clustering, and prediction method using nonlinear dynamics on micro pmu measurements,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 72, p. 1–9, 2023. [Online]. Available: http://dx.doi.org/10.1109/TIM.2023.3327481
  * [120] V. Le, X. Yao, C. Miller, and B.-H. Tsao, “Series dc arc fault detection based on ensemble machine learning,” _IEEE Transactions on Power Electronics_ , vol. 35, no. 8, pp. 7826–7839, 2020.
  * [121] J. Yan, Q. Li, and S. Duan, “A simplified current feature extraction and deployment method for dc series arc fault detection,” _IEEE Transactions on Industrial Electronics_ , vol. 71, no. 1, pp. 625–634, 2024.
  * [122] X. Han, D. Li, L. Huang, H. Huang, J. Yang, Y. Zhang, X. Wu, and Q. Lu, “Series arc fault detection method based on category recognition and artificial neural network,” _Electronics_ , vol. 9, no. 9, p. 1367, 2020.
  * [123] Y. Wang, D. Sheng, H. Hu, K. Han, J. Zhou, and L. Hou, “A novel series arc fault detection method based on mel-frequency cepstral coefficients and fully connected neural network,” _IEEE Access_ , vol. 10, pp. 97 983–97 994, 2022.
  * [124] Y. Wang, L. Hou, K. C. Paul, Y. Ban, C. Chen, and T. Zhao, “Arcnet: Series ac arc fault detection based on raw current and convolutional neural network,” _IEEE Transactions on Industrial Informatics_ , vol. 18, no. 1, pp. 77–86, 2021.
  * [125] R. Chu, P. Schweitzer, and R. Zhang, “Series ac arc fault detection method based on high-frequency coupling sensor and convolution neural network,” _Sensors_ , vol. 20, no. 17, p. 4910, 2020.
  * [126] E. Calderon-Mendoza, P. Schweitzer, and S. Weber, “Kalman filter and a fuzzy logic processor for series arcing fault detection in a home electrical network,” _International Journal of Electrical Power & Energy Systems_, vol. 107, pp. 251–263, 2019.
  * [127] W. Jiang, B. Liu, Z. Yang, H. Cai, X. Lin, and D. Xu, “Non-intrusive arc fault detection and localization method based on the mann–kendall test and current decomposition,” _Energies_ , vol. 16, no. 10, p. 3988, 2023.
  * [128] Y. Wang, F. Zhang, and S. Zhang, “A new methodology for identifying arc fault by sparse representation and neural network,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 67, no. 11, pp. 2526–2537, 2018.
  * [129] W. Luan, J. Lin, B. Liu, and B. Zhao, “Arc fault detection and identification via non-intrusive current disaggregation,” _Electric Power Systems Research_ , vol. 210, p. 108113, 2022.
  * [130] H. Zhao, J. Liu, and J. Lou, “Series arc fault detection based on current fluctuation and zero-current features,” _Electric Power Systems Research_ , vol. 202, p. 107626, 2022.
  * [131] J. Li, G. Zou, W. Wang, N. Shao, B. Han, and L. Wei, “Low-voltage series arc fault detection based on ecmc and vb-scn,” _Electric Power Systems Research_ , vol. 218, p. 109222, 2023.
  * [132] Y. Meng, Q. Yang, S. Chen, Q. Wang, and X. Li, “Multi-branch ac arc fault detection based on iceemdan and lightgbm algorithm,” _Electric Power Systems Research_ , vol. 220, p. 109286, 2023. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S037877962300175X
  * [133] A. Chabert, M. Bakkay, P. Schweitzer, S. Weber, and J. Andrea, “A transformer neural network for ac series arc-fault detection,” _Engineering Applications of Artificial Intelligence_ , vol. 125, p. 106651, 2023.
  * [134] C. Han, Z. Wang, A. Tang, H. Gao, and F. Guo, “Recognition method of ac series arc fault characteristics under complicated harmonic conditions,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 70, pp. 1–9, 2021.
  * [135] H. Lala and S. Karmakar, “Detection and experimental validation of high impedance arc fault in distribution system using empirical mode decomposition,” _IEEE Systems Journal_ , vol. 14, no. 3, pp. 3494–3505, 2020.
  * [136] Y. A. Mohammed Alsumaidaee, C. T. Yaw, S. P. Koh, S. K. Tiong, C. P. Chen, C. H. Tan, K. Ali, and Y. A. Balasubramaniam, “Detecting arcing faults in switchgear by using deep learning techniques,” _Applied Sciences_ , vol. 13, no. 7, p. 4617, 2023.
  * [137] Y. F. Wong, Y. A. Şekercioğlu, T. Drummond, and V. S. Wong, “Recent approaches to non-intrusive load monitoring techniques in residential settings,” in _2013 IEEE Computational Intelligence Applications in Smart Grid (CIASG)_. IEEE, 2013, pp. 73–79.
  * [138] I. P. Colo, C. S. Sueldo, M. De Paula, and G. G. Acosta, “Intelligent approach for the industrialization of deep learning solutions applied to fault detection,” _Expert Systems with Applications_ , p. 120959, 2023.
  * [139] H. Lala and S. Karmakar, “Detection and experimental validation of high impedance arc fault in distribution system using empirical mode decomposition,” _IEEE Systems Journal_ , vol. 14, no. 3, pp. 3494–3505, Sep. 2020.
  * [140] X. Chen, W. Gao, C. Hong, and Y. Tu, “A novel series arc fault detection method for photovoltaic system based on multi-input neural network,” _International Journal of Electrical Power & Energy Systems_, vol. 140, p. 108018, 2022. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S014206152200062X


# Cartan geometries on complex manifolds of algebraic dimension zero

Indranil Biswas, Sorin Dumitrescu and Benjamin McKay

###### Abstract

We prove that every compact complex manifold of algebraic dimension zero
bearing a holomorphic Cartan geometry of algebraic type must have infinite
fundamental group. This generalizes the main Theorem in [DM] where the same
result was proved for the special cases of holomorphic affine connections and
holomorphic conformal structures.

