Stability of the tangent bundle and existence of a Kähler-Einstein metric
type: Item
instance of: scholarly article
title: Stability of the tangent bundle and existence of a Kähler-Einstein metric
zbMATH Open document ID: 0766.14009
DOI: 10.1007/BF01445227
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164886
review text: The goal of this paper is to answer the following question: Under what conditions is the tangent bundle of a complex projective manifold \(X\) stable? --- Thanks to results of Kobayashi, Lübke, Uhlenbeck/Yau and \textit{S. K. Donaldson} [see e.g. Duke Math. J. 54, 231-247 (1987; Zbl 0627.53052)] this is equivalent to the existence of a Kähler-Einstein metric on \(X\) (if the tangent bundle is irreducible). For a survey on this problem see e.g. \textit{G. Tian} [Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 587-598 (1991; Zbl 0747.53038)]. --- Semi-stability (and the existence of a Kähler-Einstein metric) is known if the canonical divisor is nef [see \textit{S.-T. Yau}, Commun. Pure Appl. Math. 31, 339-411 (1978; Zbl 0362.53049) and \textit{H. Tsuji}, Topology 27, No. 4, 429-442 (1988; Zbl 0698.14008)]. The author studies the case where the anticanonical divisor is ample, i.e. \(X\) is a Fano manifold. The two- dimensional case (Del Pezzo surfaces) was treated by \textit{R. Fahlaoui} [Math. Ann. 283, No. 1, 171-176 (1989; Zbl 0672.14009)] and \textit{G. Tian} and \textit{S.-T. Yau} [Commun. Math. Phys. 112, 175-203 (1987; Zbl 0631.53052)]. For arbitrary dimension the case of Fermat hypersurfaces of degree \(n-1\) and \(n\) was investigated by \textit{G. Tian} [Invent. math. 89, 225-246 (1987; Zbl 0599.53046)]. He also investigated certain complement intersections of such hypersurfaces in his thesis (Harvard). In the paper under review the author shows the stability of the tangent bundle for at least three dimensional smooth complete intersections of multidegree \((d_ 1,\dots,d_ k)\) in projective \(n\)-space under the assumption: \(d_ 1\geq d_ 2\geq\cdots\geq d_ k\) and \(d_ 1+d_ 2+\cdots+d_{k-1}+2d_ k>n+1\). The theorem is deduced from the Akizuki- Nakano vanishing theorem using standard methods from vector bundle theory and induction on \(k\). --- Unfortunately, even in the three-dimensional case not all Fano manifolds with Picard number one are covered by the…
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 32Q20
zbMATH DE Number: 59940
zbMATH Keywords: hypersurface
zbMATH Keywords: existence of a Kähler-Einstein metric
zbMATH Keywords: Fano manifold
zbMATH Keywords: stability of the tangent bundle
zbMATH Keywords: three dimensional smooth complete intersections
author: S. Subramanian
reviewed by: Bernd Kreußler
MaRDI profile type: Publication
cites work: Q3657016
cites work: Infinite determinants, stable bundles and curvature
cites work: Q4143433
cites work: Stability of Einstein-Hermitian vector bundles
cites work: Q3767949
cites work: On Kähler-Einstein metrics on certain Kähler manifolds with \(C_ 1(M)>0\)
cites work: Stability of tangent bundles of minimal algebraic varieties
description: scientific article; zbMATH DE number 59940