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Complete Kähler manifolds with zero Ricci curvature. I. (English) Zbl 0719.53041
Let (M,g) be a complete Kähler manifold and $$\omega_ g$$ be the Kähler form associated to the metric g. Consider the following complex Monge-Ampère equation on M: $(*)\quad (\omega_ g+\frac{\sqrt{- 1}}{2\pi}\partial {\bar \partial}\phi)^ n=e^ f\omega^ n_ g,\quad \omega_ g+\frac{\sqrt{-1}}{2\pi}\partial {\bar \partial}\phi >0,\quad \phi \in C^{\infty}(M,{\mathbb{R}}^ 1),$ where $$\omega^ n_ g=\omega_ g\wedge...\wedge \omega_ g$$ and f is a given smooth function satisfying the integrability condition $\int_{M}(e^ f- 1)\omega^ n_ g=0.$ The main purpose of this paper is to study the solvability of (*) when M is noncompact. The authors prove an existence theorem for (*) under certain assumptions of the decay of f at infinity. This existence theorem is applied to construct complex Ricci flat metrics and complete Kähler metrics with positive Ricci curvature on many complete Kähler manifolds. A typical theorem the authors prove is the following. Let D be a neat, almost ample smooth divisor in a projective manifold $$\bar M.$$ Let $$\Omega$$ be any (1,1)-form representing the first Chern class of $$K_{\bar M}^{-1}\otimes D^{-1}$$. Then there is a complete Kähler metric with $$\Omega$$ as its Ricci form.