The biinvariant diagonal class for Hamiltonian torus actions
Abstract
Description
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 $\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\}$ defines a nonzero equivariant cohomology class $[Δ_G]\in H^*_{G\times G}(X\times X)$. 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.
A substatially revised version of the paper "A right inverse to the Kirwan map". Improved exposition. Singular quotients are also considered in the new version. 23 pages. Accepted for publication in Adv. in Math
A substatially revised version of the paper "A right inverse to the Kirwan map". Improved exposition. Singular quotients are also considered in the new version. 23 pages. Accepted for publication in Adv. in Math