On the cohomology of almost complex and symplectic manifolds and proper surjective maps

TL;DR

This paper investigates the relationships between specialized cohomology groups in almost complex and symplectic manifolds, especially under proper surjective maps, and offers new insights into the Hard Lefschetz condition in four dimensions.

Contribution

It extends the understanding of cohomology behavior under proper surjective maps in almost complex and symplectic manifolds, introducing new characterizations of the Hard Lefschetz condition.

Findings

Cohomology groups are related via proper surjective pseudo-holomorphic maps.

New characterization of the Hard Lefschetz condition in dimension 4.

Results unify and extend previous cohomological studies in almost complex and symplectic geometry.

Abstract

Let $(X,J)$ be an almost-complex manifold. In \cite{li-zhang} Li and Zhang introduce $H^{(p,q),(q,p)}_J(X)_{\rr}$ as the cohomology subgroups of the $(p+q)$-th de Rham cohomology group formed by classes represented by real pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds we study the relationship among such cohomology groups. Similar results are proven in the symplectic setting for the cohomology groups introduced in \cite{tsengyauI} by Tseng and Yau and a new characterization of the Hard Lefschetz condition in dimension $4$ is provided.