Examples and counter-examples of log-symplectic manifolds

We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably, we show that several symplectic manifolds do not admit bona fide log-symplectic structures and several bona fide log-symplectic manifolds do not admit symplectic structures; for example, #mcP2#ncP2 has bona fide log-symplectic structures if and only if m, n > 0, while they only have symplectic structures for m = 1. We introduce surgeries that produce log-symplectic manifolds out of symplectic manifolds and show that any compact oriented log-symplectic 4-manifold can be transformed into a collection of symplectic manifolds by reversing these surgeries. Finally, we show that if a compact manifold admits an achiral Lefschetz fibration with homologically essential fibres, then the manifold admits a logsymplectic structure. Then, using results of Etnyre and Fuller (Int. Math. Res. Not. (2006), art. ID 70272), we conclude that if M is a compact, simply connected 4-manifold then M#(S2 × S2) and M#cP2#cP2 have log-symplectic structures