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Title:
The space of Kähler metrics on singular varieties
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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. Mabuchi then proposed a Riemannian structure on the space of Kähler metrics under which it (formally) becomes a non-positively curved infinite dimensional space. Chen later proved that this is a metric space of non-positive curvature in the sense of Alexandrov; its metric completion was characterized only recently by Darvas.
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