Stable generalized complex structures

Abstract

A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view such structures as symplectic forms. This allows us to construct new examples of stable structures, and also to define period maps for their deformations in which the background three-form flux is either fixed or not, proving the unobstructedness of both deformation problems. We then use the same tools to establish local normal forms for the degeneracy locus and for Lagrangian branes. Applying our normal forms to the four-dimensional case, we prove that any compact stable generalized complex 4-manifold has a symplectic completion, in the sense that it can be modified near its degeneracy locus to produce a compact symplectic 4-manifold.

References in corpus (3)

• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
• New aspects of the ddc-lemma
• Poisson modules and generalized geometry

Cited by in corpus (11)

• Elliptic singularities on log symplectic manifolds and Feigin--Odesskii Poisson brackets
• Poisson structures of divisor-type
• Type one generalized Calabi--Yaus
• Linearization of Poisson groupoids
• Fibrations and stable generalized complex structures
• Fibrations in semi-toric and generalized complex geometry
• A local Torelli theorem for log symplectic manifolds
• Obstructions for Symplectic Lie Algebroids
• Hamiltonian facets of classical gauge theories on $E$-manifolds
• Scattering manifolds and symplectic fillings
• Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry