Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations

Abstract

We study zero entropy automorphisms of a compact Kähler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X,\mathbb{R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m{)}^{\ast }\circlearrowleft H^2(X,\mathbb{C})$ where $g$ is a zero entropy automorphism, in terms of $\dim X$ only. We also give an upper bound for the (essential) derived length ${\ell }_{\mathrm{ess}}(G,X)$ for every zero entropy subgroup $G$, again in terms of the dimension of $X$ only. We propose a conjectural upper bound for the essential nilpotency class $c_{\mathrm{ess}}(G,X)$ of a zero entropy subgroup $G$. Finally, we construct examples showing that our upper bound of the polynomial growth (as well as the conjectural upper bound of $c_{\mathrm{ess}}(G,X)$) are optimal.

Geometric and Functional Analysis (2022), to appear; Title shortened a bit; Part of the content is moved to another paper in progress

Cited by in corpus (3)

• Automorphism groups of compact complex surfaces: T-Jordan property, Tits alternative and solvability
• An upper bound for polynomial volume growth of automorphisms of zero entropy
• Polynomial volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang)