The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group. (English) Zbl 0651.53035

Siu, Yum-Tong

Ann. Math. (2) 127, No. 3, 585-627 (1988).

In this interesting article, the author gives the first existence proof for Kähler-Einstein metrics on some compact complex manifolds with positive first Chern classes. Following an analytic approach, he uses the presence of a large enough symmetry group in order to establish the necessary zero-th order estimate needed to get the continuity method started. (It was known before that this was the missing point to adapt the proof of the Calabi conjecture to that case.) The necessary analytic estimates are rather involved, and require working on appropriate complex curves passing through special points of the manifold with respect to the unknown Kähler potential. Substantial geometric examples are covered by this result, e.g., the Fermat cubic surface given by the equation \(\sum_{0\leq i\leq 3}(z^ i)^ 3=0\) in homogeneous coordinates in \({\mathbb{C}}P^ 3\), and the surface obtained by blowing up three points in \({\mathbb{C}}P^ 2\).

Reviewer: J.P.Bourguignon

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55	Global differential geometry of Hermitian and Kählerian manifolds
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

Keywords:

Kähler-Einstein metrics; complex manifolds; Chern classes; Calabi conjecture; Fermat cubic surface