Holomorphic Cartan geometries and Calabi–Yau manifolds

Biswas, Indranil; McKay, Benjamin

doi:10.1016/j.geomphys.2009.12.011

Cited by 9 publications

(11 citation statements)

References 5 publications

(6 reference statements)

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Section: Review Of the Literaturementioning

confidence: 88%

“…Let us contrast our results in this paper with those of [3,5,10]. In [3], we proved that if a smooth complex projective variety with c 1 = 0 bears a holomorphic Cartan geometry, then the smooth projective variety has holomorphic unramified covering map by an Abelian variety. In [5], we generalized this result to prove that if a compact Kähler manifold with c 1 = 0 bears a holomorphic Cartan geometry, then the compact Kähler manifold has a holomorphic unramified covering map by a complex torus.…”

Section: Review Of the Literaturementioning

confidence: 88%

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Holomorphic Parabolic Geometries and Calabi-Yau Manifolds

McKay¹

2011

SIGMA

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Section: Review Of the Literaturementioning

confidence: 88%

Section: Review Of the Literaturementioning

confidence: 88%

Holomorphic Parabolic Geometries and Calabi-Yau Manifolds

McKay¹

2011

SIGMA

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“…In this case it is known that M admits a finite unramified cover which is a compact complex torus [IKO] (the pull-back, to the torus, of such a global representative affine connection is a translation invariant holomorphic torsionfree affine connection). This type of results are also valid in the broader context of holomorphic Cartan geometries [BM2,BM3,Du3,BD4]; see [Sha] for holomorphic Cartan geometries.…”

Section: Introductionmentioning

confidence: 63%

“…The following proposition (statement (ii)) studies holomorphic projective connections on compact complex tori. Statement (i), which was already known in the broader context of holomorphic Cartan geometries (see for example, [BM2,BM3,BD4,Du3]), shows that compact complex tori cover the case of Kähler manifolds with trivial first Chern class.…”

Section: The Case Of Holomorphic Projective Connections On Quotients ...mentioning

confidence: 81%

Holomorphic projective connections on compact complex threefolds

Biswas,

Dumitrescu

2020

Preprint

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We prove that a holomorphic projective connection on a complex projective threefold is either flat, or it is a translation invariant holomorphic projective connection on an abelian threefold. In the second case, a generic translation invariant holomorphic affine connection on the abelian variety is not projectively flat. We also prove that a simply connected compact complex threefold with trivial canonical line bundle does not admit any holomorphic projective connection.

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“…With our assumption on the first Chern class, X is a projective Calabi-Yau manifold. But a projective Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type is covered by a compact complex torus [BM1,Du2]: a contradiction.…”

Section: Fujiki Class C and Geometric Structuresmentioning

confidence: 99%

Fujiki class 𝒞 and holomorphic geometric structures

Biswas

Dumitrescu²

2020

Int. J. Math.

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For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class C, whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class C and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.Here we prove that Question 1.1 has a positive answer for compact complex torus principal bundles over compact Kähler Calabi-Yau manifolds (Theorem 1.2).Since compact complex surfaces with trivial canonical bundle are either complex tori, or K3 surfaces, or primary Kodaira surfaces (elliptic principal bundles over elliptic curves) [BHPV, Chapter 6], from Theorem 1.2 it follows that the answer to Question 1.1 is yes when the dimension of the manifold is two.More precisely, our result in this direction is:2010 Mathematics Subject Classification. 53B35, 53C55, 53A55.

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Holomorphic Cartan geometries and Calabi–Yau manifolds

Cited by 9 publications

References 5 publications

Holomorphic Parabolic Geometries and Calabi-Yau Manifolds

Holomorphic Parabolic Geometries and Calabi-Yau Manifolds

Holomorphic projective connections on compact complex threefolds

Fujiki class 𝒞 and holomorphic geometric structures

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