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

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 o...

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Bibliographic Details
Published in:Compositio mathematica 2018-08, Vol.154 (8), p.1593-1632
Main Authors: Di Nezza, Eleonora, Guedj, Vincent
Format: Article
Language:English
Subjects:
Geodesy
Mathematics
Topology
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Summary: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.
ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X18007170