18.969 Topics in geometry: Fall 2006

Graduate course in generalized geometry

Instructor: Marco Gualtieri

Tuesday + Thursday 11-12:30, room 2-255

COURSE OUTLINE (PDF)

Completed course notes (several errors, should be viewed as preliminary) (PDF)

PROBLEM SET 1 (PDF)

PROBLEM SET 2 (PDF)

PROBLEM SET 3 (PDF)

PROBLEM SET 4 (PDF)

PROBLEM SET 5 (PDF)

PROBLEM SET 6 (PDF)

This was an introductory (i.e. first years are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.

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 and baby String theory;
• 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; the Picard group;
• Kodaira-Spencer-Kuranishi deformation theory for generalized complex structures;
• Local structure theory for generalized complex structures;
• Generalized Kahler geometry;
• Hodge decomposition theorem for Generalized Kahler structures;
• Hermitian geometry; the Gray-Hervella classification
• Equivalence theorem Generalized Kahler=Bihermitian
• Generalized Calabi-Yau structures and the Hitchin functional
• Ramond-Ramond versus Neveu-Schwarz fluxes; D-branes.

I will attempt to provide periodic exercise sheets to guide your study, as well as indicate which papers and preprints are relevant for the upcoming classes.