Classification of boundary Lefschetz fibrations over the disc

Stable generalized complex structures can be constructed out of boundary Lefschetz fibrations. On 4-manifolds, these are essentially genus one Lefschetz fibrations over surfaces, except that generic fibres can collapse to circles over a codimension 1 submanifold, which is often the boundary of the surface. We show that a 4-manifold admits a boundary Lefschetz fibration over the disc degenerating over its boundary if and only if it is diffeomorphic to S ¹ ×S ³ #nℂP ² , #mℂP ² #nℂP ² or #m(S ² ×S ² ).We conclude that the 4-manifolds S ¹ ×S ³ #nℂP ² , #(2m+1)ℂP ² #nℂP ² and #(2m+1)S ² ×S ² admit stable generalized complex structures whose type change locus has a single component.