  *     * Keywords. Cartan geometry, almost homogeneous space, algebraic dimension, Killing vector field, semistability

2010 Mathematics Subject Classification. 53C56, 14J60, 32L05

[Français]

Titre. Géométries de Cartan sur les variétés complexes de dimension algébrique
nulle Résumé. Nous montrons que toute variété complexe compacte de dimension
algébrique nulle possédant une géométrie de Cartan holomorphe de type
algébrique doit avoir un groupe fondamental infini. Il s’agit d’une
généralisation du théorème principal de [DM] où le même résultat était
démontré dans le cas particulier des connexions affines holomorphes et des
structures conformes holomorphes.

###### Contents

  1. 1\. Introduction
  2. 2\. Cartan geometries
  3. 3\. Cartan geometries on manifolds of algebraic dimension zero
  4. 4\. Equivariant compactifications of abelian groups and stability
  5. References

##  1\. Introduction

Let $G$ be connected complex Lie group and $H\,\subset\,G$ a closed complex
Lie subgroup. A holomorphic Cartan geometry of type $(G,\,H)$ on a complex
manifold $X$ is an infinitesimal structure on $X$ modeled on $G/H$ (details
are in Section 2). Holomorphic affine connections, holomorphic projective
connections and holomorphic conformal structures are among the important
geometric examples of holomorphic Cartan geometries [Sh]. Contrary to the real
setting, compact complex manifolds admitting a holomorphic Cartan geometry are
rather rare. All known examples of compact complex manifolds admitting a
holomorphic Cartan geometry actually admit a flat holomorphic Cartan geometry.
It should be clarified that the model of the flat Cartan geometry (see
Definition 2.1) could be different from the given one. Indeed, compact complex
parallelizable manifolds [Wa] biholomorphic to quotients $G/\Gamma$, where $G$
is a complex semi-simple Lie group and $\Gamma\,\subset\,G$ a uniform lattice,
are known to admit holomorphic affine connections, but no flat holomorphic
affine connections [Du2]. Nevertheless, the following was conjectured in [DM]:
Compact complex simply connected manifolds admitting a holomorphic Cartan
geometry with model $(G,\,H)$ are biholomorphic to the model $G/H$. (See [DM,
Section 6].) This question is open even for the very special case of
holomorphic projective connections: accordingly to this conjecture, complex
compact simply connected manifolds bearing a holomorphic projective connection
should be complex projective spaces (endowed with the standard flat projective
geometry); this is not known even for smooth complex projective varieties
(except for the complex dimensions one, two [KO1, KO2] and three [PR1]). A
special case of this conjecture is that compact complex simply connected
manifolds $X$ do not admit any holomorphic affine connection. This is known to
be true for Kähler manifolds; indeed, in this case the rational Chern classes
of the holomorphic tangent bundle $TX$ must vanish, [At], and using Yau’s
theorem proving Calabi’s conjecture it can be shown that $X$ admits a finite
unramified cover which is a complex torus [IKO]. According to the conjecture,
the only simply connected compact complex manifold which admits a holomorphic
conformal structure is the smooth nondegenerate quadric hypersurface in
complex projective space. However, this is not known even for smooth complex
projective varieties (except for the complex dimensions one, two [KO3] and
three [PR2]). It is known that the smooth quadric hypersurface admits
precisely one holomorphic conformal structure (the standard one) [BMc,
Corollary 3]. Examples of non–Kähler complex compact manifolds of algebraic
dimension zero admitting holomorphic Riemannian metrics [Le, Gh] of constant
sectional curvature were constructed by Ghys in [Gh]. If the sectional
curvatures of a holomorphic Riemannian metric are constant, then the
associated Levi-Civita holomorphic affine connection is projectively flat.
They also admit the corresponding flat holomorphic conformal structure. The
examples in [Gh] are deformations of parallelizable manifolds covered by ${\rm
SL}(2,\mathbb{C})$. In the direction of the conjecture, the following was
proved in [DM, Theorem 1]: Compact complex simply connected manifolds with
algebraic dimension zero admit neither holomorphic affine connections, nor
holomorphic conformal structures. Generalizing the main theorem in [DM]
(Theorem 1), here we prove the following: Compact complex manifolds of
algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic
type have infinite fundamental group (Theorem 4.1).

##  2\. Cartan geometries

Let $X$ be a complex manifold, $G$ a complex connected Lie group and $H$ a
closed complex Lie subgroup of $G$. The Lie algebras of the Lie groups $G$ and
$H$ will be denoted by $\mathfrak{g}$ and $\mathfrak{h}$ respectively.

###### Definition 2.1

A holomorphic Cartan geometry $(P,\,\omega)$ on $X$ with model $(G,\,H)$ is a
holomorphic principal (right) $H$-bundle

$\pi\,:\,P\,\longrightarrow\,X$ (2.1)

endowed with a holomorphic $\mathfrak{g}$-valued 1-form $\omega$ satisfying
the following three conditions:

  1. 1.

$\omega_{p}\,:\,T_{p}P\,\longrightarrow\,\mathfrak{g}$ is a complex linear
isomorphism for all $p\,\in\,P$,

  2. 2.

the restriction of $\omega$ to every fiber of $\pi$ coincides with the left
invariant Maurer-Cartan form of $H$, and

  3. 3.

$(R_{h})^{*}\omega\,=\,{\rm Ad}(h)^{-1}\omega$, for all $h\,\in\,H$, where
$R_{h}$ is the right action of $h$ on $P$ and ${\rm Ad}$ is the adjoint
representation of $G$ on $\mathfrak{g}$.

