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X, e.g. algebraically on a projective manifold. The following is a basic question from geometric invariant theory whose answer is unknown even if.Vis projective. PROBLEM. Classify all T-invariant open subsets U of X such that the geometricquotient U —> U/Texists with U/Ta compact complex space (necessarily algebraic if Xis). In this paper a simple to state and use solution to this problem is given. The classification of U is reduced to finite combinatorics. Associated to the T action on A"is a certain finite 2-complex 'ë(X). Certain {0,1} valued functions, called moment measures, are defined in the set of 2-cells of 'ë(X). There is a natural one-to-one correspondence between the U with compact quotients, U/T, and the moment measures. Let X be a connected compact Kahler manifold with a meromorphic action of F « (C*)k: T X X- * X. The following is a basic question from geometric invariant theory whose answer i

Year: 2013

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