On the Betti numbers of irreducible compact hyperkähler manifolds of complex dimension four
type: Item
instance of: scholarly article
title: On the Betti numbers of irreducible compact hyperkähler manifolds of complex dimension four
zbMATH Open document ID: 1011.53039
published in: Mathematical Research Letters
publication date: 2003-05-20T00:00:00Z
review text: The author shows that there is only a small number of possibilities for Betti numbers of an irreducible \(4\)-dimensional hyper-Kähler manifold \(M\). All Betti numbers of such a manifold are determined by \(b_2\) and \(b_3\). Examples are known only for \((b_2,b_3)=(23,0)\) or \((7,8)\). The author proves, in particular, that if \(b_2\geq 9\) then \(b_2=23\) and \(b_3=0\). First he proves that \(b_2\leq 23\). The proof is an easy consequence of \textit{S. M. Salamon}'s Riemann-Roch formula [see Topology 35, No. 1, 137-155 (1996; Zbl 0854.58004)] and \textit{M. Verbitsky}'s result saying that \(S^2H^2(M)\hookrightarrow H^4(M)\) [see Geom. Funct. Anal. 6, No. 4, 601-611 (1996; Zbl 0861.53069)]. Then the author uses some results of \textit{N. Hitchin} and \textit{J. Sawon} in [Duke Math. J. 106, 599-615 (2001; Zbl 1024.53032)] and \textit{F. A. Bogomolov} in Geom. Funct. Anal. 6, No. 4, 612-618 (1996; Zbl 0862.53050) to bound \(b_3\).
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 1744224
zbMATH Keywords: hyper-Kähler manifolds
zbMATH Keywords: Betti numbers
author: Daniel Guan
MaRDI profile type: Publication
full work available at URL: mrl.2001.v8.n5.a8
OpenAlex ID: W2113321794
DOI: 10.4310/MRL.2001.V8.N5.A8
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description: scientific article; zbMATH DE number 1744224