Symplectic Boundary Conditions and Cohomology
arXiv:1710.03741
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 Kähler structures of Kähler manifolds with boundary.
arXiv comment62 pages, typos corrected, Section 4.1-4.2 improved, Appendix added