Finite abelian group actions on weakly Lefschetz cohomologically symplectic manifolds

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.