Type: Article
Publication Date: 2020-07-11
Citations: 46
DOI: https://doi.org/10.1515/crelle-2020-0019
Abstract Let X be a compact Kähler manifold. Given a big cohomology class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{\theta\}} , there is a natural equivalence relation on the space of θ-psh functions giving rise to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒮</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{S}(X,\theta)} , the space of singularity types of potentials. We introduce a natural pseudo-metric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mi mathvariant="script">𝒮</m:mi> </m:msub> </m:math> {d_{\mathcal{S}}} on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒮</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\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–Ampère equations with varying singularity type converge as governed by the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mi mathvariant="script">𝒮</m:mi> </m:msub> </m:math> {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.