  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.
