Boundary behavior of Kähler-Einstein metric of negative Ricci curvature over quasi-projective manifolds with boundary of general type
The article is concerned with the asymptotic boundary behavior of a complete Kähler-Einstein metric of negative Ricci curvature on a quasi-projective algebraic manifold $X$ with a smooth boundary divisor $D$. \textit{R. Kobayashi} showed in [Osaka J. Math. 21, 399--418 (1984; Zbl 0582.32011)] that the ampleness of the log canonical bundle $K_{\overline{X}} +D$ implies the existence of a complete Kähler-Einstein metric on $X$ with a Poincaré growth condition in normal direction, where $X= \overline{X}\backslash D$. A differential geometric version of the adjunction formula $K_D=(K_{\overline{X}} + D)|D$ states that the complete Kähler-Einstein metric on $X$ converges to the Kähler-Einstein metric on $D$ in tangent direction [the reviewer, Math. Ann. 311, No. 4, 631--645 (1998; Zbl 0915.32002)]. A line bundle $L$ is called ample modulo $D$, if the non-ample locus $\mathbf{B}_+(L)$ is contained in $D$. If the log-canonical bundle is nef, big, and ample modulo $D$ (cf. Theorem 2.1) the initial results of \textit{G. Tian} and \textit{S. T. Yau} [Adv. Ser. Math. Phys. 1, 574--629 (1987; Zbl 0682.53064)], \textit{H. Tsuji} [Duke Math. J. 57, No. 2, 537--553 (1988; Zbl 0671.14002)], \textit{S. Bando} [Adv. Stud. Pure Math. 18, No. 2, 105--136 (1990; Zbl 0754.53048)], and \textit{S.-T. Yau} [Commun. Anal. Geom. 1, No. 3, 473--486 (1993; Zbl 0842.53035)] amount to the existence of a unique, singular, complete Kähler-Einstein metric. In this involved situation asymptotics of (almost-)complete, singular Kähler-Einstein metrics, which are also called (log)canonical Kähler-Einstein currents, are investigated and determined by the author. \textbf{Theorem 1.1.} (Kikuta): If $K_{\overline{X}}+D$ is semiample and ample modulo $D$ (in particular big), and big when restricted to $D$, then along the directions tangential to $D$, the (almost) complete Kähler-Einstein metric of negative Ricci curvature on $X$ asymptotically converges to the singular Kähler-Einstein metric on $D$ as currents. Furthermore outside the non-ample locus $\mathbf{B}_+(K_D)$ this convergence holds also in the $C^\infty_{loc}$-topology with a volume given by the log-canonical polarization. The author's work together with results by \textit{R. Kobayashi} [Tohoku Math. J. (2) 37, 43--77 (1984; Zbl 0582.53046)] and \textit{N. Mok} [Prog. Math. 296, 331--354 (2012; Zbl 1253.32006)] resp.\ on the case $dim(X)=2$ and toroidal compactifications resp.\ justify the following conjecture. \textbf{Conjecture} (Kikuta): The asymptotic property holds, if $K_{\overline{X}} + D$ is semiample on $\overline{X}$ and ample modulo $D$. For the proof of the main theorem details of the construction of singular Kähler-Einstein metrics in terms of the Kähler-Ricci flow, in particular on the approach by \textit{J. Lott} and \textit{Z. Zhang} [Duke Math. J. 156, No. 1, 87--123 (2011; Zbl 1248.53050)], \textit{J. Song} and \textit{G. Tian} [J. Am. Math. Soc. 25, No. 2, 303--353 (2012; Zbl 1239.53086)], \textit{V. Tosatti} et al. [Am. J. Math. 140, No. 3, 653--698 (2018; Zbl 1401.53055)], and \textit{A. Chau} [J. Differ. Geom. 66, No. 2, 211--232 (2004; Zbl 1082.53070)] are necessary in the first place. However, stronger convergence properties of the Ricci flow are needed, namely convergence in $C^\infty(K)$ for compact subsets $K\subset X=\overline{X}\backslash D$ (Theorem 3.3). Next, under the extra assumption that $K_D$ is big the convergence is shown to be uniform of class $C^\infty$ outside any neighborhood of $\mathbf{B}_+(K_D)$ in $X$ (Theorem 3.4). For the proof of the asymptotic boundary behavior a suitable reference metric needs to be constructed so that the Monge-Ampere equation can be solved with boundary estimates. In particular the modified Kähler-Ricci flow of \textit{D. Coman} et al. [J. Reine Angew. Math. 676, 33--49 (2013; Zbl 1269.32018)] is being applied. The article concludes with two longer sections containing details of the complicated proofs of the necessary $C^0$- and higher order estimates.
(almost-)complete Kähler-Einstein metric of negative curvature
scientific article; zbMATH DE number 7370872
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