Descent with algebraic structures for symplectic cohomology
Umut Varolgunes
TL;DR
The paper develops a chain-level descent framework for symplectic cohomology with supports $SH_M^*(K)$ over involutive covers, incorporating the $KSV$ operad and BV-algebra structures. By modeling $SC^*_M(K)$ as a degreewise $T$-adic completion of a hocolim of Floer data and employing Thom-Whitney descent, it proves linear descent for weakly involutive covers and promotes descent to the level of $KSV$-algebras, yielding $BV_\infty$-quasi-isomorphisms in the involutive case. The work then sketches a mirror symmetry program where local BV_infty data glue to a global mirror via a Thom-Whitney construction, with the cohomology isomorphism $QH^*(M;\Lambda) \cong H^*(Y,\wedge TY)$ as a key expected outcome. Overall, the framework provides a principled path from local Floer-theoretic constructions to global algebraic structures and their geometric/mirror-symmetric implications, contingent on compatibility of restriction maps and higher homotopies. It lays groundwork for further refinement into a full Plumbers' PROP or $A_\infty$-enhanced setting relevant to nontrivial mirror symmetry scenarios.
Abstract
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.