Let X be a compact Kähler manifold. Given a big cohomology class , there is a natural equivalence relation on the space of θ-psh functions giving rise to , the space of singularity types of potentials. We introduce a natural pseudo-metric on 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 -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.
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The Dirichlet problem for a complex Monge–Ampère equation,
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T. Darvas, E. Di Nezza and C. H. Lu,
Monotonicity of non-pluripolar products and complex Monge–Ampère equations with prescribed singularity,
Anal. PDE 11 (2018), no. 8, 2049–2087.
J.-P. Demailly and J. Kollár,
Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds,
Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), no. 4, 525–556.
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