Classification of boundary Lefschetz fibrations over the disc
Abstract
We show that a fourmanifold 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 fourmanifolds whose stable structure arise from boundary Lefschetz fibrations over the disc.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 DOI:
 10.48550/arXiv.1706.09207
 arXiv:
 arXiv:1706.09207
 Bibcode:
 2017arXiv170609207B
 Keywords:

 Mathematics  Differential Geometry;
 53D18;
 53D17
 EPrint:
 18 pages, 8 figures. Paper for the proceedings of the conference in honour of Prof. Nigel Hitchin on the occasion of his 70th birthday