Compactification of complete Kähler surfaces with negative Ricci curvature (Q1263819)

The author generalizes a theorem of N. Mok about the compactification of complete Kähler surfaces of finite volume satisfying certain curvature conditions [\textit{N. Mok}, Ann. Math., II. Ser. 129, No.2, 383-425 (1989; Zbl 0672.32012)] by dropping Mok's assumption, that X is of seminegative sectional curvature. The main result of the paper is as follows: Let (X,g) be a complete Kähler surface of finite volume and negative Ricci curvature. If the sectional curvatures are bounded, then X is biholomorphically equivalent to a Zariski-open subset of a projective algebraic surface. The analogue of this theorem was proved to be true in all dimensions by \textit{N. Mok} and \textit{J.-Q. Zhong} under the additional assumption that X is homotopic to a finite CW-complex [Ann. Math., II. Ser. 129, No.3, 427- 470 (1989)].