Isolated torus invariants and automorphism groups of rigid varieties

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.