2023/06/29 by Indranil Biswas, Biswas, Indranil, Junyan Cao +5 · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2306.16729
openalex publication_date 2023/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a compact complex manifold such that its canonical bundle KX 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.