###### Definition 2.2

The _kernel_ $N$ of a model $(G,\,H)$ is the largest subgroup of $H$ normal in
$G$; the kernel is clearly a closed complex normal subgroup, and we denote its
Lie algebra by $\mathfrak{n}$. A model $(G,\,H)$ is _effective_ if its kernel
is $N\,=\,\left\\{1\right\\}$. In other words, $(G,\,H)$ is effective if $H$
does not contain any nontrivial subgroup normal in $G$. A Cartan geometry is
_effective_ if its model is effective. Any Cartan geometry $(P,\,\omega)$ has
an _induced effective_ Cartan geometry $(\overline{P},\,\overline{\omega})$
where $\overline{P}\,=\,P/N$ and $\overline{\omega}$ is the unique
$\mathfrak{g}/\mathfrak{n}$-valued 1-form on $\overline{P}$ which pulls back,
via the natural quotient map $P\,\longrightarrow\,\overline{P}$, to the
$\mathfrak{g}/\mathfrak{n}$-valued one-form given by $\omega$.

###### Definition 2.3

If $\text{Ad}(H/N)\,\subset\,\text{GL}({\mathfrak{g}}/\mathfrak{n})$ is an
algebraic subgroup, then the holomorphic Cartan geometries with model
$(G,\,H)$ are said to be of algebraic type.

Holomorphic affine connections, holomorphic projective connections, and
holomorphic conformal structures are among the examples of effective
holomorphic Cartan geometries of algebraic type [Pe, Sh]. Notice that
Definition 2.1 implies, in particular, that the complex dimension of the
homogeneous model space $G/H$ coincides with the complex dimension $n$ of the
manifold $X$. Let us give some important examples. A holomorphic affine
connection corresponds to the model
(${\mathbb{C}}^{n}\rtimes\text{GL}(n,{\mathbb{C}}),\,\text{GL}(n,{\mathbb{C}})$)
of the complex affine space acted on by the group
${\mathbb{C}}^{n}\rtimes\text{GL}(n,{\mathbb{C}})$ of complex affine
transformations. A holomorphic projective connection is a Cartan geometry with
model $(\text{PGL}(n+1,{\mathbb{C}}),\,Q)$, where
$\text{PGL}(n+1,{\mathbb{C}})$ is the complex projective group acting on the
complex projective space ${\mathbb{C}}P^{n}$ and where
$Q\,\subset\,\text{PGL}(n+1,{\mathbb{C}})$ is the maximal parabolic subgroup
that fixes a given point in ${\mathbb{C}}P^{n}$. Holomorphic conformal
structures are modeled on the quadric
$z_{0}^{2}+z_{1}^{2}+\ldots+z_{n+1}^{2}\,=\,0$ in ${\mathbb{C}}P^{n+1}$. Here
$G$ is the subgroup ${\rm PO}(n+2,\mathbb{C})$ of the complex projective group
$\text{PGL}(n+2,{\mathbb{C}})$ preserving the above quadric, while $H$ is the
stabilizer of a given point in the quadric. For more details about those
Cartan geometries and for many other examples the reader may consult [Sh].

###### Definition 2.4

Let $X$ be equipped with a holomorphic Cartan geometry $(P,\,\omega)$ of type
$(G,\,H)$. A (local) biholomorphism between two open subsets $U$ and $V$ in
$X$ is a (local) automorphism of the Cartan geometry $(P,\,\omega)$ if it
lifts to a holomorphic principal $H$-bundle isomorphism between $\pi^{-1}(U)$
and $\pi^{-1}(V)$ that preserves the form $\omega$; here $\pi$ is the
projection in (2.1). A (local) holomorphic vector field on $X$ is a (local)
Killing field of $(P,\,\omega)$ if its (local) flow acts by (local)
automorphisms of $(P,\,\omega)$. The Cartan geometry $(P,\,\omega)$ is called
locally homogeneous on an open subset $U\,\subset\,X$ if the tangent space
$T_{u}X$ is spanned by local Killing fields of $(P,\,\omega)$ for every point
$u\,\in\,U$.

The curvature of $(P,\,\omega)$ is defined as a $2$-form $\Omega$ on $P$, with
values in $\mathfrak{g}$, given by the following formula

$\Omega(Y,\,Z)\,=\,d\omega(Y,\,Z)-[\omega(Y),\,\omega(Z)]_{\mathfrak{g}}$
(2.2)

for all local holomorphic vector fields $Y,\,Z$ on $P$. When at least one of
$Y$ and $Z$ is a vertical vector field (meaning lies in the kernel of the
differential $d\pi$ of $\pi$ in (2.1)), then $\Omega(Y,\,Z)\,=\,0$ (see [Sh,
5.3.10]). From this it follows that $\Omega$ is basic and it descends on $X$.
More precisely we have the following classical interpretation of $\Omega$. Let

$P_{G}\,:=\,P\times^{H}G\,\stackrel{{\scriptstyle\pi_{G}}}{{\longrightarrow}}\,X$
(2.3)

be the holomorphic principal $G$–bundle over $X$ obtained by extending the
structure group of the holomorphic principal $H$–bundle $P$ using the
inclusion of $H$ in $G$. Recall that $P_{G}$ is the quotient of $P\times G$
where two points $(p_{1},\,g_{1}),\,(p_{2},\,g_{2})\,\in\,P\times G$ are
identified if there is an element $h\,\in\,H$ such that $p_{2}\,=\,p_{1}h$ and
$g_{2}\,=\,h^{-1}g_{1}$. The projection $\pi_{G}$ in (2.3) is induced by the
map $P\times G\,\longrightarrow\,X$, $(p,\,g)\,\longmapsto\,\pi(p)$. The
action of $G$ on $P_{G}$ is induced by the action of $G$ on $P\times G$ given
by the right–translation action of $G$ on itself. Consider the adjoint vector
bundle ${\rm ad}(P_{G})\,=\,P_{G}\times^{G}{\mathfrak{g}}$ of $P_{G}$. We
recall that ${\rm ad}(P_{G})$ is the quotient of $P_{G}\times\mathfrak{g}$
where two points $(c_{1},\,v_{1})$ and $(c_{2},\,v_{2})$ are identified if
there is an element $g\,\in\,G$ such that $c_{2}\,=\,c_{1}g$ and $v_{2}$ is
the image of $v_{1}$ through the automorphism of the Lie algebra
$\mathfrak{g}$ defined by automorphism of $G$ given by
$z\,\longmapsto\,g^{-1}zg$. Starting with $(P,\,\omega)$ there is a natural
Ehresmann connection on the principal bundle $P_{G}$ given by the following
$\mathfrak{g}$-valued form on $P\times G$:

