# Generalized Kahler geometry

## Course information

Code: MAT1312F

Instructor: Marco Gualtieri

Class schedule: W12-1 F10-12 BA6180

Evaluation: There will be some assignments and a project assigned
by the instructor. Students in year > 2 of the PhD who wish to audit the course, please
contact the instructor.

This will be an introductory graduate course in generalized geometry, with a special emphasis on generalized K\"ahler geometry. The main references for this class are the published papers on generalized complex and K\"ahler geometry, but we will also draw from more recent developments in the physics literature.

A basic familiarity with manifolds will be assumed; here is a list of topics which will be covered in the lecture course:

#### Gerbes, B-fields, and exact Courant algebroids;

#### Relation to sigma models in physics;

#### Linear algebra of a split-signature real bilinear form; pure spinors;

#### Generalized Riemannian structures and the generalized Hodge star;

#### Integrability, Dirac structures, Lie algebroids and bialgebroids;

#### Generalized complex structures; examples of such;

#### Generalized holomorphic bundles

#### Generalized K\"ahler geometry;

#### Hodge decomposition theorem for Generalized K\"ahler structures;

#### Equivalence theorem Generalized K\"ahler=Bihermitian

#### The generalized K\"ahler potential

#### Generalized K\"ahler reduction

#### The generalized K\"ahler–Ricci flow of Streets and Tian

## Assignments

## Resources

Articles

*“Generalized Calabi-Yau manifolds”*, Hitchin.- “Generalized complex geometry”, Gualtieri.
- “Generalized Kahler geometry”, Gualtieri.
- “Reduction of Courant algebroids and generalized complex structures”, Bursztyn, Cavalcanti, and Gualtieri.
- “Branes on Poisson varieties”, Gualtieri.
- “Instantons, Poisson structures and generalized Kaehler geometry”, Hitchin.
- “Goto’s generalized Kahler stability theorem”, Cavalcanti.
- “Topological sigma-models with H-flux and twisted generalized complex manifolds”, Kapustin and Li.
- “Variation of Hodge structure for generalized complex manifolds”, Baraglia.
- Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type
- Morita equivalence and the generalized Kähler potential

## 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.