Complete Kähler manifolds with zero Ricci curvature. I

Abstract

The problem of constructing complete manifolds with zero Ricci curvature is important for both physicists and geometers.When the manifold is compact and Kahler, this problem was solved satisfactorily by the second author in 1976.While it is not difficult to construct explicit examples of noncompact manifolds with zero Ricci curvature, a complete understanding of complete noncompact manifolds with zero Ricci curvature is still needed. .Therefore, immediately after the work in 1976, the second author proposed a scheme to classify these manifolds.This is the first part of the papers being written by the authors on a systematic research on the existence of these metrics.They have natural applications to algebraic geometry which shall be reported on later.A typical theorem we prove is the following.Let D be a neat, almost ample smooth divisor in a projective manifold M. Let Q be any (1, 1 )-form representing the first Chern class of K~l ® D-1 • Then there is a complete Kahler metric with Q as its Ricci form.(We define a divisor to be neat if no compact algebraic curve in M\D is homologous to a linear sum of curves supported in D.) In particular, if D is the anticanonical divisor, there is a complete Ricci flat Kahler metric on M\D.If K~l ®D-1 is ample, M\D carries a complete Kahler metric with positive Ricci curvature.Naturally, such theorems immediately imply theorems on the topology of M\D.For example, D is connected and the fundamental group of M\D is almost nilpotent.The assumption on the smoothness of D can be removed.When D has normal crossing and higher multiplicity, the situation is more complicated and we shall deal with it later.This paper can be looked upon as a study of the solutions of a global complex Monge-Ampere equation on a Kahler manifold.The uniqueness of such an equation is very interesting and largely unknown.It is presumably related to the automorphism of the manifold.