Lagrangian branes with boundary and symplectic methods for stable generalized complex manifolds

Abstract

Generalized complex (GC) geometry interpolates between ordinary symplectic and complex geometry. Stable generalized complex manifolds (first introduced by Cavalcanti, Gualtieri in 2015) carry a Poisson structure which is generically symplectic, but degenerates on a (real) codimension-2 submanifold. Up to gauge equivalence, the stable generalized complex structure is thus determined by what is called an elliptic symplectic form, which allows the extension of a number of techniques and results from symplectic geometry to stable GC geometry. This paper introduces a new type of submanifold in stable GC manifolds: Lagrangian branes with boundary, which are generically Lagrangian and intersect the degeneracy locus in their boundary. By relating stable GC manifolds to log symplectic manifolds, we are able to prove results on local neighbourhoods and small deformations of such branes. We further investigate stable generalized complex Lefschetz fibrations, where Lagrangian branes with boundary arise as Lefschetz thimbles. These objects are thus expected to be part of a Fukaya category for stable GC manifolds, which we hope to develop in future work and which would allow the application of Floer theory techniques to a larger class of manifolds.