Asymptotics of complete Kähler-Einstein metrics -- negativity of the holomorphic sectional curvature (Q1856383)

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 \(C.\)