We formulate and prove a chain level descent property of symplectic cohomology for involutive covers by compact subsets that take into account the natural algebraic structures that are present. The notion of an involutive cover is reviewed. We indicate the role that the statement plays in mirror symmetry.
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.
This article uses relative symplectic cohomology, recently studied by the second author, 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 M whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of M exhibits strong rigidity properties akin to super-heavy 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 behaviour of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.
We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the symplectic cohomology of the complement of a simple crossings symplectic divisor. We also prove rigidity results for the skeleton of the divisor complement.
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