Homogénéité locale pour les métriques riemanniennes holomorphes en dimension 3

We prove a Bochner type vanishing theorem for compact complex manifolds Y in Fujiki class C, with vanishing first Chern class, that admit a cohomology class [α] ∈ H 1,1 (Y, R) which is numerically effective (nef) and has positive self-intersection (meaning Y α n > 0, where n = dim C Y ). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold Y are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of Y must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold Y admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.2010 Mathematics Subject Classification. 53B35, 53C55, 53A55.

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“…− dy 2 n ). For more details about the geometry of holomorphic Riemannian metrics the reader is referred to [Du2,Gh2].…”

Section: Geometric Structures On Manifolds In Class Cmentioning

confidence: 99%

A Bochner principle and its applications to Fujiki class 𝒞 manifolds with vanishing first Chern class

Biswas

Dumitrescu²,

Guenancia

2019

Commun. Contemp. Math.

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“…We expect the answer for Question 1.1 to be yes for all holomorphic geometric structures of affine type on compact complex manifolds in the Fujiki class C that have trivial canonical bundle. We also expect holomorphic Riemannian metrics to be always locally homogeneous on compact complex manifolds (this was proved in complex dimension three [Du4]) and holomorphic affine connections to be always locally homogeneous on compact complex manifolds with trivial canonical bundle; it may be noted that contrary to the case of holomorphic Riemannian metrics, here the condition on the triviality of the canonical bundle is not automatically satisfied and is in fact even necessary: there exists non-locally homogeneous holomorphic affine connections on principal elliptic bundles with odd first Betti number (hence non-Kähler) over Riemann surfaces of genus g ≥ 2 [Du5].…”

Section: Introductionmentioning

confidence: 97%

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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“…In this case, the Killing form is a bi‐invariant holomorphic Riemannian metric of constant negative curvature (see Ghys ). Therefore, the quotients

j \times ρ false(normalΓ false) ∖ {prefixSO}_{0} false(3, 1 false)

inherit such holomorphic Riemannian metrics (see the work of Dumitrescu and Dumitrescu‐Zeghib for classification results for these structures in low dimensions). Ghys studied such quotients in the case that