Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate

TL;DR

The paper proves that in uniruled symplectic manifolds, contact hypersurfaces must always separate, using Gromov-Witten invariants and avoiding the need for virtual moduli cycles.

Contribution

It establishes that contact hypersurfaces in uniruled symplectic manifolds always separate, removing previous assumptions and applying a novel approach to Gromov-Witten invariants.

Findings

Contact hypersurfaces in uniruled symplectic manifolds always separate.

All contact manifolds embedded as contact type hypersurfaces satisfy the Weinstein conjecture.

The proof uses the Cieliebak-Mohnke approach without virtual moduli cycles.

Abstract

We observe that nonzero Gromov-Witten invariants with marked point constraints in a closed symplectic manifold imply restrictions on the homology classes that can be represented by contact hypersurfaces. As a special case, contact hypersurfaces must always separate if the symplectic manifold is uniruled. This removes a superfluous assumption in a result of G. Lu, thus implying that all contact manifolds that embed as contact type hypersurfaces into uniruled symplectic manifolds satisfy the Weinstein conjecture. We prove the main result using the Cieliebak-Mohnke approach to defining Gromov-Witten invariants via Donaldson hypersurfaces, thus no semipositivity or virtual moduli cycles are required.