$\widetilde{\omega}(p,g)\,=\,Ad(g^{-1})\pi_{1}^{*}(\omega)+\pi_{2}^{*}(\omega_{G})\,,$

where $\pi_{1}$ and $\pi_{2}$ are the projections of $P\times G$ on the first
and the second factor respectively, while $\omega_{G}$ is the left-invariant
Maurer-Cartan form on $G$. The above form $\widetilde{\omega}$ is invariant by
the previous $H$-action on $P\times G$ and it vanishes when restricted to the
fibers of the fibration $P\times G\,\longrightarrow\,P_{G}$. This implies that
$\widetilde{\omega}$ is basic and it descends on $P_{G}$. Let us denote also
by $\widetilde{\omega}$ this one-form on $P_{G}$ with values in
$\mathfrak{g}$: it is an Ehresmann connection on the principal bundle $P_{G}$
(see [BD], Proposition 3.4). Recall that the curvature of the connection
$\widetilde{\omega}$ on the principal bundle $P_{G}$ is a tensor of the
following type:

$\text{Curv}(\widetilde{\omega})\,\in\,H^{0}(X,\,\text{ad}(P_{G})\otimes\Omega^{2}_{X})\,.$
(2.4)

Moreover, the curvature $\Omega$ of $\omega$ in (2.2) vanishes if and only if
$\text{Curv}(\widetilde{\omega})$ vanishes. If $\Omega\,=\,0$, then $\omega$
produces local isomorphisms of $X$ to the homogeneous space $G/H$. This way we
get a developing map from the universal cover of $X$ to $G/H$, which is a
local biholomorphism (see for example [Sh] or [BD, p. 9]).

##  3\. Cartan geometries on manifolds of algebraic dimension zero

As a preparation for the proof of Theorem 4.1, we first prove the following
Lemma 3.1 which is an adaptation, for Cartan geometries, of [DM, Proposition
2] on rigid geometric structures. Recall that a compact complex manifold is of
algebraic dimension zero if it does not admit any nonconstant meromorphic
function [Ue]. These manifolds are far from being algebraic. Indeed, a
manifold is bimeromorphic with an algebraic manifold if and only if its field
of meromorphic functions separates points [Moi, Ue].

###### Lemma 3.1

Let $X$ be a compact complex simply connected manifold of algebraic dimension
zero endowed with a holomorphic effective Cartan geometry $(P,\,\omega)$. Then
the connected component of the automorphism group of the identity of
$(P,\,\omega)$ is a connected complex abelian Lie group $L$ acting on $X$ with
an open dense orbit. The open dense orbit is the complement of an
anticanonical divisor. Moreover, the natural map from $L$ to this open dense
orbit, constructed by fixing a point of this orbit, is a biholomorphism.
Furthermore, $L$ is covered by $({\mathbb{C}}^{*})^{n}$.

  * Proof.

By Theorem 1.2 in [Du] (see also [Du1]) the Cartan geometry $(P,\,\omega)$ is
locally homogeneous on an open dense subset in $X$. Recall that on simply
connected manifolds, local Killing vector fields of Cartan geometries extend
to all of the manifold $X$ [Am, No, Gr, Pe]. Therefore, it follows that there
exists a Lie algebra $\mathfrak{a}$ of globally defined Killing vector fields
on $X$ which span $TX$ at the generic point. Let us fix a basis
$\\{X_{1},\,\cdots,\,X_{k}\\}$ of $\mathfrak{a}$ and consider the generalized
Cartan geometry $(P,\omega,\mathfrak{a})$ (in the sense of Definition 4.11 in
[Pe]) which is a juxtaposition of $(P,\omega)$ with the family of vector
fields $\\{X_{1},\,\cdots,\,X_{k}\\}$. The proof of Theorem 1.2 in [Du]
applies to the generalized geometry $(P,\,\omega,\,\mathfrak{a})$ which must
also be locally homogeneous on an open dense set. Notice that local Killing
vector fields of $(P,\omega,\mathfrak{a})$ are restrictions of elements in
$\mathfrak{a}$ which commute with the vector fields $X_{i}$. It follows that
local vector fields of $(P,\,\omega,\,\mathfrak{a})$ are elements in the
centralizer $\mathfrak{a^{\prime}}$ of $\mathfrak{a}$, and
$\mathfrak{a^{\prime}}$ is transitive on an open dense subset. Since
$\mathfrak{a^{\prime}}$ is commutative, the action is simply transitive on the
open dense subset (note that $\mathfrak{a^{\prime}}$ has the same dimension as
that of $X$). A basis of $\mathfrak{a^{\prime}}$ is a family of commuting
vector fields spanning $TX$ at the generic point. Since the vector fields
$X_{i}$ commute with elements in this basis, they must be a linear combination
of elements in $\mathfrak{a^{\prime}}$ with constant coefficients (on the open
dense subset and hence on all of $X$). In particular, we have
$X_{i}\,\in\,\mathfrak{a^{\prime}}$, which implies that
$\mathfrak{a}\,=\,\mathfrak{a^{\prime}}$ is an abelian algebra; this also
implies that $k$ is the complex dimension of $X$. In particular, the
corresponding connected Lie group $L$ (the connected component of the identity
of the automorphism group of $(P,\,\omega)$) is an abelian group of the same
dimension as that of $X$ acting with an open dense orbit. This open dense
orbit is the complement of the vanishing set $S$ of the holomorphic section
$X_{1}\bigwedge\cdots\bigwedge X_{k}$ of the anticanonical bundle. Since $L$
is abelian, any $\ell_{0}\,\in\,L$ stabilizing some point $x_{0}$ of the open
orbit stabilizes all points $\ell x_{0}$ for $\ell\,\in\,L$:
$\ell_{0}x_{0}\,=\,x_{0}$, so $\ell_{0}\ell
x_{0}\,=\,\ell\ell_{0}x_{0}\,=\,\ell x_{0}$. By density of the open dense
orbit, $\ell_{0}$ stabilizes all points of $X$. So the stabilizer of any point
in the open orbit is trivial, i.e., $X$ is an $L$-equivariant compactification
of $L$. The following arguments which prove that $L$ is covered by
$({\mathbb{C}}^{*})^{n}$ are borrowed from [DM] (Proposition 2). Let us define
nontrivial meromorphic 1-forms $\omega_{i}$ on $X$ by

