Volumes of foliations birationally bounded by algebraically integrable families

TL;DR

This paper proves that the set of volumes of certain log canonical foliations, which are bounded by algebraically integrable families, satisfies the descending chain condition, advancing understanding in algebraic geometry.

Contribution

It establishes the DCC for volumes of log canonical foliations bounded by algebraically integrable families and introduces a deformation invariance result for relative log canonical volumes.

Findings

Volumes of the specified foliations satisfy the DCC.

Deformation invariance of relative log canonical volumes is proven.

Answers a special case of a question by Cascini, Hacon, and Langer.

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.