Coregularity of Fano varieties

Abstract

The regularity of a Fano variety, denoted by $\mathrm{reg}(X)$, 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.

50 pages

Cited by in corpus (1)

• Fundamental groups of log Calabi-Yau surfaces