Symplectic Structures from Almost Symplectic Structures

In this talk, we will consider a stabilized version of the fundamental existence problem of symplectic structures (cf. Open Problem 1 in McDuff & Salamon). Given a formal symplectic manifold, i.e. a closed manifold M

with a non-degenerate 2-form and a non-degenerate second cohomology class, we investigate when its natural stabilization to M × T2 can be realized by a symplectic form.

We show that this can be done whenever the formal symplectic manifold admits a positive symplectic divisor. It follows that if a formal symplectic 4-manifold, which either satisfies that its positive/negative second betti numbers are both at least 2, or that is simply connected, then M × T2 is symplectic.This is joint work with Fabio Gironella, Fran Presas, Lauran Touissant.