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Kähler-Einstein metrics and integral invariants. (English) Zbl 0646.53045
Lecture Notes in Mathematics, 1314. Berlin etc.: Springer-Verlag. iv, 140 p. (1988).
These notes mainly concern the nonexistence and existence problem of Kähler-Einstein metrics on compact complex manifolds of positive first Chern class. There are three known obstructions to the existence of Kähler-Einstein metrics on those manifolds. The first two were found by Y. Matsushima [Nagoya Math. J. 11, 145-150 (1957; Zbl 0091.348)] and the author himself [Invent. Math. 73, 437-443 (1983; Zbl 0506.53030)], respectively and are related to the structure of the automorphism groups of those manifolds. The third one is the stability of the tangent bundles due to S. Kobayashi [Proc. Japan Acad., Ser. A 58, 158-162 (1982; Zbl 0546.53041)] and M. Lübke [Manuscr. Math. 42, 245-257 (1983; Zbl 0558.53037)].
In these notes, the author gathers recent results on the first two with more details on the one due to the author. The author’s obstruction is defined as a linear map f from the Lie algebra of the automorphism group to \({\mathbb{C}}\) on a compact Kähler manifold with positive first Chern class. He proves that such a manifold admits a Kähler-Einstein metric only if \(f\equiv 0\). In these notes, the author gives the formulation of f, the generalizations of it and the interpretations of it as an obstruction, a classical invariant, a moment map, etc. The author also discusses the uniqueness theorem of S. Bando and T. Mabuchi [Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 11-40 (1987; Zbl 0641.53065)] on Kähler-Einstein metrics with positive scalar curvature and some existence theorems of Kähler- Einstein metrics.
Reviewer: G.Tian

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry