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The metric geometry of singularity types

  • Tamás Darvas , Eleonora Di Nezza and Hoang-Chinh Lu EMAIL logo
Published/Copyright: July 11, 2020

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 d 𝒮 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 d 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.

Award Identifier / Grant number: DMS-1610202

Award Identifier / Grant number: DMS-1846942 (CAREER)

Funding statement: The first named author has been partially supported by NSF grants DMS-1610202 and DMS-1846942 (CAREER).

Acknowledgements

This work was finished while the three authors participated in the “Research in Paris” program of Institut Henri Poincaré, and we would like to thank the institute for the hospitality and support. We thank the referees for the useful remarks/comments/clarifications which improved the presentation of the paper.

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Received: 2019-09-05
Revised: 2020-03-14
Published Online: 2020-07-11
Published in Print: 2021-02-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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