Metrics of nonnegative scalar curvature on surfaces of Kähler type
type: Item
instance of: scholarly article
title: Metrics of nonnegative scalar curvature on surfaces of Kähler type
zbMATH Open document ID: 0897.53030
DOI: 10.1007/BF02940831
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1998-02-08T00:00:00Z
review text: It is known (see \textit{A. L. Besse} [`Einstein manifolds' (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 10, Springer Verlag, Berlin etc.) (1987; Zbl 0613.53001)]) that every compact manifold of dimension \(n\geq 3\) carries a Riemannian metric of (constant) negative scalar curvature, but in general there are obstructions for the existence of metrics of positive scalar curvature. Using Seiberg-Witten invariants, the author proves that the only compact complex surfaces \(X\) with \(\text{kod} (X)\geq 0\) and \(b_1(X)\) even, which carry a Riemannian metric of nonnegative scalar curvature, are minimal with \(\text{kod} (X)=0\). In this case, every such metric is Ricci-flat.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1114174
zbMATH Keywords: Riemannian metrics
zbMATH Keywords: Ricci-flat metrics
zbMATH Keywords: Seiberg-Witten theory
zbMATH Keywords: compact complex surfaces
zbMATH Keywords: nonnegative scalar curvature
author: Paul Lupascu
reviewed by: Alberto Cavicchioli
MaRDI profile type: Publication
cites work: Q4175362
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description: scientific article; zbMATH DE number 1114174