Generalized Geometry and its Applications

Generalized geometry, whose definition was inspired in particular by duality symmetries in String Theory, was recently put into a rigorous mathematical framework by Hitchin [Hi] and his students [Ca, Gu]. At the most basic level it amounts to replacing structures defined on the tangent bundle of a manifold by similarly defined structures of the direct sum of tangent and cotangent bundle. As such there exist generalizations of complex manifolds, Kahler manifolds, Calabi-Yau manifolds, etc. This project aims to review these developments and their applications.

[Ca] G. Cavalcanti, New aspects of the dd^c-lemma, PhD thesis, University of Oxford, 2004, [arXiv:math.DG/0501406]
[Gu] M. Gualtieri, Generalized complex geometry, PhD thesis, University of Oxford, 2003, [arXiv:math.DG/0401221]
[Hi] N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281-308, [arXiv:math.DG/0209099]