Completed Symplectic Cohomology and Liouville Cobordisms

摘要

Symplectic cohomology is an algebraic invariant of filled symplectic cobordisms that encodes dynamical information. In this thesis we define a modified symplectic cohomology theory, called action-completed symplectic cohomology, that exhibits quantitative behavior. We illustrate the non-trivial nature of this invariant by computing it for annulus subbundles of line bundles over complex projective space. The proof relies on understanding the symplectic cohomology of the complex fibers and the quantum cohomology of the projective base. We connect this result to mirror symmetry and prove a non-vanishing result in the presence of Lagrangian submanifolds with non-vanishing Floer homology. The proof uses Lagrangian quantum cohomology in conjunction with a closed-open map.