### Geometry and Topology of the space of Kähler metrics on singular varieties

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 Title: Geometry and Topology of the space of Kähler metrics on singular varieties Authors: Nezza, EDGuedj, V Item Type: Working Paper Abstract: Let $Y$ be a compact K\"ahler normal space and $\alpha \in H^{1,1}(Y,\mathbb{R})$ a K\"ahler class. We study metric properties of the space $\mathcal{H}_\alpha$ of K\"ahler metrics in $\alpha$ using Mabuchi geodesics. We extend several results by Calabi, Chen, Darvas previously established when the underlying space is smooth. As an application we analytically characterize the existence of K\"ahler-Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties. Issue Date: 31-Aug-2018 URI: http://hdl.handle.net/10044/1/53416 Sponsor/Funder: Commission of the European Communities Funder's Grant Number: Marie Skłodowska-Curie Action 660940 — KRF-CY (MSCA–IF) Keywords: math.DGmath.CV Notes: 44 pages. arXiv admin note: substantial text overlap with arXiv:1401.7857. v2: Section 1 and the statement of Theorem B have been modified Appears in Collections: Faculty of Natural Sciences