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

A Hamiltonian action of a complex torus on a symplectic complex manifold is said to be {\it multiplicity free} if a general orbit is a lagrangian submanifold. To any multiplicity free Hamiltonian action of a complex torus $T\cong ({\text{C}}^{\times }{)}^n$ on a Stein manifold $X$ we assign a certain 5-tuple consisting of a Stein manifold $Y$, an '{e}tale map $Y\to {\text{t}}^{\ast }$, a set of divisors on $Y$ and elements of $H^2(Y,\mathbb{Z}{)}^{\oplus n},H^2(Y,\text{C})$. We show that $X$ is uniquely determined by this invariants. Furthermore, we describe all 5-tuples arising in this way.