Quantum cohomology as a deformation of symplectic cohomology

Construction of New Symplectic Cohomology

In this article, we present new symplectic 4-manifolds with same integral cohomology as S × S. The generalization of this construction is given as well, an infinite family of symplectic 4-manifolds cohomology equivalent to #(2g−1)(S 2 × S) for any g ≥ 2. We also compute the Seiberg-Witten invariants of these manifolds.

Cohomology rings of symplectic cuts

The “symplectic cut” construction [Le] produces, from a symplectic manifold M with a Hamiltonian circle action, two symplectic orbifolds C − and C+. We compute the rational cohomology ring of C+ in terms of those of M and C − . 1 Statement of the results Let M be a symplectic n-manifold endowed with an Hamiltonian S1-action with moment map f : M → R. Suppose that 0 is a regular value of f and c...

A Biased View of Symplectic Cohomology

Symplectic Cohomology for Stable Fillings

We discuss a generalisation of symplectic cohomology for symplectic manifolds which weakly fill their contact boundary and satisfy an additional stability condition. Furthermore, we develop a geometric setting for proving maximum principles for Floer trajectories, and prove a Moser-type result for weak fillings. This is a preliminary version of the paper.

Wall crossing for symplectic vortices and quantum cohomology

We derive a wall crossing formula for the symplectic vortex invariants of toric manifolds. As an application, we give a proof of Batyrev’s formula for the quantum cohomology of a monotone toric manifold with minimal Chern number at least two.