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Volumes of foliations birationally bounded by algebraically integrable families

Zhixiu Fan

TL;DR

This work proves that the volumes of lc foliations that are birationally bounded by algebraically integrable families satisfy the descending chain condition. The authors combine a fixed-model DCC argument with deformation invariance of relative log canonical volumes in families of weak semistable morphisms to transfer discreteness from leaves to the whole family. A key technical tool is the ACSS framework, enabling existence of suitable models and adjunction along lc centres. The results extend the classical DCC for varieties to a foliated setting and offer a path toward understanding moduli and birational geometry of foliations in higher dimensions.

Abstract

We prove that for log canonical foliations which are birationally bounded by algebraically integrable families, the set of their volumes satisfies the DCC. This answers a special case of a question posed by Cascini, Hacon, and Langer. As a key ingredient, we establish the deformation invariance of relative log canonical volumes for a family of weak semistable morphisms, which can be viewed as a relative version of the classical result proved by Hacon, McKernan, and Xu.

Volumes of foliations birationally bounded by algebraically integrable families

TL;DR

This work proves that the volumes of lc foliations that are birationally bounded by algebraically integrable families satisfy the descending chain condition. The authors combine a fixed-model DCC argument with deformation invariance of relative log canonical volumes in families of weak semistable morphisms to transfer discreteness from leaves to the whole family. A key technical tool is the ACSS framework, enabling existence of suitable models and adjunction along lc centres. The results extend the classical DCC for varieties to a foliated setting and offer a path toward understanding moduli and birational geometry of foliations in higher dimensions.

Abstract

We prove that for log canonical foliations which are birationally bounded by algebraically integrable families, the set of their volumes satisfies the DCC. This answers a special case of a question posed by Cascini, Hacon, and Langer. As a key ingredient, we establish the deformation invariance of relative log canonical volumes for a family of weak semistable morphisms, which can be viewed as a relative version of the classical result proved by Hacon, McKernan, and Xu.
Paper Structure (9 sections, 13 theorems, 51 equations)

This paper contains 9 sections, 13 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Special notation
  4. Toroidal geometry and weak semistable reduction
  5. Foliations
  6. Special algebraically integrable foliations
  7. DCC of volumes of triples with a fixed birational model
  8. Deformation invariance of relative log canonical volumes
  9. Proof of the main theorem

Key Result

Theorem 1

Fix a positive integer $d$ and a DCC set $I\subset{[0,1]}$. Let $\mathcal{D}$ be the set of log canonical pairs $(X,\Delta)$ such that $\dim X=d$, the coefficients of $\Delta$ belong to $I$, and $K_X+\Delta$ is big. Then we have

Theorems & Definitions (52)

  • Theorem : HMXacc
  • Theorem 1.2
  • Theorem 1.3: see Theorem \ref{['inv vol']} for a more general form
  • Remark 1.5
  • Definition 2.2: Relative log smoothness
  • Definition 2.3
  • Definition 2.4: Toroidal couple and toroidal embedding, cf. KKMS73 and AK00
  • Remark 2.5
  • Definition 2.7: Quasi-smooth toroidal couple, cf. AK00
  • Remark 2.8
  • ...and 42 more