Cohomology and Hodge Theory on Symplectic Manifolds: II

Understand

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.