Distinguishing between exotic symplectic structures

TL;DR

This paper explores the uniqueness of exotic symplectic structures on certain manifolds by examining their behavior under small deformations, revealing distinctions beyond classical invariants like symplectic cohomology.

Contribution

It introduces methods to distinguish exotic symplectic structures that share identical classical invariants, and constructs examples with multiple distinct exotic structures.

Findings

Distinguished between two exotic structures using deformation analysis.

Constructed manifolds with multiple exotic structures indistinguishable by symplectic cohomology.

Showed that classical invariants can fail to distinguish certain exotic structures.

Abstract

We investigate the uniqueness of so-called exotic structures on certain exact symplectic manifolds by looking at how their symplectic properties change under small nonexact deformations of the symplectic form. This allows us to distinguish between two examples based on those found in \cite{maydanskiy,maydanskiyseidel}, even though their classical symplectic invariants such as symplectic cohomology vanish. We also exhibit, for any $n$, an exact symplectic manifold with $n$ distinct but exotic symplectic structures, which again cannot be distinguished by symplectic cohomology.