Locality of relative symplectic cohomology for complete embeddings

TL;DR

This paper investigates the behavior of relative symplectic cohomology under complete embeddings, establishing isomorphisms under certain conditions and introducing a new construction technique called integrable anti-surgery.

Contribution

It proves the locality of truncated relative symplectic cohomology for complete embeddings and introduces integrable anti-surgery for constructing such embeddings.

Findings

Isomorphism of truncated relative symplectic cohomology under certain conditions

Introduction of integrable anti-surgery technique

Application to symplectic topology and mirror symmetry of cluster manifolds

Abstract

A complete embedding is a symplectic embedding $\iota:Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $\iota(K)$ inside $M$. Under the assumption that the torsion exponents of $K$ are bounded we deduce the same result for relative symplectic cohomology. We introduce a technique for constructing complete embeddings using what we refer to as integrable anti-surgery. We apply these to study symplectic topology and mirror symmetry of symplectic cluster manifolds and other examples of symplectic manifolds with singular Lagrangian torus fibrations satisfying certain completeness conditions.