Generalized complex geometry

TL;DR

This paper explores the foundational aspects of generalized complex geometry, including its symmetries, deformation theory, relation to Poisson structures, local structure, and the concept of branes, unifying complex and symplectic geometries.

Contribution

It provides a comprehensive study of generalized complex geometry's properties, including symmetry groups, deformation theory, and the introduction of generalized complex branes, advancing the understanding of this unifying geometric framework.

Findings

Enhanced symmetry group identified

Elliptic deformation theory developed

Generalized complex branes characterized

Abstract

Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define and study generalized complex branes, which interpolate between flat bundles on Lagrangian submanifolds and holomorphic bundles on complex submanifolds.