$\omega_{i}(\xi)\,=\,\frac{X_{1}\wedge X_{2}\wedge\cdots\wedge
X_{i-1}\wedge\xi\wedge X_{i+1}\wedge\cdots\wedge X_{n}}{X_{1}\wedge
X_{2}\wedge\cdots\wedge X_{n}}.$

Notice that the previous meromorphic 1-forms $\omega_{i}$ are $L$-invariant
and are holomorphic when restricted to $X\setminus S$. Since $L$ is abelian
the classical Lie-Cartan formula implies that the forms $\omega_{i}$ are
closed (on the open dense orbit $X\setminus S$ and hence on all of $X$). The
indefinite integral of any nonzero closed meromorphic 1-form is a nonconstant
meromorphic function on some covering space of the complement of the simple
poles of the 1-form. Since $X$ is simply connected and does not admit
nontrivial meromorphic functions, every nonzero meromorphic 1-form on $X$ must
have a simple pole on some component of the complement $S$ of the open dense
orbit of $L$. Define

$\Delta\,=\,H_{1}(X\setminus S,\,\mathbb{Z})\,=\,H_{1}(L,\,\mathbb{Z})\,.$

Then $\Delta$ identifies with a discrete subgroup in the Lie algebra
$\mathfrak{a}$ of $L$ and $L$ is isomorphic to the quotient
$\mathfrak{a}/\Delta$. The restriction to $X\setminus S$ of the $L$-invariant
meromorphic $1$-forms $\omega_{i}$ identify with a basis of
$\mathfrak{a}^{*}$. Pair any
$\gamma\,\in\,\Delta,\,\omega\,\in\,\mathfrak{a}^{*}$ as follows:

$\gamma,\,\omega\,\longmapsto\,\int_{\gamma}\omega\,\in\,\mathbb{C}{}\,.$
(3.1)

Assume that for a given $\omega$ this pairing vanishes for every $\gamma$.
Then $\omega$ integrates around each component of $S$ to define a meromorphic
function on $X$. This meromorphic function must be constant and therefore we
have $\omega\,=\,0$. Consequently, the pairing in (3.1) is nondegenerate, and
hence $\Delta\,\subset\,\mathfrak{a}$ spans $\mathfrak{a}$ over $\mathbb{C}$.
This implies that $L\,=\,\mathfrak{a}/\Delta$ is covered by
$({\mathbb{C}}^{*})^{n}$. $\Box$

Recall that a large and important class of non-Kähler smooth equivariant
compactifications of complex abelian Lie groups (namely the so-called LMBV
manifolds) was constructed in [BM, Mer] (see also [PUV]).

##  4\. Equivariant compactifications of abelian groups and stability

In this section we prove the main result of the article (Theorem 4.1). Let us
first recall the notion of slope stability for holomorphic vector bundles
complex manifolds which will be used in the sequel. Let $X$ be a compact
complex manifold of complex dimension $n$. Let us fix a Gauduchon metric $g$
on it. Recall that the $(1,1)$-form $\alpha_{g}$ associated to a Gauduchon
metric $g$ satisfies the equation

$\partial\overline{\partial}\alpha_{g}^{n-1}\,=\,0\,.$

By a result of Gauduchon, any hermitian metric on $X$ is conformally
equivalent to a Gauduchon metric (satisfying the above equation) which is
uniquely defined up to a positive constant [Ga]. The degree of a torsionfree
coherent analytic sheaf $F$ on $X$ is defined to be

$\text{deg}(F)\,:=\,\frac{\sqrt{-1}}{2\pi}\int_{M}K(\det
F)\wedge\alpha_{g}^{n-1}\,\in\,{\mathbb{R}}\,,$ (4.1)

where $\det F$ is the determinant line bundle for $F$ [Ko, Ch. V, § 6] (or
Definition 1.34 in [Br]) and $K$ is the curvature for a hermitian connection
on $\det F$ compatible with $\overline{\partial}_{\det F}$. This degree is
independent of the hermitian metric since any two such curvature forms differ
by a $\partial\overline{\partial}$-exact $2$–form on $X$:

$\int_{X}(\partial\overline{\partial}u)\wedge\alpha_{g}^{n-1}\,=\,-\int_{X}u\wedge\partial\overline{\partial}\alpha_{g}^{n-1}\,=\,0\,.$

This notion of degree is independent of the choice of the hermitian
connection, but depends on the Gauduchon metric (it is not a topological
invariant). In the particular case where $d\alpha_{g}=0$ we find the classical
degree for Kähler manifolds (and in this case the degree is a topological
invariant). Define

$\mu(F)\,:=\,\frac{\text{deg}(F)}{\text{rank}(F)}\,\in\,{\mathbb{R}}\,,$

which is called the slope of $F$ (with respect to $g$). A torsionfree coherent
analytic sheaf $F$ on $X$ is called stable (respectively, semi-stable) if for
every coherent analytic subsheaf $W\,\subset\,F$ such the
$\text{rank}(W)\,\in\,[1\,,\text{rank}(F)-1]$, the inequality
$\mu(W)\,<\,\mu(F)$ (respectively, $\mu(W)\,\leq\,\mu(F)$) holds (see [LT, p.
44, Definition 1.4.3], [Ko, Ch. V, § 7]). The rest of the section is devoted
to the proof of the main theorem of the article:

###### Theorem 4.1

Compact complex manifolds with algebraic dimension zero bearing a holomorphic
Cartan geometry of algebraic type have infinite fundamental group.

