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

Abstract

Let $(X,\omega )$ be a compact normal Kähler space, with Hodge metric $\omega$. 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 ${\mathcal{H}}_{\omega }(X)$, the space of Kähler potentials, whose completion is the finite energy space ${\mathcal{E}}_{\omega }^1(X)$. 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.

arXiv commentv2 Final version. J. Streets discovered that the approximation result of [EGZ2] has incomplete proof. As a result, the approximation result of [CGZ] has to be used, that applies only in the case of Hodge Kahler metrics. We modified the statements of Theorem 2.1 and Theorem 3.5 accordingly. The main applications regarding canonical Kahler metrics are unchanged