Potential functions and actions of tori on Kaehler manifolds

Abstract

Let M be a Kaehler manifold with a free, holomorphic and Hamiltonian action of the standard n-torus T. We give a simple, explicit and canonical formula for the Kaehler potential on the Kaehler reduction of M. As a consequence we can derive improvements of several classical results known for more general Hamiltonian reductions. Among these are a forms-level proof of the Duistermaat-Heckman theorem; an elementary proof of Atiyah's proof of the convexity of the moment image of a complexified T-orbit; another formula due to Biquard-Gauduchon for the Kaehler potential; and a formula in terms of moment data for the Kaehler metric on a toric variety, due originally to the second author.