# Generalized complex geometry

## Course information

Code: MAT1312F

Instructor: Marco Gualtieri

Class schedule: W3-6 BA6180

Evaluation: Class participation and reading project

This is an introductory course in Generalized geometry. We will study the origins of the subject in the study of geometric structures defined by differential forms, and in the study of Dirac structures. We will cover generalized complex, generalized Kahler and Calabi-Yau structures, Poisson and twisted Poisson structures and their relatives, Courant algebroids and their relation to gerbes, the 2-dimensional sigma model in physics, and D-branes, among other topics. The goal of the course is to bring people close to the research level in the subject.

The course will also be a good opportunity to learn the basics of complex and symplectic geometry. The evaluation of the course will be by class participation and by a written research report on one of the main papers in the field.

## Notes

## Resources

### Articles

*“Generalized Calabi-Yau manifolds”*, Hitchin.- “Generalized complex geometry”, Gualtieri.
- “Generalized Kahler geometry”, Gualtieri.
- “Pure Spinors on Lie groups”, Alekseev, Bursztyn, and Meinrenken.
- “Dirac Manifolds”, Courant.
- “Reduction of Courant algebroids and generalized complex structures”, Bursztyn, Cavalcanti, and Gualtieri.
- “Branes on Poisson varieties”, Gualtieri.
- “Stable generalized complex structures”, Cavalcanti and Gualtieri.
- “Instantons, Poisson structures and generalized Kaehler geometry”, Hitchin.
- “Generalized holomorphic bundles and the B-field action”, Hitchin.
- “Brackets, forms and invariant functionals”, Hitchin.
- “Goto’s generalized Kahler stability theorem”, Cavalcanti.
- “Generalized Calabi-Yau structures, K3 surfaces, and B-fields”, Huybrechts.
- “Topological sigma-models with H-flux and twisted generalized complex manifolds”, Kapustin and Li.
- “Leibniz algebroids, twistings and exceptional generalized geometry”, Baraglia.
- “Variation of Hodge structure for generalized complex manifolds”, Baraglia.

## Lectures

- “Lectures on generalized geometry”, Hitchin.
- “Generalized complex geometry and T-duality”, Cavalcanti and Gualtieri.
- “Introduction to Generalized Complex Geometry”, Cavalcanti.
- “Lectures on Generalized Complex Geometry and Supersymmetry”, Zabzine.
- “Low-dimensional geometry–a variational approach”, Hitchin.

## Research project

The research project will consist of a written exposition/resume of a research paper in the field, whether in the physics or mathematics literature.