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arXiv · 2406.16604

Volume of algebraically integrable foliations and locally stable families

Jingjun Han·Junpeng Jiao·Mengchu Li·Jihao Liu
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Abstract

In this paper, we study the volume of algebraically integrable foliations and locally stable families. We show that, for any canonical algebraically integrable foliation, its volume belongs to a discrete set depending only on its rank and the volume of its general leaves. In particular, if the foliation is of general type, then its volume has a positive lower bound depending only on its rank and the volume of its general leaves. This implies some special cases of a question posed by Cascini, Hacon, and Langer. As a consequence, we show that the relative volume of a stable family with a normal generic fiber belongs to a discrete set if the dimension and the volume of its general fibers are bounded. Log versions of the aforementioned theorems are also provided and proved.

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Jingjun Han, Junpeng Jiao, Mengchu Li, Jihao Liu. 2024-06-24. Volume of algebraically integrable foliations and locally stable families. https://arxiv.org/abs/2406.16604

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