Symplectic manifolds and cohomological decomposition

TL;DR

This paper investigates conditions under which the Lefschetz decomposition from symplectic geometry induces a cohomological decomposition, focusing on the second cohomology and manifolds satisfying the Hard Lefschetz Condition.

Contribution

It characterizes when the Lefschetz decomposition yields a cohomological decomposition, especially for the second cohomology and under the Hard Lefschetz Condition.

Findings

Decomposition always holds for the second de Rham cohomology.

Decomposition occurs if the symplectic manifold satisfies the Hard Lefschetz Condition.

Abstract

Given a closed symplectic manifold, we study when the Lefschetz decomposition induced by the $\mathfrak{sl}(2;\mathbb{R})$-representation yields a decomposition of the de Rham cohomology. In particular, this holds always true for the second de Rham cohomology group, or if the symplectic manifold satisfies the Hard Lefschetz Condition.