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

  • Eleonora Di Nezza (a1) and Vincent Guedj (a2)

Abstract

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.

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Current address: IHES, Université Paris Saclay, 91400 Bures sur Yvette, France email dinezza@ihes.fr

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

  • Eleonora Di Nezza (a1) and Vincent Guedj (a2)

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