Hyperbolicity of coarse moduli spaces and isotriviality for certain families

Ya Deng

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arXiv · 1908.08372

Hyperbolicity of coarse moduli spaces and isotriviality for certain families

Ya Deng

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Abstract

In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense of Satake). As an application, we prove the following hyperbolic version of Campana's isotriviality conjecture: for the smooth family of canonically polarized or polarized Calabi-Yau manifolds, when the Kobayashi pseudo-distance of the base vanishes identically, the family must be isotrivial, that is, any two fibers are isomorphic. We also prove that for the smooth projective family of polarized Calabi-Yau manifolds, its variation of the family is less than or equal to the essential dimension of the base.

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Ya Deng. 2019-08-22. Hyperbolicity of coarse moduli spaces and isotriviality for certain families. https://arxiv.org/abs/1908.08372

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