Stability of the tangent bundle and existence of a Kähler-Einstein metric
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 above conditions, e.g. the quadric in \(\mathbb{P}^ 4\) or the smooth intersection of two quadrics in \(\mathbb{P}^ 5\).
three dimensional smooth complete intersections
scientific article; zbMATH DE number 59940