Non-standard Symplectic Structures via Symplectic Cohomology

TL;DR

This paper explores how symplectic cohomology can distinguish different symplectic structures on Liouville domains, demonstrating the existence of infinitely many non-standard structures in higher dimensions.

Contribution

It proves the existence of infinitely many non-standard symplectic structures on finite type Liouville manifolds for dimensions at least six, advancing understanding of symplectic topology.

Findings

Existence of infinitely many non-standard structures in dimension ≥6

Development of tools relating symplectic cohomology to structure classification

Construction of examples using Lefschetz fibrations

Abstract

We examine how symplectic cohomology may be used as an invariant on symplectic structures, and investigate the non-uniqueness of these structures on Liouville domains, a field which has seen much development in the past decade. Notably, we prove the existence of infinitely many non-standard symplectic structures on finite type Liouville manifolds for dimensions $n\geq 6$. To do this, we build up notions of Liouville domains, Lefschetz fibrations, and symplectic cohomology.