Every K 3 surface is Kähler
type: Item
instance of: scholarly article
title: Every K 3 surface is Kähler
zbMATH Open document ID: 0557.32004
DOI: 10.1007/BF01393829
author: Yum-Tong Siu
published in: Inventiones Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143039
review text: Kodaira made the conjecture that every compact complex surface with even first Betti number is Kähler. The only open case was that of K-3 surfaces; so the result of this paper completes the confirmation of Kodaira's conjecture. A previous proof by \textit{A. N. Todorov} [Invent. Math. 61, 251-265 (1980; Zbl 0472.14006)] of the same result presents some serious gaps so that this is the first complete proof. Two important ingredients of the approach followed here are the surjectivity of the period map for K-3 surfaces due to Todorov (loc. cit.) and \textit{E. Looijenga} [Geometry, Proc. Symp., Utrecht 1980, Lect. Notes Math. 894, 107-112 (1981; Zbl 0473.53041)] and the global Torelli theorem for Kähler K-3 surfaces due to \textit{D. Burns} jun. and \textit{M. Rapoport} [Ann. Sci. Éc. Norm. Super., IV. Sér. 8, 235-273 (1975; Zbl 0324.14008)]. One has to prove that a K-3 surface which has the same periods as a Kähler K-3 surface is necessarily analytically isomorphic to it. The program has serious difficulties that the author overcomes with deep and subtle arguments. First he proves the existence of a closed 2-form \(\xi\) on any K-3 surface M with positive definite (1,1)-component. Then that the orthogonal projections \(\theta\) to \(H_ R^{1,1}(M)\) (with respect to the quadratic form defined by the cup product) of the cohomology class of \(\xi\) belongs to the positive cone of \(H_ R^{1,1}(M)\) and moreover if M is Kähler \(\theta\) lies in the same component of the positive cone of \(H_ R^{1,1}(M)\) as any Kähler class of M. The derived result is obtained from the previous facts and a clever use of the before recalled local and global Torelli theorems for K-3 surfaces.
reviewed by: F.Gherardelli
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14J15
zbMATH DE Number: 3888045
zbMATH Keywords: every K3 surface is Kähler
zbMATH Keywords: Kodaira conjecture
zbMATH Keywords: compact complex surface with even first Betti number
zbMATH Keywords: surjectivity of the period map for K-3 surfaces
zbMATH Keywords: global Torelli theorem
Wikidata QID: Q56386934
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3888045