Finite abelian group actions on weakly Lefschetz cohomologically symplectic manifolds

TL;DR

This paper investigates finite abelian group actions on a class of manifolds called weakly Lefschetz cohomologically symplectic manifolds, establishing bounds on their symmetry and analyzing the cohomology of associated abelian covers.

Contribution

It proves bounds on the Betti numbers for free actions of large abelian groups and provides a structure theorem for effective actions of large prime order abelian groups on these manifolds.

Findings

Existence of a bound C such that for m ≥ C, free ({Z}/m)^k actions imply Betti sum ≥ 2^k.

Structure theorem for effective actions of ({Z}/p)^r with large prime p.

Some abelian covers have finitely generated cohomology with spectral sequence degeneration.

Abstract

We study finite abelian group actions on weakly Lefschetz cohomologically symplectic (WLS) manifolds, a collection of manifolds that includes all compact connected Kaehler manifolds. We prove that for any WLS manifold $X$ there exists a number $C$ such that, for any integer $m\geq C$, if $({\mathbf Z}/m)^k$ acts freely on $X$, then $\sum_j b_j(X;{\mathbf Q})\geq 2^k$. We also prove a structure theorem for effective actions on WLS manifolds of $({\mathbf Z}/p)^r$, where $p$ is a big enough prime, analogous to some results for tori of Lupton and Oprea, and we find bounds on the discrete degree of symmetry of WLS manifolds. Our technique, which may be of independent interest, is based on studying the cohomology of abelian covers of WLS manifolds $X$ associated to certain maps $\pi:X\to T^k$. We prove that, in the presence of actions of arbitrarily big finite abelian groups, some of these abelian covers have finitely generated cohomology, and the spectral sequence associated to $\pi$ degenerates at the second page over the rationals.