Calculus on symplectic manifolds

TL;DR

This paper develops new differential complexes on symplectic manifolds, especially on complex projective space, by coupling elliptic complexes with vector bundles under specific curvature conditions, extending classical geometric structures.

Contribution

It introduces a novel class of differential complexes on symplectic manifolds when the connection curvature is proportional to the symplectic form, generalizing BGG complexes.

Findings

Construction of a new elliptic complex on symplectic manifolds.

Application to complex projective space with Fubini-Study form.

Extension of BGG complexes in symplectic geometry.

Abstract

On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its Fubini-Study form and connection, we can build a series of differential complexes akin to the Bernstein-Gelfand-Gelfand complexes from parabolic differential geometry.