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Regularizing Properties of the Kähler–Ricci Flow

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An Introduction to the Kähler-Ricci Flow

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2086))

Abstract

These notes present a general existence result for degenerate parabolic complex Monge–Ampère equations with continuous initial data, slightly generalizing the work of Song and Tian on this topic. This result is applied to construct a Kähler–Ricci flow on varieties with log terminal singularities, in connection with the Minimal Model Program. The same circle of ideas is also used to prove a regularity result for elliptic complex Monge–Ampère equations, following Székelyhidi–Tosatti.

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Notes

  1. 1.

    The authors state their result assuming that ψ 0 is merely bounded, but they use in an essential way the continuity of ψ 0 , which is nevertheless known in this context by Kołodziej [Kol98].

  2. 2.

    This of course assumes that c 1(X) has a definite sign.

  3. 3.

    In the Kähler–Einstein Fano case, a celebrated result of Bando and Mabuchi [BM87] asserts that any two solutions are connected by the flow of a holomorphic vector field.

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, CNRS-Université Pierre et Marie Curie, 4 place Jussieu, 75251, Paris, France

    Sébastien Boucksom

  2. Institut de Mathématiques de Toulouse and Institut Universitaire de France, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Vincent Guedj

Authors
  1. Sébastien Boucksom
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  2. Vincent Guedj
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Correspondence to Sébastien Boucksom .

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Editors and Affiliations

  1. Institut de Mathématiques de Jussieu, CNRS-Université Pierre et Marie Curie, Paris, France

    Sebastien Boucksom

  2. Institut Fourier and Institut Universitaire de France, Université Joseph Fourier, Saint-Martin d'Hères, France

    Philippe Eyssidieux

  3. Institut de Mathématiques de Toulouse and Institut Universitaire de France, Université Paul Sabatier, Toulouse, France

    Vincent Guedj

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Boucksom, S., Guedj, V. (2013). Regularizing Properties of the Kähler–Ricci Flow. In: Boucksom, S., Eyssidieux, P., Guedj, V. (eds) An Introduction to the Kähler-Ricci Flow. Lecture Notes in Mathematics, vol 2086. Springer, Cham. https://doi.org/10.1007/978-3-319-00819-6_4

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