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arXiv · 1604.07127

Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics

Tamás Darvas
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Abstract

Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the finite energy space $\mathcal E^1_ω(X)$. Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of Kähler-Einstein metrics on log Fano pairs is equivalent to properness of the K-energy in a suitable sense. To our knowledge, this result represents the first characterization of general log Fano pairs admitting Kähler-Einstein metrics. We also discuss the analogous result for Kähler-Ricci solitons on Fano varieties.

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Tamás Darvas. 2016-10-25. Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics. https://arxiv.org/abs/1604.07127

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