Generalized complex geometry

TL;DR

This thesis explores generalized complex geometry, revealing new phenomena, examples, and structures, including a local normal form, deformation theory, and connections to string theory D-branes.

Contribution

It introduces novel examples, proves a generalized Darboux theorem, establishes deformation theory, and links generalized complex geometry to bi-Hermitian and string theory concepts.

Findings

New examples on manifolds without known complex or symplectic structures

A generalized Darboux theorem with local normal form

Existence of a Kuranishi moduli space for deformations

Abstract

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.