The Lefschetz Hyperplane Theorem is mostly wrong (symplectically speaking)

Princeton/IAS Symplectic Geometry Seminar
Topic:The Lefschetz Hyperplane Theorem is mostly wrong (symplectically speaking)
Speaker:Mark McLean
Affiliation:Stony Brook University
Date:Friday, November 14
Time/Room:1:30pm - 2:30pm/Math 407, Columbia University

We show that for any symplectic manifold of dimension \(2n > 4\), there exists a symplectic hypersurface Poincare dual to some positive multiple of the symplectic form whose \((n - 2)\)th Betti number is as large as we like. The idea here is to find an appropriate Liouville domain inside each of these symplectic manifolds and use Donaldson's asymptotically holomorphic techniques from his 1996 paper to find a symplectic hypersurface not intersecting some deformation of this Liouville domain.