Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917771998134272 |
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| author | Biswas, Indranil Cao, Junyan Dumitrescu, Sorin Guenancia, Henri |
| author_facet | Biswas, Indranil Cao, Junyan Dumitrescu, Sorin Guenancia, Henri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_16729 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures Biswas, Indranil Cao, Junyan Dumitrescu, Sorin Guenancia, Henri Differential Geometry Algebraic Geometry 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. |
| title | Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures |
| topic | Differential Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2306.16729 |