  * Proof.

Assume, by contradiction, that a complex manifold $X$ of algebraic dimension
zero admits a holomorphic Cartan geometry $(P,\,\omega)$ and has finite
fundamental group. Considering the universal cover of $X$ and pulling back the
Cartan geometry $(P,\,\omega)$ to it, we would assume that $X$ is simply
connected. Replace the Cartan geometry by the induced effective Cartan
geometry, so we can assume that it is effective. By Lemma 3.1, the maximal
connected subgroup of the automorphism group of $(P,\,\omega)$ is a complex
connected abelian Lie group $L$ acting on $X$ with an open dense orbit (it is
the complement of an anticanonical divisor on $X$). The complement of this
anticanonical divisor is biholomorphic to $L$ (see Lemma 3.1 again). Let us
now prove the following lemma describing the geometry of simply connected
smooth equivariant compactifications of complex abelian Lie groups.

###### Lemma 4.2

Let $X$ be a simply connected smooth equivariant compactification of a complex
abelian Lie group $L$. Then the following four hold:

    1. 1.

There is no nontrivial holomorphic $k$-form on $X$, for $k\,\geq\,1$.

    2. 2.

Any holomorphic line bundle $E_{1}$ over $X$ admitting a holomorphic
connection is holomorphically trivial.

    3. 3.

If $E$ is a holomorphic vector bundle over $X$ admitting a holomorphic
connection, then $E$ is semi-stable of degree zero.

    4. 4.

If $F$ is a coherent analytic subsheaf of the holomorphic tangent bundle $TX$,
such that $TX/F$ is nonzero and torsionfree, then the quotient ${\rm
deg}(TX/F)\,>\,0$.

    * Proof of Lemma 4.2.

(1) Let us first consider the special case of $k\,=\,1$. Let $\eta$ be a
holomorphic one-form on $X$. Let $\\{X_{1},\,\cdots,\,X_{n}\\}$ be a family of
commuting holomorphic vector fields on $X$ which span $TX$ at the generic
point. By the Lie-Cartan formula we get that for any
$i,\,j\,\in\,\\{1,\,\cdots,\,n\\}$:

$d\eta(X_{i},\,X_{j})\,=\,X_{i}\cdot\eta(X_{j})-X_{j}\cdot\eta(X_{i})-\eta([X_{i},X_{j}])\,=\,0\,.$
(4.2)

Since any holomorphic function on $X$ is a constant one and
$[X_{i},\,X_{j}]\,=\,0$ for all $i,\,j$, it follows from (4.2) that the form
$\eta$ is closed. Now since $X$ is simply connected, $\eta$ coincides with the
differential of a global holomorphic function $u$ on $X$. Since $X$ is
compact, $u$ is constant and hence $\eta\,=\,0$. Now consider a $k$-form
$\eta$ on $X$ with $k\,>\,1$. For any
$i_{1},\,i_{2},\,\cdots,\,i_{k-1}\,\in\,\\{1,\,\cdots,\,n\\}$, the one-form on
$X$

$v\,\longmapsto\,\eta(X{{}_{i_{1}}},\,\cdots,\,X_{i_{k-1}},\,v)$

vanishes by the previous proof. Consequently, $\eta$ must vanish. (2) Let
$\nabla$ be a holomorphic connection on a holomorphic line bundle $E_{1}$. The
curvature ${\rm Curv}(\nabla)$ of $\nabla$ is a holomorphic two-form on $X$.
Hence ${\rm Curv}(\nabla)\,=\,0$ by (1). So $\nabla$ is flat. This implies
that $E_{1}$ is the trivial holomorphic line bundle, because $X$ is simply
connected. (3) Let $\nabla$ be a holomorphic connection of $E$. The
determinant line bundle ${\det}(E)$ of $E$ has a holomorphic connection
induced by $\nabla$. So ${\det}(E)$ is holomorphically trivial by (2). Hence
the degree of $E$ is zero (see (4.1)). Assume, by contradiction, that $E$ is
not semi-stable. Then there exists a maximal semi-stable coherent analytic
subsheaf $W$ (the smallest nonzero term of the Harder–Narasimhan filtration of
$E$) [HL, pp. 14–15, Theorem 1.3.1]. The maximal semistable subsheaf $W$ has
the following property:

$H^{0}(X,\,\text{Hom}(W,\,E/W))\,=\,0\,.$ (4.3)

Indeed, this follows immediately from the facts that the slopes of the graded
pieces for the Harder–Narasimhan filtration of $E/W$ are strictly less than
$\mu(W)$, and there is no nonzero homomorphism from a semistable sheaf to
another semistable sheaf of strictly smaller slope. Now consider the second
fundamental form

$S\,:\,W\,\longrightarrow\,(E/W)\otimes\Omega^{1}_{X}$

of $W$ for the holomorphic connection $\nabla$, which associates to any
locally defined holomorphic section $s$ of $W$ the projection, to
$(E/W)\otimes\Omega^{1}_{X}$, of $\nabla(s)$ (which is a section of
$E\otimes\Omega^{1}_{X}$). For any global holomorphic vector field $X_{i}$ on
$X$ we have the homomorphism $W\,\longrightarrow\,E/W$ defined by
$w\,\longmapsto\,S(w)(X_{i})$. From (4.3) we know that this homomorphism
vanishes identically. Since the vector fields $X_{i}$ span $TX$ at the generic
point, we have $S\,=\,0$. Therefore, the connection $\nabla$ preserves $W$. In
other words, $\nabla$ induces a holomorphic connection on $W$. This implies
that the coherent analytic sheaf $W$ is locally free (see Lemma 4.5 in [BD]).
Since $W$ admits a holomorphic connection, from (2) we have ${\rm
deg}(W)\,=\,0$. This contradicts the assumption that $W$ destabilizes $E$.
Consequently, $E$ is semistable. (4) Take $F\,\subset\,TX$ as in (4). Let
$TX/F$ be of rank $r\geq 1$. Consider the projection
$q\,:\,TX\,\longrightarrow\,TX/F$ and the associated homomorphism

