Coregularity of Fano varieties

TL;DR

This paper introduces the concept of coregularity for Fano varieties, exploring its geometric significance, reviewing related history and theorems, and proposing open problems to deepen understanding of this invariant.

Contribution

It defines the coregularity invariant for Fano varieties, surveys its theoretical background, and suggests new research directions in the study of Fano geometry.

Findings

Coregularity relates to the dual complex of log Calabi--Yau structures.

The paper reviews key theorems and examples of Fano varieties.

Several open problems about coregularity are proposed.

Abstract

The regularity of a Fano variety, denoted by ${\rm reg}(X)$, is the largest dimension of the dual complex of a log Calabi--Yau structure on $X$. The coregularity is defined to be \[ {\rm coreg}(X):= \dim X - {\rm reg}(X)-1. \] The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of $X$. In this note, we review the history of Fano varieties, give some examples, survey some important theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.