Cohomology and Hodge Theory on Symplectic Manifolds: I

TL;DR

This paper introduces new finite-dimensional cohomologies on symplectic manifolds, featuring Lefschetz decomposition, harmonic representatives, and a primitive cohomology linked to lagrangians and coisotropic submanifolds.

Contribution

It develops novel cohomological theories on symplectic manifolds, including primitive cohomology and dualities with lagrangian and coisotropic submanifolds.

Findings

Existence of finite-dimensional cohomologies with Lefschetz decomposition

Unique harmonic representatives in each cohomology class

Identification of dual currents of lagrangians and coisotropic submanifolds with primitive cohomology

Abstract

We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagrangians and more generally coisotropic submanifolds with elements of a primitive cohomology, which dualizes to a homology on coisotropic chains.