The metric geometry of singularity types

  • 1 Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park,, USA
  • 2 Sorbonne Université, 4 place Jussieu, Paris, France
  • 3 Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Bât. 307 (IMO), Orsay, France
Tamás Darvas, Eleonora Di Nezza and Hoang-Chinh Lu
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  • Laboratoire de Mathématiques d’Orsay, Bât. 307 (IMO), Université Paris-Saclay, 91405, Orsay, Cedex, France
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Abstract

Let X be a compact Kähler manifold. Given a big cohomology class {θ}, there is a natural equivalence relation on the space of θ-psh functions giving rise to 𝒮(X,θ), the space of singularity types of potentials. We introduce a natural pseudo-metric d𝒮 on 𝒮(X,θ) 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 d𝒮-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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