Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate
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.
Keywords
Cite
@article{arxiv.1202.4685,
title = {Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate},
author = {Chris Wendl},
journal= {arXiv preprint arXiv:1202.4685},
year = {2017}
}
Comments
24 pages, 1 figure; v.3 is a substantial expansion in which the semipositivity condition has been removed by implementing Cieliebak-Mohnke transversality; it also includes a new appendix to explain why the forgetful map in the Cieliebak-Mohnke context is a pseudocycle; v.4 has one short remark added; to appear in J. London Math. Soc