Free actions of large finite abelian groups on weakly Lefschetz cohomologically symplectic manifolds force total Betti number at least 2^k, with a torus-like structure theorem for elementary abelian p-groups.
Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps
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Let $X$ be a complex projective variety. Suppose that the group of birational automorphisms of $X$ contains finite subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for $r$ fixed and $N$ arbitrarily large. We show that $r$ does not exceed $2\dim(X)$. Moreover, the equality holds if and only if $X$ is birational to an abelian variety. We also show that an analogous result holds for groups of bimeromorphic automorphisms of compact K\"ahler spaces, under some additional assumptions.
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Finite abelian group actions on weakly Lefschetz cohomologically symplectic manifolds
Free actions of large finite abelian groups on weakly Lefschetz cohomologically symplectic manifolds force total Betti number at least 2^k, with a torus-like structure theorem for elementary abelian p-groups.