On the cohomological action of automorphisms of compact K\"ahler threefolds

TL;DR

This paper extends known results on surface automorphisms to threefolds, providing bounds on cohomological actions and spectra of automorphisms of compact Kähler threefolds, with examples demonstrating optimality.

Contribution

It offers new bounds and spectral descriptions for automorphisms of compact Kähler threefolds, generalizing surface results and including optimality examples.

Findings

Bound on growth rate for virtually unipotent automorphisms: at most cn^4

Description of the spectrum of automorphism actions on cohomology

Examples on complex tori demonstrate the bounds are optimal

Abstract

Extending well-known results on surfaces, we give bounds on the cohomological action of automorphisms of compact K\"ahler threefolds. More precisely, if the action is virtually unipotent we prove that the norm of $(f^n)^*$ grows at most as $cn^4$; in the general case, we give a description of the spectrum of $f^*$, and bounds on the possible conjugates over $\mathbb Q$ of the dynamical degrees $\lambda_1(f),\lambda_2(f)$. Examples on complex tori show the optimality of the results.