Fujiki class 𝒞 and holomorphic geometric structures

Abstract

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.

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

“…• In the Fujiki case (Corollary 4.5), this is a consequence of Theorem A (see second remark below the theorem) since the case of non-maximal algebraic dimension had been treated before in [BD2]. As an interesting consequence, we establish Corollary 4.6 asserting that on a compact complex manifold with trivial canonical bundle, any holomorphic rigid geometric structure of affine type φ admits non-zero locally Killing vector fields.…”

“…Earlier works, [BD1,BD2,BD3,Du1,Du2,BDG], which were inspired by [Gro, DG], aimed to adapt Gromov's ideas and arguments to holomorphic geometric structures on compact complex manifolds. In that vein, the third-named author proved the following theorem:…”

Holomorphic projective connections on compact complex threefolds

“…Theorem 1.1 of [BD3] implies that the holomorphic affine connection on M is locally homogeneous, and hence the fundamental group of M is infinite [BD3, Corollary 1.1]: a contradiction. Hence M does not admit any holomorphic projective connection.…”

“…All simply connected Kähler manifolds, and, more generally, all simply connected manifolds in the Fujiki class C [Fu] (i.e., compact complex manifolds bimeromorphic to a Kähler manifold [Va]), bearing a holomorphic projective connection are actually complex projective manifold [BD3,Theorem 4.3]. In view of this, Corollary 3 also gives a positive answer to Conjecture 4 for simply connected threefolds belonging to the Fujiki class C. More precisely, a compact simply connected complex threefold in the Fujiki class C equipped with a holomorphic projective connection is isomorphic to CP 3 endowed with its standard flat projective connection.…”

Holomorphic Riemannian metric and fundamental group

“…• the holomorphic tangent bundle of the manifold is polystable with respect to some Gauduchon metric on it [BD1]; • the manifold is Moishezon [BD2]; • the manifold is a complex torus principal bundle over a compact Kähler (Calabi-Yau) manifold [BD2].…”