Seminar Algebra and Geometry: Eleonora Di Nezza (Imperial College London)

The space of Kähler metrics on singular varieties

The geometry and topology of the space of Kähler metrics on a compact Kähler manifold is a classical subject, first systematically studied by Calabi in relation with the existence of extremal Kähler metrics. Then, Mabuchi proposed a Riemannian structure on the space of Kähler metrics under which it (formally) becomes a non-positive curved infinite dimensional space. Chen later proved that this is a metric space of non-positive curvature in the sense of Alexandrov and its metric completion was characterized only recently by Darvas. In this talk we will talk about the extension of such a theory to the setting where the compact Kähler manifold is replaced by a compact singular normal Kähler space. As one application we give an analytical criterion for the existence of Kähler-Einstein metrics on certain mildly singular Fano varieties, an analogous to a criterion in the smooth case due to Darvas and Rubinstein.
This is based on a joint work with Vincent Guedj.

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