Holomorphic $$\text {GL}_2({\mathbb C})$$-geometry on compact complex manifolds

Biswas, Indranil; Dumitrescu, Sorin

doi:10.1007/s00229-020-01252-9

Cited by 3 publications

(3 citation statements)

References 36 publications

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“…They are not necessarily flat. It was proved in [BD6] that generic (translation invariant) Cartan geometries with model the complex projective space on compact complex tori are not flat. Theorem 2.3 goes into the direction to prove that Kähler Calabi-Yau manifolds admitting generalized Cartan geometries with model (G, H), also admit flat generalized Cartan geometries with model (G, H).…”

Section: Let Us First Definementioning

confidence: 99%

Principal Bundles with Holomorphic Connections Over a Kahler Calabi-Yau Manifold

Biswas,

Dumitrescu

2023

Preprint

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Section: Let Us First Definementioning

confidence: 99%

Principal Bundles with Holomorphic Connections Over a Kahler Calabi-Yau Manifold

Biswas,

Dumitrescu

2023

Preprint

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“…In other words, the index l is the maximal positive integer such that there exists a holomorphic line bundle L over M such that L ⊗l = K M . Theorem 3.4 below, proved in [BD2], shows, in particular, that among the quadrics Q n , n ≥ 3, only Q 3 admits a holomorphic GL(2, C)-structure.…”

Section: Gl(2)-geometry and Sl(2)-geometrymentioning

confidence: 99%

“…The methods used in [BD2] to prove Theorem 3.4 and Theorem 3.5 does not use the results in [BM,HM,KO2,KO3,Ye]: they are specific to the case of GL(2, C)-geometry and unify the twisted holomorphic symplectic case (even dimensional case) and the holomorphic conformal case (odd dimensional case).…”

Section: Theorem 35 ([Bd2]mentioning

confidence: 99%

Holomorphic G-structures and foliated Cartan geometries on compact complex manifolds

Biswas,

Dumitrescu

2021

Preprint

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Holomorphic G-Structures and Foliated Cartan Geometries on Compact Complex Manifolds

Biswas¹,

Dumitrescu²

2022

Surveys in Geometry I

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Let X be a compact complex manifold such that its canonical bundle K X is numerically trivial. Assume additionally that X is Moishezon or X is Fujiki with dimension at most four. Using the MMP and classical results in foliation theory, we prove a Beauville-Bogomolov type decomposition theorem for X. We deduce that holomorphic geometric structures of affine type on X are in fact locally homogeneous away from an analytic subset of complex codimension at least two, and that they cannot be rigid unless X is an étale quotient of a compact complex torus. Moreover, we establish a characterization of torus quotients using the vanishing of the first two Chern classes which is valid for any compact complex n-folds of algebraic dimension at least n − 1. Finally, we show that a compact complex manifold with trivial canonical bundle bearing a rigid geometric structure must have infinite fundamental group if either X is Fujiki, X is a threefold, or X is of algebraic dimension at most one.

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Holomorphic $$\text {GL}_2({\mathbb C})$$-geometry on compact complex manifolds

Cited by 3 publications

References 36 publications

Principal Bundles with Holomorphic Connections Over a Kahler Calabi-Yau Manifold

Principal Bundles with Holomorphic Connections Over a Kahler Calabi-Yau Manifold

Holomorphic G-structures and foliated Cartan geometries on compact complex manifolds

Holomorphic G-Structures and Foliated Cartan Geometries on Compact Complex Manifolds

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