Stable generalized complex structures

TL;DR

This paper introduces a new framework for understanding stable generalized complex structures using Lie algebroids, constructs examples, studies their deformations, and proves normal forms and symplectic completions in four dimensions.

Contribution

It develops a Lie algebroid approach to stable generalized complex structures, constructs new examples, and establishes deformation theory and normal forms, including symplectic completions in four dimensions.

Findings

Constructed new examples of stable structures.

Proved unobstructedness of deformation problems.

Established local normal forms and symplectic completions.

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.