Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps

TL;DR

This paper investigates the structure of finite abelian subgroups in the groups of birational and bimeromorphic automorphisms of complex varieties, establishing bounds on their rank relative to the variety's dimension and characterizing cases of equality.

Contribution

It proves an upper bound on the rank of finite abelian subgroups in automorphism groups of complex varieties and characterizes when this bound is achieved.

Findings

The rank of finite abelian subgroups is at most twice the dimension of the variety.

Equality occurs if and only if the variety is birational to an abelian variety.

An analogous result holds for bimeromorphic automorphisms of compact Kähler spaces under certain conditions.

Abstract

Let $X$ be a complex projective variety. Suppose that the group of birational automorphisms of $X$ contains finite subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for $r$ fixed and $N$ arbitrarily large. We show that $r$ does not exceed $2\dim(X)$. Moreover, the equality holds if and only if $X$ is birational to an abelian variety. We also show that an analogous result holds for groups of bimeromorphic automorphisms of compact K\"ahler spaces, under some additional assumptions.