Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures

TL;DR

This paper proves a decomposition theorem for certain complex manifolds with trivial canonical bundle, showing that holomorphic geometric structures are mostly locally homogeneous and characterizing torus quotients.

Contribution

It establishes a Beauville-Bogomolov type decomposition for $K$-trivial Moishezon manifolds and characterizes rigid geometric structures and torus quotients in this context.

Findings

Holomorphic geometric structures of affine type are locally homogeneous outside a codimension two subset.

Rigid geometric structures imply infinite fundamental group under certain conditions.

Characterization of torus quotients via vanishing of first two Chern classes.

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 \'etale 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.