Comparison of K\"{a}hler quotients of torus actions

Abstract

Let $T$ be a torus with the complexification $T^{\mathbb{C}}$ and $(X, ds^{2})$ a compact K\"{a}hler Hamiltonian $T$-manifold with the moment map $\Phi$ such that $T^{\mathbb{C}}$ acts on $X$ holomorphically. For each $\alpha$ in the moment body $\Phi(X)$, the K\"{a}hler quotient $X_{\alpha}=\Phi^{-1}(\alpha)/T$ is a reduced normal complex analytic space admitting a unique K\"{a}hler structure $\kappa_{\alpha}$ induced from $ds^{2}$. Inspired by the theory of variation of Geometric Invariant Theory, when $\alpha$ moves from a subpolytope (a connected component of the set of regular values of $\Phi$) to another one in the interior of $\Phi(X)$, we show that the quotient $X_{\alpha}$ undergoes a bimeromorphic transformation, and this enables us to compare the K\"{a}hler classes of the different quotients. In particular, as applications, we prove that each nondegenerate singular K\"{a}hler quotient has a partial and rational desingularisation which is obtained by shifting the moment map; moreover, we obtain a formula on the Riemann--Roch numbers of singular K\"{a}hler quotients.