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Pré-Publication, Document De Travail Année : 2023

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

Résumé

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.
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Dates et versions

hal-04145233 , version 1 (29-06-2023)

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  • HAL Id : hal-04145233 , version 1

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Indranil Biswas, Junyan Cao, Sorin Dumitrescu, Henri Guenancia. Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures. 2023. ⟨hal-04145233⟩
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