Generalized complex geometry
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 Bfield. 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 biHermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4dimensional biHermitian 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 subobjects correspond to Dbranes in the topological A and Bmodels of string theory.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2004
 arXiv:
 arXiv:math/0401221
 Bibcode:
 2004math......1221G
 Keywords:

 Differential Geometry;
 53C15;
 53C80
 EPrint:
 Oxford University DPhil thesis, 107 pages