Comparison of Kähler quotients of torus actions

Abstract

Let $T$ be a torus with the complexification $T^{\mathbb{C}}$ and $(X, ds^{2})$ a compact Kähler Hamiltonian $T$-manifold with the moment map $Φ$ such that $T^{\mathbb{C}}$ acts on $X$ holomorphically. For each $α$ in the moment body $Φ(X)$, the Kähler quotient $X_α=Φ^{-1}(α)/T$ is a reduced normal complex analytic space admitting a unique Kähler structure $κ_α$ induced from $ds^{2}$. Inspired by the theory of variation of Geometric Invariant Theory, when $α$ moves from a subpolytope (a connected component of the set of regular values of $Φ$) to another one in the interior of $Φ(X)$, we show that the quotient $X_α$ undergoes a bimeromorphic transformation, and this enables us to compare the Kähler classes of the different quotients. In particular, as applications, we prove that each nondegenerate singular Kähler 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 Kähler quotients.