Cohomology and Hodge Theory on Symplectic Manifolds: II

TL;DR

This paper introduces new primitive cohomologies on symplectic manifolds by decomposing the exterior derivative into operators similar to Dolbeault operators, leading to finite-dimensional invariants that vary with the symplectic form.

Contribution

It develops a novel decomposition of the exterior derivative on symplectic manifolds and constructs primitive cohomologies analogous to Dolbeault cohomology, with explicit calculations on nilmanifolds.

Findings

Primitive cohomologies are finite-dimensional due to elliptic complex structure.

Dimensions of primitive cohomologies vary with the symplectic form class.

Explicit computation of primitive cohomologies on nilmanifolds.

Abstract

We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calculate the known primitive cohomologies on a nilmanifold and show that their dimensions can vary depending on the class of the symplectic form.