Quotients by $\textbf {C}^\ast \times \textbf {C}^\ast$ actions

Abstract:

Let $T \approx {{\mathbf {C}}^\ast } \times {{\mathbf {C}}^\ast }$ act meromorphically on a compact Kähler manifold $X$, e.g. algebraically on a projective manifold. The following is a basic question from geometric invariant theory whose answer is unknown even if $X$ is projective. PROBLEM. Classify all $T$-invariant open subsets $U$ of $X$ such that the geometric quotient $U \to U/T$ exists with $U/T$ a compact complex space (necessarily algebraic if $X$ is). 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 $X$ is a certain finite $2$-complex $\mathcal {C}(X)$. Certain $\{ 0,1\}$ valued functions, called moment measures, are defined in the set of $2$-cells of $\mathcal {C}(X)$. There is a natural one-to-one correspondence between the $U$ with compact quotients, $U/T$, and the moment measures.