Volume of algebraically integrable foliations and locally stable families

TL;DR

This paper investigates the volume properties of algebraically integrable foliations and stable families, establishing discreteness and bounds related to their ranks and general leaves, with implications for a question by Cascini, Hacon, and Langer.

Contribution

It proves that the volume of canonical algebraically integrable foliations and stable families belongs to a discrete set determined by their rank and general leaves' volume, and provides log versions of these results.

Findings

Volumes of canonical algebraically integrable foliations are discrete.

Positive lower bounds for volumes of foliations of general type.

Relative volumes of stable families are discrete under bounded conditions.

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.