Boundedness of Fano type fibrations

TL;DR

This paper investigates the boundedness and singularities of Fano and Fano type fibrations, extending the theory within the broader context of log Calabi-Yau fibrations, with implications for various central topics in birational geometry.

Contribution

It establishes new results on the boundedness and singularities of Fano fibrations and generalizes the theory to log Calabi-Yau fibrations, encompassing many key concepts in birational geometry.

Findings

Proved results on boundedness of Fano fibrations.

Analyzed singularities in Fano type fibrations.

Extended the theory to log Calabi-Yau fibrations.

Abstract

In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has "good" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geometry such as Fano varieties, Mori fibre spaces, flipping and divisorial contractions, crepant models, germs of singularities, etc. We develop the theory in the more general framework of log Calabi-Yau fibrations. Dans cet article, nous prouvons divers r\'esultats sur les limites et les singularit\'es de fibrations de Fano et les fibrations de type Fano. Une fibration de Fano est un morphisme projectif $X\to Z$ de vari\'et\'es alg\'ebriques \`a fibres connexes tel que $X$ est Fano sur $Z$, c'est-\`a-dire que $X$ a de "bonnes" singularit\'es et $-K_X$ est ample sur $Z$. Une fibration de type Fano est d\'efinie de fa\c{c}on similaire quand $X$ est suppos\'e \^etre proche d'\^etre Fano sur $Z$. Cette classe comprend de nombreux ingr\'edients centraux de g\'eom\'etrie birationnelle tels que les vari\'et\'es de fano, les espaces de fibres Mori, le flip et les contractions divisorielles, les mod\`eles r\'ep\'etiteurs, les germes de singularit\'es, etc. Nous d\'eveloppons la th\'eorie dans le cadre plus g\'en\'eral des log-fibrations de Calabi-Yau.