The metric geometry of singularity types

TL;DR

This paper introduces a new metric on the space of singularity types of potentials on compact Kähler manifolds, proving its properties and applications to complex Monge-Ampère equations and multiplier ideal sheaves.

Contribution

It defines a natural pseudometric on the space of singularity types, analyzes its properties, and applies it to convergence of solutions and semicontinuity of multiplier ideal sheaves.

Findings

The metric space of singularity types is complete with positive mass.

Solutions to Monge-Ampère equations converge in the $d_ ext{S}$-topology.

Semicontinuity of multiplier ideal sheaves is established.

Abstract

Let $X$ be a compact K\"ahler manifold. Given a big cohomology class $\{\theta\}$, there is a natural equivalence relation on the space of $\theta$-psh functions giving rise to $\mathcal S(X,\theta)$, the space of singularity types of potentials. We introduce a natural pseudometric $d_{\mathcal {S}}$ on $\mathcal S(X,\theta)$ that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge-Amp\`ere equations with varying singularity type converge as governed by the $d_\mathcal S$-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.