Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties
Let \(X\) be a Fano manifold, that is a complex projective manifold such that the anticanonical divisor \(-K_X\) is ample. By the work of \textit{X. Chen} et al. [J. Am. Math. Soc. 28, No. 1, 199--234 (2015; Zbl 1312.53097)] we know that \(X\) admits a Kähler-Einstein metric if and only if \(X\) is \(K\)-stable. In order to define compactifications of moduli spaces of Kähler-Einstein Fano manifolds, it is necessary to generalize the theory to varieties with some mild singularities. In this paper the authors consider \(\mathbb Q\)-Fano varieties, that is normal projective varieties such that some multiple of the anticanonical divisor \(-K_X\) is Cartier and ample. They show that if \(X\) admits a Kähler-Einstein metric, the singularities of \(X\) are at most log-terminal in the sense of the minimal model program. Then they define a Mabuchi functional Mab and \(J\)-functional \(J\) for log-terminal \(\mathbb Q\)-Fano varieties, and as in the smooth setting, one says that the Mabuchi functional is proper if \(\mbox{Mab} \to \infty\) if \(J \to \infty\). The first main result of the paper generalizes classical results of \textit{S. Bando} and \textit{T. Mabuchi} [Adv. Stud. Pure Math. 10, 11--40 (1987; Zbl 0641.53065)] and \textit{W. Ding} and \textit{G. Tian} [Invent. Math. 110, No. 2, 315--335 (1992; Zbl 0779.53044)] to this singular setting: the identity component \(\mbox{Aut}^0(X)\) of the automorphism group acts transitively on the set of Kähler-Einstein metrics. Moreover, if the Mabuchi functional is proper, then \(\mbox{Aut}^0(X)=1\) and the unique minimizer of the Mabuchi functional is a Kähler-Einstein metric \(\omega_{KE}\). \newline In the second part of the paper the authors describe the behaviour of the Ricci iteration (introduced in [\textit{J. Keller}, J. Inst. Math. Jussieu 8, No. 4, 743--768 (2009; Zbl 1178.32020); \textit{Y. A. Rubinstein}, Adv. Math. 218, No. 5, 1526--1565 (2008; Zbl 1143.53065)]) and the Kähler-Ricci flow (see [\textit{J. Song} and \textit{G. Tian}, J. Am. Math. Soc. 25, No. 2, 303--353 (2012; Zbl 1239.53086)]): assume that \(X\) is a \(\mathbb Q\)-Fano variety with log-terminal singularities. Given a a smooth form \(\omega_0\) in the anticanonical class \(c_1(X)\), there exists a unique sequence of closed positive currents \(\omega_j \in c_1(X)\) with continuous potentials such that the restriction to the smooth locus of \(X\) is smooth and satisfies \(\mbox{Ric}(\omega_{j+1})=\omega_j\). Moreover, if the Mabuchi functional is proper, this Ricci iteration converges to \(\omega_{KE}\), uniformly on \(X\) at the level of potentials. Under the same assumption, the normalised Kähler Ricci flow \(\omega_t\) converges to \(\omega_{KE}\) for \(t \to \infty\) and \(\lim_{t \to \infty} \omega_t^n = \omega_{KE}^n\).
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