Manchester Geometry Seminar 2007/2008

Frobenius manifolds were introduced by Dubrovin to give a geometric reformulation of the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) system of differential equations, which describes deformations of topological field theories.

Generalized complex structures are a simultaneous generalization of complex and symplectic structures. One of their characteristic features is that they can be, say, symplectic on a dense open set and complex along a submanifold. This phenomenon is referred to as "type change". Studying neighbourhood theorems for type change points and generalized Lagrangian submanifolds of a generalized complex 4-manifold we show that one can always blow up a nondegenerate point in the type change locus and conversely, a generalized Lagrangian sphere intersecting the type-change locus at one point can be blown down. We use these results to show that manifolds of the form n CP² + m (CP²)‾ (bar for complex conjugation) have generalized complex structures iff they have almost complex structures.