Locally stable degenerations of log Calabi-Yau pairs

TL;DR

This paper investigates the birational boundedness of special fibers in log Calabi-Yau and Fano fibrations, establishing conditions under which components of special fibers are bounded when general fibers meet certain criteria.

Contribution

It introduces new boundedness results for special fibers of log Calabi-Yau and Fano fibrations in locally stable families, extending understanding of their birational properties.

Findings

Special fibers are birationally bounded under natural conditions.

Boundedness depends on the properties of the general fiber.

Results apply to families over curves with stable conditions.

Abstract

We study the birational boundedness of special fibers of log Calabi-Yau fibrations and Fano fibrations. We show that for a locally stable family of Fano varieties or polarised log Calabi-Yau pairs over a curve, if the general fiber satisfies some natural boundedness conditions, then every irreducible component of the special fiber is birationally bounded.