Differential Geometry Seminar on
Generalized Complex Geometry


Fontanka River, Saint Petersburg

A generalized complex structure J on a smooth manifold M is an assignment to each fiber of the extended tangent bundle TM⊕T*M of an orthogonal linear complex structure in a locally compatible manner. The resulting formalism extends both complex and symplectic geometry, and was introduced in 2003 by Nigel Hitchen with an eye to string theory.

The aim of this learning seminar is first to review the foundation material, and then to acquaint ourselves with the state of the art and open questions.


1. Linear generalized complex structures

2. Clifford algebras and spinors

3. Canonical Lines of pure forms and Lie algebroids



4. The Courant bracket



5. Further properties of the Courant bracket



6. B-field transformations and generalized submanifolds



7. Review