Symmetric and K\"ahler--Einstein Fano polygons

TL;DR

This paper proves that all symmetric Fano polytopes are K"ahler--Einstein and explores their automorphism groups, showing every finite subgroup of GL_2(Z) can be realized as such an automorphism group.

Contribution

It generalizes previous results by establishing that all symmetric Fano polytopes are K"ahler--Einstein and characterizes their automorphism groups in detail.

Findings

Every symmetric Fano polytope is K"ahler--Einstein.

Every finite subgroup of GL_2(Z) is an automorphism group of a K"ahler--Einstein Fano polygon.

Detailed analysis of automorphism groups of 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.