Isolated torus invariants and automorphism groups of rigid varieties

TL;DR

This paper proves that the neutral component of automorphism groups of certain rigid affine varieties is a torus, confirming a conjecture for specific classes and exploring automorphism groups of suspensions.

Contribution

It confirms Perepechko and Zaidenberg's conjecture for toric and complexity-one torus action varieties, and provides a criterion for rigidity of suspensions.

Findings

The neutral component of automorphism groups is a torus for these varieties.

A criterion for rigidity of m-suspensions over rigid varieties is established.

Automorphism groups of suspensions satisfying the criterion are characterized.

Abstract

Perepechko and Zaidenberg conjectured that the neutral component of the automorphism group of a rigid affine variety is a torus. We prove this conjecture for toric varieties and varieties with a torus action of complexity one. We also obtain a criterion for an $m$-suspension over a rigid variety to be rigid (for every rigid variety and every regular function). Additionally, we study the automorphism group of $m$-suspensions satisfying this criterion.