$\widehat{q}\,:\,\bigwedge\nolimits^{r}TX\,\longrightarrow\,{\det}(TX/F)\,.$
(4.4)

Note that $q$ produces a homomorphism
$\bigwedge^{r}TX\,\longrightarrow\,{\det}(TX/F)$ over the subset
$U\,\subset\,X$ where the torsionfree coherent analytic sheaf $TX/F$ is
locally free; now this homomorphism extends to entire $X$ by Hartogs’ theorem,
because ${\det}(TX/F)$ is locally free and the complex codimension of
$X\setminus U$ is at least two. Choose
$i_{1},\,\cdots,\,i_{r}\,\in\,\\{1,\,2,\,\cdots,\,n\\}$ for which
$\\{q(X_{i_{1}}),\,\cdots,\,q(X_{i_{r}})\\}$ generate $TX/F$ at the generic
point. Then $q(X_{i_{1}})\bigwedge\cdots\bigwedge q(X_{i_{r}})$ defines a
holomorphic section of ${\det}(TX/F)$ which is not identically zero; we shall
denote this section of ${\det}(TX/F)$ by $\sigma$. So we have ${\rm
deg}(TX/F)\,\geq\,0$, because ${\det}(TX/F)$ admits a nonzero holomorphic
section namely $\sigma$. Assume by contradiction ${\rm deg}(TX/F)\,=\,0$. Then
the divisor ${\rm div}(\sigma)\,\subset\,X$ of the above section $\sigma$ is
the zero divisor. Indeed, the degree of $TX/F$ is the volume of ${\rm
div}(\sigma)$ with respect to the given Gauduchon metric (see [Br, Proposition
5.23]). Consequently, the line bundle ${\det}(TX/F)$ is holomorphically
trivializable. Once a holomorphic trivialization of ${\det}(TX/F)$ is fixed,
the homomorphism $\widehat{q}$ in (4.4) becomes a holomorphic $1$-form on $X$
which is not identically zero. This is in contradiction with (1). This proves
that ${\rm deg}(TX/F)\,>\,0$. $\Box$

Continuing with the proof of Theorem 4.1, let us now consider the curvature of
the Cartan geometry $(P,\,\omega)$

$\text{Curv}(\widetilde{\omega})\,\in\,H^{0}(X,\,\text{ad}(P_{G})\otimes\Omega^{2}_{X})$

constructed in (2.4). When contracted with any global holomorphic vector field
$X_{i}$ on $X$, this curvature form $\text{Curv}(\widetilde{\omega})$ produces
a holomorphic homomorphism

$\varphi(X_{i})\,:\,TX\,\longrightarrow\,\text{ad}(P_{G})\,,\ \
v\,\longmapsto\,\text{Curv}(\widetilde{\omega})(X_{i},\,v)\,.$ (4.5)

The holomorphic connection $\widetilde{\omega}$ on $P_{G}$ induces a
holomorphic connection on the vector bundle $\text{ad}{(P_{G})}$ associated to
the principal $G$–bundle $P_{G}$. Hence from Lemma 4.2(3) we know that the
vector bundle $\text{ad}{(P_{G})}$ is semistable of degree zero. Consequently,
from Lemma 4.2(3) it follows immediately that

$H^{0}(X,\,\text{Hom}(TX,\,\text{ad}(P_{G}))\,=\,0\,;$

indeed, the image of a nonzero homomorphism
$TX\,\longrightarrow\,\text{ad}(P_{G})$ would contradict the semistability
condition for $\text{ad}(P_{G})$. In particular, the homomorphism
$\varphi(X_{i})$ in (4.5) vanishes identically. Since the vector fields
$X_{i}$ generate the tangent bundle of $X$ over a nonempty open subset of $X$,
from the vanishing of the homomorphisms $\varphi(X_{i})$ we conclude that
$\text{Curv}(\widetilde{\omega})\,=\,0$. Consequently, the Cartan geometry
$(P,\,\omega)$ is flat. Since $X$ is simply connected, the developing map of
the Cartan geometry is a holomorphic biholomorphism between $X$ and $G/H$. But
$G/H$ is algebraic and $X$ has algebraic dimension zero: a contradiction. This
completes the proof of Theorem 4.1. $\Box$

Recall that a generalized Cartan geometry in the sense of [BD] is given by a
pair $(P,\,\omega)$ satisfying conditions (2) and (3) in Definition 2.1, while
the homomorphism in Definition 2.1(1) is not required to be an isomorphism
anymore. Consequently, $X$ and $G/H$ do not necessarily have the same
dimension. The proof of Theorem 4.1 shows that on simply connected smooth
equivariant compactifications $X$ of complex abelian Lie groups, all
(generalized) holomorphic Cartan geometries in the sense of [BD] are flat
(notice that the condition on the algebraic dimension of $X$ is not needed for
this part of the proof; neither the condition on the algebraicity of $G/H$).
In conclusion all (generalized) holomorphic Cartan geometries on $X$ are given
by holomorphic maps $X\,\longrightarrow\,G/H$.

## Acknowledgements

The authors are very grateful to Laurent Meersseman for many enlightening
discussions on the subject. We thank the referee for helpful comments.
Indranil Biswas is partially supported by a J. C. Bose Fellowship. Sorin
Dumitrescu wishes to thank T.I.F.R. Mumbai for hospitality. This research was
supported in part by the International Centre for Theoretical Sciences (ICTS)
during a visit for participating in the program - Analytic and Algebraic
Geometry (Code: ICTS/aag2018/03).

