Boundary behavior of Kähler-Einstein metric of negative Ricci curvature over quasi-projective manifolds with boundary of general type
type: Item
instance of: scholarly article
title: Boundary behavior of Kähler-Einstein metric of negative Ricci curvature over quasi-projective manifolds with boundary of general type
zbMATH Open document ID: 1479.32009
author: Shin Kikuta
published in: Kodai Mathematical Journal
publication date: 2021-07-12T00:00:00Z
review text: 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 curren…
reviewed by: Georg Schumacher
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53E30
zbMATH DE Number: 7370872
zbMATH Keywords: (almost-)complete Kähler-Einstein metric of negative curvature
zbMATH Keywords: Kähler-Ricci flow
zbMATH Keywords: asymptotic boundary behavior
zbMATH Keywords: singular Kähler-Einstein metric
MaRDI profile type: Publication
full work available at URL: kmj44106
OpenAlex ID: W3179605726
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DOI: 10.2996/KMJ44106
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description: scientific article; zbMATH DE number 7370872