Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

TL;DR

This paper demonstrates the invariance of symplectic cohomology under certain deformations of the symplectic form, extending previous results to non-exact and non-compactly supported cases, with applications to magnetic geodesics.

Contribution

It establishes the invariance of symplectic cohomology under broader deformations and applies this to twisted cotangent bundles over surfaces, revealing new existence results for magnetic geodesics.

Findings

Symplectic cohomology remains invariant under non-exact deformations.

Explicit computation of symplectic cohomology for twisted cotangent bundles.

Existence of multiple periodic magnetic geodesics on surfaces.

Abstract

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be non-exact and non-compactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.