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Volume of algebraically integrable foliations and locally stable families

Jingjun Han, Junpeng Jiao, Mengchu Li, Jihao Liu

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.

Volume of algebraically integrable foliations and locally stable families

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.

Paper Structure

This paper contains 14 sections, 21 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Locally stable families
  3. Preliminaries
  4. Locally stable families
  5. Moduli of stable families
  6. Base change
  7. Maximal variation and bigness
  8. Algebraically integrable foliations
  9. Basic definitions
  10. Simple, core, and ACC modifications
  11. Minimal model program
  12. Finite covers
  13. ACC for foliated pseudo-effective thresholds
  14. Proof of the main theorems

Key Result

Theorem 1.2

Let $\mathcal{F}$ be an algebraically integrable foliation with canonical singularities on a normal projective variety $X$. Let $L$ be a general leaf of $\mathcal{F}$. Then there exists a discrete set $\Gamma_0$ depending only on $\operatorname{rank}\mathcal{F}$ and $\operatorname{vol}(K_L)$ such th

Theorems & Definitions (64)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 54 more