The biinvariant diagonal class for Hamiltonian torus actions

Details The biinvariant diagonal class for Hamiltonian torus actions The biinvariant diagonal class for Hamiltonian torus actions

2004-12-10

arxiv.org/abs/math/0412218v2

Mathematics

Symplectic Geometry

A substatially revised version of the paper "A right inverse to the Kirwan map". Improved exposition. Singular quotients are a

Scientific paper

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 $[\Delta_G]\in H^*_{G\times G}(X\times X)$. 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 $[\Delta_G]$ to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.

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