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

Abstract

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.

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

“…In this case the pull-back of F to the universal cover M of M is defined by a submersion M −→ G/H called the developing map of the transverse flat Cartan geometry (see section 3.2 in [BD2]). The more general notion of transversal generalized Cartan geometry was worked out in [BD4].…”

“…Let us present the bundle theoretic definition of a transversal generalized Cartan geometry (as defined in [BD2,BD4]).…”

“…Since M is simply connected, it follows that the developing map dev is a submersion from M to G/H (such that T F coincides with the kernel of the differential of dev) [BD2,BD4]. The fibers of this submersion must be also Kähler manifolds with trivial first Chern class.…”

“…Lemma 5.1 ( [BD2,BD4]). The flat partial connection λ on E H induces a unique flat partial connection along F on At(E H )/λ ′ (T F) such that the homomorphisms in the exact sequence (5.3) are partial connection preserving (where ad(E H ) is endowed with the canonical connection induced from (E H , λ) and T M/T F is endowed with its canonical Bott connection).…”

“…If the condition (2) in the above Definition is weakened to the condition (2') requiring that β is an isomorphism over on an open dense set in M, we get the notion of transversal branched Cartan geometry as defined in [BD2]. Moreover, if one completely drops the condition (2) in the above Definition, this leads to the more general notion of a transversal generalized Cartan geometry as defined in [BD4].…”