Differential Geometry Seminar on

Generalized Complex Geometry

venue SPbU, Dept. of Mathematics & Computer Science

time TBD

organizer Casey Blacker (cblacker271@gmail.com)

PDMI coordinator Sylvain Lavau (lavau@math.univ.lyon1.fr)

A generalized complex structure

on a smooth manifold

is an assignment to each fiber of the

extended tangent bundle

of a linear complex structure in a locally compatible manner. The

resulting formalism extends both complex and symplectic geometry, and was introduced in 2003 by

Nigel Hitchin with an eye to string theory.

The aim of this learning seminar is first to review the foundational material, and then to acquaint

ourselves with the state of the art and open questions.

Related constructions include,

• Lie algebroids

• Dirac structures

• Courant brackets

• moment maps

• generalized K ̈ahler structures

• T-duality

This seminar should appeal to students and researchers with interests in differential geometry and

mathematical physics.

Mathematical references:

• Gil Cavalcanti. Introduction to generalized complex geometry. Publica ̧c ̃oes Matem ́aticas do IMPA.

Instituto Nacional de Matem ́atica Pura e Aplicada (IMPA), Rio de Janeiro, 2007. 26o Col ́oquio

Brasileiro de Matem ́atica, https://impa.br/wp-content/uploads/2017/04/26CBM_05.pdf

• Marco Gualtieri. Generalized complex geometry. PhD thesis, University of Oxford, November

2003, https://arxiv.org/abs/math/0401221

• Nigel Hitchin. Lectures on generalized geometry. In Surveys in differential geometry. Volume

XVI. Geometry of special holonomy and related topics, volume 16 of Surv. Differ. Geom., pages

79–124. Int. Press, Somerville, MA, 2011

• Nigel Hitchin. Generalized Calabi-Yau manifolds. Q. J. Math., 54(3):281–308, 2003,

https://proxy.library.spbu.ru:2060/10.1093/qmath/hag025

Physical references:

• Paul Koerber. Lectures on generalized complex geometry for physicists. Fortschr. Phys., 59(3-

4):169–242, 2011, https://proxy.library.spbu.ru:2150/doi/abs/10.1002/prop.201000083

• Maxim Zabzine. Lectures on generalized complex geometry and supersymmetry. Arch. Math.

(Brno), 42(suppl.):119–146, 2006, https://www.emis.de/journals/AM/06-S/zabzine.pdf