Contact hypersurfaces in uniruled symplectic manifolds always separate

Abstract

We observe that non-zero 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 Lu [Math. Res. Lett. 7 (2000) 383–387], 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.