## References

  * [Am] A. M. Amores, _Vector fields of a finite type $G$-structure_,J. Diff. Geom. 14 (1980), 1–6. MR-0577874
  * [At] M. F. Atiyah, _Complex analytic connections in fibre bundles_ , Trans. Amer. Math. Soc. 85 (1957), 181–207. MR-MR0086359
  * [BD] I. Biswas and S. Dumitrescu, _Generalized holomorphic Cartan geometries_ , Eur. J. Math. (2019), https://doi.org/10.1007/s40879-019-00327-6.
  * [BMc] I. Biswas and B. McKay, _Holomorphic Cartan geometries and rational curves_ , Complex Manifolds 3 (2016), 145–168. MR-3489612
  * [BM] F. Bosio and L. Meersseman, _Real quadrics in ${\mathbb{C}}^{n}$, complex manifolds and convex polytops_, Acta Math. 197 (2006), no. 1, 53–127. MR-2285318
  * [Br] V. Brînzanescu, _Holomorphic vector bundles over compact complex surfaces_ , Lectures Notes in Mathematics, 1624. Springer, Berlin Heidelberg, 1996. x+170 pp. MR-1439504
  * [Du] S. Dumitrescu, _Killing fields of holomorphic Cartan geometries_ , Monatsh. Math. 161 (2010), no. 2, 145–154. MR-2680003
  * [Du1] S. Dumitrescu, _Meromorphic almost rigid geometric structures_ , Geometry, Rigidity and Group Actions. Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, 32–58. MR-2807828
  * [Du2] S. Dumitrescu, _Une caractérisation des variétés complexes compactes parallélisables admettant des structures affines_ , C. R. Math. Acad. Sci. Paris 347 (2009), no. 19-20, 1183–1187. MR-2567000
  * [DM] S. Dumitrescu and B. McKay, _Affine connections on complex manifolds of algebraic dimension zero_ , Moscow Math. Journal 16 (2016), no. 4, 675–689. MR-3598502
  * [Ga] P. Gauduchon, _Sur la $1$-forme de torsion d’une variété hermitienne compacte_, Math. Ann., 267 (1984), no. 4, 495–518. MR-0742896
  * [Gh] E. Ghys, _Déformations des structures complexes sur les espaces homogènes de ${\rm SL}(2,\mathbb{C})$_, J. Reine Angew. Math. 468 (1995), 113–138. MR-1361788
  * [Gr] M. Gromov, _Rigid Transformations Groups_ , Géométrie Différentielle (Paris, 1986), 65–139. Travaux en Cours, 33, Hermann, Paris, 1988. MR-0955852
  * [HL] D. Huybrechts and M. Lehn, _The geometry of moduli spaces of sheaves_ , Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, 1997. xiv+269 pp. MR-1450870
  * [IKO] M. Inoue, S. Kobayashi and T. Ochiai, _Holomorphic affine connections on compact complex surfaces_ , J. Fac. Sci. Univ. Tokyo 27 (1980), no. 2, 247–264. MR-0586449
  * [Ko] S. Kobayashi, _Differential geometry of complex vector bundles_ , Publications of the Mathematical Society of Japan, 15. Princeton University Press, Princeton, NJ, 1987. xii+305 pp. MR-0909698
  * [KO1] S. Kobayashi and T. Ochiai, _Holomorphic projective structures on compact complex surfaces_ , Math. Ann. 249 (1980), no. 1, 75–94. MR-0575449
  * [KO2] S. Kobayashi and T. Ochiai, _Holomorphic projective structures on compact complex surfaces. II_ , Math. Ann. 255 (1981), no. 4, 519–521. MR-0618182
  * [KO3] S. Kobayashi and T. Ochiai, _Holomorphic structures modeled after hyperquadrics_ , Tôhoku Math. J. 34 (1982), no. 4, 587–629. MR-0685426
  * [Le] C. LeBrun, _$\mathcal{H}$ -spaces with cosmological constant_, Proc. Royal Soc. London Ser. A 380 (1982), no. 1778, 171–185. MR-0652038
  * [LT] M. Lübke and A. Teleman, _The Kobayashi-Hitchin correspondence_ , World Scientific Publishing Co., Inc., River Edge, NJ, 1995. x+254 pp. MR-1370660
  * [Mer] L. Meersseman, _A new geometric construction of compact complex manifolds in any dimension_ , Math. Ann. 317 (2000), no. 1, 79–115. MR-1760670
  * [Moi] B. Moishezon, _On $n$ dimensional compact varieties with $n$ independent meromorphic functions_, Amer. Math. Soc. Transl. (2) 63 (1967), 51–177. MR-0216522
  * [No] K. Nomizu, _On local and global existence of Killing vector fields_ , Ann. of Math. (2) 72 (1960), 105–120. MR-0119172
  * [Pe] V. Pecastaing, _On two theorems about local automorphisms of geometric structures_ , Ann. Inst. Fourier 66 (2016), no. 1, 175–208. MR-3477874
  * [PUV] T. Panov, Y. Ustinovsky and M. Verbitsky, _Complex geometry of moment angles manifolds_ , Math. Z. 284 (2016), no. 1-2, 309–333. MR-3545497
  * [PR1] J. Priska and I. Radloff, _Threefolds with holomorphic normal projective connections_ , Math. Ann. 329 (2004), no. 3, 379–400. MR-2127983
  * [PR2] J. Priska and I. Radloff, _Projective threefolds with holomorphic conformal structure_ , Internat. J. Math. 16 (2005), no. 6, 595–607. MR-2153485
  * [Sh] R. Sharpe, _Differential Geometry. Cartan’s generalization of Klein’s Erlangen program_ , Graduate Texts in Mathematics, 166. Springer-Verlag, New York, 1997. xx+421 pp. MR-1453120
  * [Ue] K. Ueno, _Classification theory of algebraic varieties and compact complex spaces_ , Lecture Notes in Mathematics 439. Springer-Verlag, Berlin-New York, 1975. xix+278 pp. MR-0506253
  * [Wa] H.-C. Wang, _Complex parallisable manifolds_ , Proc. Amer. Math. Soc. 5 (1954), 771–776. MR-0074064