On the connectedness of the automorphism group of an affine toric variety

TL;DR

This paper provides a combinatorial criterion for the connectedness of the automorphism group of affine toric varieties and describes its component group, showing that the automorphism group has finitely many connected components.

Contribution

It introduces a new combinatorial and divisor class group criterion to determine the connectedness of automorphism groups in affine toric varieties.

Findings

Automorphism group connectedness characterized combinatorially.

Component group of automorphism group described explicitly.

Number of connected components is finite.

Abstract

We obtain a criterion for the automorphism group of an affine toric variety to be connected in combinatorial terms and in terms of the divisor class group of the variety. The component group of the automorphism group of a non-degenerate affine toric variety is described. In particular, we show that the number of connected components of the automorphism group is finite.