Super-rigidity of certain skeleta using relative symplectic cohomology

Abstract

This paper uses relative symplectic cohomology, recently studied by Varolgunes, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi–Yau symplectic manifold [Formula: see text] whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of [Formula: see text] exhibits strong rigidity properties akin to superheavy subsets of Entov–Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also develop what we call the contact Fukaya trick, concerning the behavior of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.