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2 results

  • Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

  • Part of
  • Kyle Broder
  • Journal:
    Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
    Published online by Cambridge University Press:
    05 January 2023, pp. 912-926
    Print publication:
    September 2023
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  • We show that the singularities of the twisted Kähler–Einstein metric arising as the longtime solution of the Kähler–Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics are intimately related to the holomorphic sectional curvature of reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross, Tosatti, and Zhang (2020, Preprint, arXiv:1911.07315) with explicit constants appearing in the divisorial pole.

  • Geometry and topology of the space of Kähler metrics on singular varieties

  • Part of
  • Eleonora Di Nezza, Vincent Guedj
  • Journal:
    Compositio Mathematica / Volume 154 / Issue 8 / August 2018
    Published online by Cambridge University Press:
    19 July 2018, pp. 1593-1632
    Print publication:
    August 2018
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  • Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.