Symplectic Boundary Conditions and Cohomology

TL;DR

This paper develops new symplectic boundary conditions enabling elliptic boundary value problems and Hodge theories for primitive form cohomologies on manifolds with boundary, linking them to Lefschetz maps and Kähler structures.

Contribution

It introduces symplectic boundary conditions that facilitate elliptic problems and Hodge theories for primitive cohomologies on manifolds with boundary, a novel approach in symplectic geometry.

Findings

Established elliptic boundary value problems for symplectic Laplacians.

Defined relative primitive cohomologies using new boundary conditions.

Connected primitive cohomologies with Lefschetz maps and Kähler structures.

Abstract

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the cohomologies of primitive forms on manifolds with boundary. We further use these boundary conditions to define a relative version of the primitive cohomologies and to relate primitive cohomologies with Lefschetz maps on manifolds with boundary. As we show, these cohomologies of primitive forms can distinguish certain K\"ahler structures of K\"ahler manifolds with boundary.