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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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