On the virtual invariants of zero entropy groups of compact K\"ahler manifolds

TL;DR

This paper investigates the structure of zero entropy automorphism groups of compact K"ahler manifolds, establishing bounds on their virtual derived length and nilpotency class related to the manifold's dimension and Kodaira dimension.

Contribution

It provides new bounds on the virtual derived length and nilpotency class of zero entropy automorphism groups, linking group structure to geometric invariants of the manifold.

Findings

Virtual derived length _{ ext{vir}}(G)  \, ext{dim} X - \, ext{kappa}(X)

Bound on virtual nilpotency class c_{ ext{vir}}(G)  \, ext{dim} X - \, ext{kappa}(X)

Geometric description of G-action when bounds are sharp

Abstract

Let $X$ be a compact K\"ahler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -\kappa(X)$, where $\kappa(X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{vir}}(G) \le \dim X -\kappa(X)$ for the virtual nilpotency class $c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds.