Complete Kähler manifolds with zero Ricci curvature. II
type: Item
instance of: scholarly article
title: Complete Kähler manifolds with zero Ricci curvature. II
zbMATH Open document ID: 0766.53053
DOI: 10.1007/BF01243902
published in: Inventiones Mathematicae
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/143931
review text: 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.
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 59791
zbMATH Keywords: quasiprojective manifold
zbMATH Keywords: Calabi conjecture
zbMATH Keywords: Ricci curvature
zbMATH Keywords: gravitational instantons
author: Gang Tian
author: Shing-Tung Yau
reviewed by: Neda Bokan
MaRDI profile type: Publication
Wikidata QID: Q56144934
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OpenAlex ID: W3015305294
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description: scientific article; zbMATH DE number 59791