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Asymptotics of complete Kähler-Einstein metrics – negativity of the holomorphic sectional curvature. (English) Zbl 1022.32008
Complete Kähler-Einstein metrics $$ds_{X}^{2}$$ of constant Ricci curvature on quasiprojective varieties are of interest in various geometric situations.
Theorem 1. Let $$\overline{X}$$ be a compact complex surface, $$C \subset \overline{X}$$ a smooth divisor satisfying the condition $K_{\overline{X}} + C > 0.$ Then the holomorphic sectional curvature of the complete Kähler-Einstein metric on $$X = \overline{X}\mathbb C$$ is bounded from above by a negative constant near the compactifying divisor. The sectional and holomorphic bisectional curvature are bounded on $$X.$$
Theorem 2. The Kähler-Einstein metric $$\omega_{X}$$ converges to the Kähler-Einstein metric $$\omega_{C},$$ when restricted to directions parallel to $$C.$$

MSC:
 32Q05 Negative curvature complex manifolds 32Q20 Kähler-Einstein manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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