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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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