# University of Toronto's Symplectic Geometry Seminar

September 19, 2005, 2:10pm

Bahen 6183

## Yi Lin

###
University Toronto

##
Symmetries in generalized kahler geometry

** Abstract: **
Generalized complex structures were introduced by Hitchin and further
developed by Gualtieri. In particular, an associated notion of generalized
Kahler structure was defined by Gualtieri, who shows this notion is
essentially equivalent to that of a bi-Hermitian structure which was first
discovered by physicists studying super-symmetric sigma-model. An
outstanding open question in the study of bi-Hermitian geometry was to
determine whether there exist bi-Hermitian strucutres on CP^2, a minimal
ruled surface admitting an effective anti-canonical divisor, or a complex
surfaces obtained from either CP^2 or a minimal ruled surface by blowing
up points along an effective anti-canonical divisor. Using the deformation
theorem he developed for generalized complex geometry, Gualtieri proved
that there exists a bi-Hermitian structure on CP^2. More recently, Hitchin
used the generalized Kahler geometry approach to obtain an explicit
construction of a bi-Hermtian structure on CP^2 as well as CP^1 x CP^1. In
this talk, we are going to discuss the definition of moment maps for a
compact Lie group acting on a generalized complex or Kahler manifold.
As an application, we give very simple explicit constructions of
bi-Hermitian structures on C\P^n, Hirzebruch surfaces, the blow
up of CP^2 at arbitrarily many points, and other toric varieties,
as well as complex Grassmannians. In paractice our method
will give one a powerful machinery of producing bi-Hermitian structures on
any manifold which can be produced as the symplectic quotient of C^N.
This is a joint work with Sue Tolman.