Compact Kahler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semi stable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hansen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kahler quotient. (C) 2009 Elsevier Inc. All rights reserved.