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Geometry and topology of the space of Kähler metrics on singular varieties
- Part of
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- 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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- Article
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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.
