Discreteness of volumes of divisors on Calabi-Yau type varieties

TL;DR

This paper proves that volumes of divisors on Calabi-Yau type varieties are discrete and depends only on the dimension and singularities, leading to a boundedness result for polarized log Calabi-Yau pairs.

Contribution

It establishes the discreteness of divisor volumes on Calabi-Yau varieties and confirms a conjecture on boundedness of polarized log Calabi-Yau pairs.

Findings

Volumes of divisors form a fixed discrete set depending on dimension and singularities.

Proves a conjecture by Birkar on boundedness of polarized log Calabi-Yau pairs.

Provides new insights into the structure of Calabi-Yau type varieties.

Abstract

We study the volumes of divisors in Calabi--Yau type varieties. We show that given a klt Calabi--Yau pair $(X,B)$ and an integral divisor $A$ on $X$, the volume of $A$ is in a fixed discrete set depending only on the dimension and singularities of $(X,B)$. As an application, we prove a boundedness result of polarized log Calabi--Yau pairs which was conjectured by Birkar.