Complete Kähler manifolds with zero Ricci curvature. II
This is a continuation of the previous paper [J. Am. Math. Soc. 3, 579--609 (1990; Zbl 0719.53041)] on settling the non-compact version of Calabi's conjecture on quasiprojective manifolds \(M\) which can be written as \(\overline{M}\setminus D\). The whole arguments in the previous paper are generalized to the case that \(\overline{M}\) is a normal Kähler orbifold and \(D\) is an admissible divisor. The authors construct complete Kähler metrics on \(M=\overline{M}\setminus D\) with prescribed Ricci form in \(C_ 1(-K_{\overline M}-\beta L_ D)\) for \(\beta>1\) under some suitable assumptions on \(\overline{M}\) and \(D\). They find complete Kähler metrics on \(M\) with either zero Ricci curvature of finite topological type or non-negative Ricci curvature. These constructions include practically all known examples of complete Kähler manifolds with zero Ricci curvature of finite topological type. Besides constructing many new examples of such manifolds which may serve as gravitational instantons, these metrics provide a bridge between metric geometry and algebraic geometry of \(M\) because the authors do have some understanding of complete manifolds with non-negative curvature.
scientific article; zbMATH DE number 59791