Classification of boundary Lefschetz fibrations over the disc

Stefan Behrens, Gil R. Cavalcanti, Ralph L. Klaasse

Research output: Chapter in Book/Report/Conference proceedingChapterAcademicpeer-review


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.

Original languageEnglish
Title of host publicationGeometry and Physics
Subtitle of host publicationA Festschrift in Honour of Nigel Hitchin
PublisherOxford University Press
Number of pages20
ISBN (Electronic)9780198802020
Publication statusPublished - 1 Jan 2018


  • 4-manifold
  • Generalized complex structure
  • Lefschetz fibration


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