Symmetry notions for toric Fanos

TL;DR

This paper surveys various symmetry concepts in toric Fano varieties, clarifies their interrelations, and provides an accessible proof of Demazure's automorphism group theorem, consolidating folklore knowledge into a coherent presentation.

Contribution

It offers a comprehensive survey of symmetry notions in toric Fano varieties and presents a simplified proof of Demazure's structure theorem, filling gaps in existing literature.

Findings

Clarifies relationships between different symmetry notions.

Provides an accessible proof of Demazure's theorem.

Consolidates folklore knowledge into a unified presentation.

Abstract

We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the relationship between these notions. While mostly folklore knowledge, this does not seem to be readily available in the literature. Finally, we take the opportunity to give an accessible and simplified proof of Demazure's 1970 structure theorem for the automorphism group of a smooth toric variety, previously considered quite inaccessible.