# University of Toronto's Symplectic Geometry Seminar

May 26, 2008, 2:10am
Bahen 6183

## Classification of multiplicity free tori Hamiltonian actions on Stein manifolds

Abstract:

The talk is based on the preprint 0706.0632v2. We consider faithful Hamiltonian actions of an $n$-dimensional complex torus $T$ on a symplectic Stein manifold $X$ (the symplectic form is holomorphic). Such an action is called {\it multiplicity free} if $dim X=2n$. To any $X$ we assign a Stein manifold of dimension $n$ equipped with $n$ holomorphic functions, a set of divisors and $n+1$ 2nd cohomology classes. We show that these data classify all mutliplicty free Hamiltonian actions of tori. This result is somewhat similar to that obtained by Delzant for actions of compact tori on compact $C^\infty$-manifolds.