Classification of boundary Lefschetz fibrations over the disc

Abstract

We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to $S^1\times S^3\#n\overline{\mathbb{C}{P}^2}$, $\#m\mathbb{C}{P}^2\#n\overline{\mathbb{C}{P}^2}$ or $\#m(S^2\times S^2)$. Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, we conclude that the manifolds $S^1\times S^3\#n\overline{\mathbb{C}{P}^2}$, $\#(2m+1)\mathbb{C}{P}^2\#n\overline{\mathbb{C}{P}^2}$ and $\#(2m+1)S^2\times S^2$ admit stable structures whose type change locus has a single component and are the only four-manifolds whose stable structure arise from boundary Lefschetz fibrations over the disc.

18 pages, 8 figures. Paper for the proceedings of the conference in honour of Prof. Nigel Hitchin on the occasion of his 70th birthday