PMATH955: Topics in geometry -- Generalized Complex Geometry

Lectures: TTh 9:00--10:20 (MC5046).
Office hours: TTh 11:00--12:00, or by appointment.
Course information: Outline.

Overview: Generalized complex geometry, which interpolates between symplectic and complex geometry, was introduced by Hitchin in 2002. Its framework has since led to important breakthroughs in bi-Hermitian geometry as well as mirror symmetry. This course will be an introduction to this exciting new research area and will cover material that any graduate student interested in geometry will find useful (such as bundles, connections, sheaf cohomology, symplectic structures, complex structures and their deformations, symplectic reductions, and moduli spaces).

The course should be accessible to students who have taken PMATH 465 (Differential Geometry) or an equivalent course.

Outline of topics:

Background from symplectic, Poisson, and complex geometry:
- Differential forms, de Rham cohomology, and Dolbeault cohomology.
- Riemannian metrics, volume forms, and the Hodge star operator.
- Vector bundles and connections: definitions and basic constructions.
- Symplectic vector spaces, Lagrangian subspaces, and symplectic bundles.
- Symplectic manifolds, Darboux's theorem, and submanifolds of symplectic manifolds.
- Poisson algebras and Poisson manifolds.
- Hamiltonian group actions, moment maps, and symplectic reductions (time permitting).
- Almost complex structures, integrability, and Kähler manifolds.
- Sheaf cohomology and deformations of complex structures.

Generalized complex geometry:
- Courant brackets and B-fields; generalized metrics and the generalized Hodge operator.
- Gerbes, exact Courant algebroids, and twisted structures.
- Generalized complex manifolds: definitions and basic examples.
- Integrability and spinors; generalized Calabi-Yau manifolds.
- The generalized Darboux theorem; generalized complex submanifolds.
- Deformations of generalized complex structures.
- Generalized Kähler manifolds and their relation to bi-Hermitian geometry.
- Lie algebroid connections and generalized holomorphic vector bundles.
- Reductions of generalized complex and generalized Kähler structures (time permitting).
- Construction of new examples of generalized complex and generalized Kähler spaces via reductions methods (time permitting).

Some references:

M. Gualtieri, Generalized complex geometry, Oxford University DPhil thesis, 2004 (arXiv:math/0401221v1).
N. Hitchin, Lectures on generalized geometry (arXiv:math/1008.0973).
J. L. Kazdan, Lecture notes on Applications of Partial Differential Equations to Some Problems in Differential Geometry (pdf).
D. McDuff and D. Salamon, Introduction to symplectic topology.
I. Vaisman, Lectures on the geometry of Poisson manifolds.
K. H. Bhaskara and K. Viswanath, Poisson algebras and Poisson manifolds.
D. Huybrechts, Complex Geometry: An Introduction.
S. Kobayashi, Differential Geometry of Complex Vector Bundles.
F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry.
J. M. Lee, Introduction to smooth manifolds.