Locality and deformations in relative symplectic cohomology

Relative symplectic cohomology is a Floer theoretic invariant associated with compact subsets K of a closed or geometrically bounded symplectic manifold M. The motivation for studying it is that it is often possible to reduce the study of global Floer theory of M to the Floer theory of a handful of local models covering M which one hopes will be easier to compute (Varolgunes’ spectral sequence). As an example, it is expected that at least in the setting of the Gross-Siebert program, the mirror can be pieced together from the relative symplectic cohomologies of neighborhoods of fibers of an SYZ fibration (singular or not). However, even when K is a well understood model, such as the Weinstein neighborhood of a Lagrangian torus, the construction of relative SH is rather unwieldy. In particular, it is not entirely obvious how to relate the symplectic cohomology of K relative to M with Floer theoretic invariants intrinsic to K. I will discuss a number of results, most of them in preparation, which aim to alleviate this difficulty in the setting Lagrangian torus fibrations with singularities.

Partly joint with U. Varolgunes.