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

TL;DR

This paper classifies multiplicity free Hamiltonian actions of complex tori on Stein manifolds by associating a unique 5-tuple of invariants, providing a complete description and characterization of such actions.

Contribution

It introduces a new classification scheme using a 5-tuple of invariants that uniquely determines the action and characterizes all possible such 5-tuples.

Findings

Each multiplicity free Hamiltonian action corresponds to a unique 5-tuple.

The paper describes all 5-tuples that can arise from such actions.

The classification provides a complete invariant-based description of these actions.

Abstract

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 $T\cong (\C^\times)^n$ on a Stein manifold $X$ we assign a certain 5-tuple consisting of a Stein manifold $Y$, an \'{e}tale map $Y\to \t^*$, a set of divisors on $Y$ and elements of $H^2(Y,\Z)^{\oplus n}, H^2(Y,\C)$. We show that $X$ is uniquely determined by this invariants. Furthermore, we describe all 5-tuples arising in this way.