Fibrations and log-symplectic structures

Abstract

Log-symplectic structures are Poisson structures $\pi$ on $X^{2n}$ for which $\bigwedge^n \pi$ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the $b$-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps. More precisely, we introduce the notion of a $b$-hyperfibration and show that they give rise to log-symplectic structures. Moreover, we link log-symplectic structures to achiral Lefschetz fibrations and folded-symplectic structures.