University of Toronto's Symplectic Geometry Seminar

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.