×

Lectures on Hermitian-Einstein metrics for stable bundles and Kähler- Einstein metrics. Delivered at the German Mathematical Society Seminar in Düsseldorf (FRG) in June, 1986. (English) Zbl 0631.53004

DMV Seminar, Bd. 8. Basel-Boston: Birkhäuser Verlag. 171 p.; DM 48.00 (1987).
This book covers the lectures on Hermitian-Einstein or Kähler-Einstein metrics delivered by the author at the German Mathematical Society Seminar in Düsseldorf in 1986. Although it consists only of 171 pages, its presentation is not merely reasonably self-contained, but also very well-organized, and moreover the wide view of the author lifts it far above the level of lecture notes.
The whole volume is divided into five rather independent chapters and they are summarized as follows: (1) The first chapter is an expository account of the recent result of Donaldson on the existence and uniqueness of Hermitian-Einstein metrics for stable vector bundles over projective algebraic manifolds. (2) The second chapter is devoted to the solution, by Aubin-Calabi-Yau, of Calabi’s conjecture on the existence and uniqueness of Kähler-Einstein metrics in the case of negative or zero anticanonical class, where the original third-order a priori estimate is replaced by the \(C^{2,\alpha}\)-estimate of Evans and Trudinger. (3) The third chapter deals with the uniqueness result, up to biholomorphisms, by Bando-Mabuchi for Kähler-Einstein metrics in the case of positive anticanonical class.
(4) The existence problem of Kähler-Einstein metrics is still open in the case of positive anticanonical class, and two obstructions to the existence are known. In the fourth chapter, the obstructions by Matsushima-Lichnerowicz and Futaki-Kazdan-Warner are discussed. (5) The last chapter is devoted to the author’s existence result for Kähler- Einstein metrics in the case where the manifold with positive anticanonical class admits a suitable finite symmetry. In particular, such existence can be shown for (i) suitable Fermat hypersurfaces or (ii) the surface obtained by blowing up three points of \({\mathbb{P}}^ 2({\mathbb{C}})\), which where, just recently, independently proved by Tian and Yau.
Reviewer: T.Mabuchi

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
32Q99 Complex manifolds
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds