Symmetric and K\"ahler--Einstein Fano polygons

Abstract

We investigate \emph{singular} symmetric and K\"ahler--Einstein Fano polytopes. More precisely, we show that every symmetric Fano polytope is K\"ahler--Einstein generalizing the work by Batyrev and Selivanova, and study the automorphism groups of symmetric and K\"ahler--Einstein Fano polygons in detail. In particular, every finte subgroup of $GL_2(\mathbb{Z})$ is an automorphism group of a K\"ahler--Einstein Fano polygon.