Algebraic varieties with automorphism groups of maximal rank

TL;DR

This paper investigates the automorphism groups of algebraic varieties, showing that varieties with the most automorphisms of positive entropy are essentially tori or their quotients, confirming a long-standing belief.

Contribution

It provides partial confirmation of the conjecture that varieties with maximal automorphism rank are birational to tori or their quotients, extending previous results.

Findings

Varieties with maximal automorphism rank are birational to tori or quotients.

Automorphism groups of positive entropy are maximized only in these cases.

Supports the conjecture relating automorphism groups and variety structure.

Abstract

We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper.