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On Calabi’s conjecture for complex surfaces with positive first Chern class. (English) Zbl 0716.32019
This substantial article closes the question on the existence of Kähler-Einstein metrics on compact complex surfaces $$F$$ with positive first Chern class: $$F$$ admits a Kähler-Einstein metric if (and only if) the Lie algebra $$\text{Lie}(F)$$ of holomorphic vector fields on $$F$$ is reductive. (By a theorem of Y. Matsushima, Nagoya Math. J. 11, 145–150 (1957; Zbl 0091.34803) a compact complex manifold $$M$$ admits a Kähler-Einstein metric only if $$\text{Lie}(M)$$ is reductive.)
It is known that $$F$$ is biholomorphically equivalent either to $${\mathbb P}_ 2$$, $${\mathbb P}_ 1\times {\mathbb P}_ 1$$ or to a surface $$F_ r$$ arising by blowing up $${\mathbb P}_ 2$$ in $$r$$ points in general position, $$1\leq r\leq 8$$. Since $$\text{Lie}(F_ r)$$ is not reductive for $$1\leq r\leq 3$$ and since $$\text{Lie}(F_ r)=\{0\}$$ for $$4\leq r\leq 8$$, $$F$$ admits a Kähler-Einstein metric iff $$F\in \{{\mathbb P}_ 2,{\mathbb P}_ 1\times {\mathbb P}_ 1, F_ r, 4\leq r\leq 8\}$$.
The proof of the main theorem is based on a partial $$C^ 0$$-estimate for the solutions of some complex Monge-Ampère equations. It uses and extends previous results of the author [Invent. Math. 89, 225–246 (1987; Zbl 0599.53046)] and of his joint work with S. T. Yau [Commun. Math. Phys. 112, 175–203 (1987; Zbl 0631.53052)]. The methods in the proof of the main theorem also yield results on the degeneration of Kähler-Einstein surfaces, which are interesting in itself.

##### MSC:
 32J15 Compact complex surfaces 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32W20 Complex Monge-Ampère operators 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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