The biinvariant diagonal class for Hamiltonian torus actions

Abstract

Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs { ( x , y ) ∈ X × X | y = g x for some g ∈ G } defines a nonzero equivariant cohomology class [ Δ G ] ∈ H G × G ∗ ( X × X ) . We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a Hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [ Δ G ] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.