Birational boundedness of stable families

TL;DR

This paper establishes birational boundedness results for stable families and algebraically integrable foliations, linking foliated geometry to classical boundedness problems and proving an ACC conjecture for foliations.

Contribution

It introduces a new framework connecting foliated geometry with boundedness problems and proves an ACC conjecture for foliations, extending classical results.

Findings

Normal projective stable families of maximal variation are birationally bounded.

Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded.

Proved M extsuperscript{c}Kernan's ACC conjecture for interpolated log canonical thresholds of foliations.

Abstract

We prove that normal projective stable families of maximal variation, of fixed dimension, and with bounded adjoint volume are birationally bounded. This is a consequence of a substantially stronger statement, formulated a priori independently of stable families: algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded. In this way, the birational geometry of foliations provides a systematic framework for approaching classical boundedness problems for fibrations. A key input is our proof of M\textsuperscript{c}Kernan's ACC conjecture for interpolated log canonical thresholds of algebraically integrable foliations. This may be viewed as the foliated analogue of Shokurov's ACC conjecture for log canonical thresholds, proved in the classical setting by Hacon--M\textsuperscript{c}Kernan--Xu. As applications, we establish two boundedness criteria for Fano algebraically integrable adjoint foliated structures: Birkar's criterion for exceptional Fanos, and Jiang's criterion for Fanos for which both Tian's $\alpha$-invariant and the anti-canonical volume are bounded away from zero. We also obtain several results on the birational geometry of algebraically integrable adjoint foliated structures, including lower bounds for adjoint volumes, boundedness of automorphism groups, and ACC theorems for pseudo-effective thresholds, $\mathbb{R}$-complementary thresholds, and the Fano spectrum.