Symplectic structures from almost symplectic structures

In this talk, we will consider a stabilized version of the fundamental existence problem of symplectic structures (McDuff--Salamon, Problem 1). 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 x T^2 can be realized by a symplectic form. We show that this can be done whenever the formal symplectic manifold admits a symplectic divisor.

It follows that the product with T^2 of an almost symplectic blow up admits a symplectic form. Another corollary is that if a formal symplectic 4-manifold M, which either satisfies that its positive/negative second betti numbers are both at least 2, or that is simply connected, then M x T^2 is symplectic. For instance (CP^2#CP^2#CP^2)xT^2 is symplectic, even though CP^2#CP^2#CP^2 is not, by work of Taubes.

These results follow from a stabilized version of Eliashberg--Murphy's h-principle for symplectic cobordisms, which makes no assumptions on overtwistedness at the boundary.

This is joint work with Fabio Gironella, Fran Presas, Lauran Toussaint.