Fibrations and stable generalized complex structures

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these to construct stable generalized complex structures out of log-symplectic structures. In particular we introduce the notion of a boundary Lefschetz fibration for this purpose and describe how they can be obtained from genus one Lefschetz fibrations over the disk.Comment: 35 pages, 2 figure

Classification of boundary Lefschetz fibrations over the disc

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

Geometric structures and Lie algebroids

In this thesis we study geometric structures from Poisson and generalized complex geometry with mild singular behavior using Lie algebroids. The process of lifting such structures to their Lie algebroid version makes them less singular, as their singular behavior is incorporated in the anchor of the Lie algebroid. We develop a framework for this using the concept of a divisor, which encodes the singularities, and show when structures exhibiting such singularities can be lifted to a Lie algebroid built out of the divisor. Once one has successfully lifted the structure, it becomes possible to study it using more powerful techniques. In the case of Poisson structures one can turn to employing symplectic techniques. These lead for example to normal form results for the underlying Poisson structures around their singular loci. In this thesis we further adapt the methods of Gompf and Thurston for constructing symplectic structures out of fibration-like maps to their Lie algebroid counterparts. More precisely, we introduce the notion of a Lie algebroid Lefschetz fibration and show when these give rise to A-symplectic structures for a given Lie algebroid A. We then use this general result to show how log-symplectic structures arise out of achiral Lefschetz fibrations. Moreover, we introduce the concept of a boundary Lefschetz fibration and show when they allow their total space to be equipped with a stable generalized complex structure. Other results in this thesis include homotopical obstructions to the existence of A-symplectic structures using characteristic classes, and splitting results for A-Lie algebroids (i.e., Lie algebroids whose anchor factors through that of a fixed Lie algebroid A), around specific transversal submanifolds.Comment: Utrecht University PhD thesis, 220 page