Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate

摘要

We observe that nonzero Gromov-Witten invariants with marked pointconstraints in a closed symplectic manifold imply restrictions on the homologyclasses that can be represented by contact hypersurfaces. As a special case,contact hypersurfaces must always separate if the symplectic manifold isuniruled. This removes a superfluous assumption in a result of G. Lu, thusimplying that all contact manifolds that embed as contact type hypersurfacesinto uniruled symplectic manifolds satisfy the Weinstein conjecture. We provethe main result using the Cieliebak-Mohnke approach to defining Gromov-Witteninvariants via Donaldson hypersurfaces, thus no semipositivity or virtualmoduli cycles are required.