Rigidity of certain finite group actions on the complex projective plane
type: Item
instance of: scholarly article
title: Rigidity of certain finite group actions on the complex projective plane
zbMATH Open document ID: 0695.57025
DOI: 10.1007/BF02564697
author: R. Lee
author: Ib Madsen
author: Ian Hambleton
published in: Commentarii Mathematici Helvetici
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140174
review text: According to \textit{I. Hambleton} and \textit{R. Lee} [J. Algebra 116, 227-242 (1988; Zbl 0659.20041)] and \textit{D. M. Wilczyński} [Trans. Am. Math. Soc. 303, 707-731 (1987; Zbl 0639.57022)], the finite groups G which act locally linearly on the complex projective plane \(P^ 2({\mathbb{C}})\), trivially on homology, are the subgroups of \(PGL_ 3({\mathbb{C}})\). Moreover, the acting groups can also act linearly (i.e., via projective representations) on \(P^ 2({\mathbb{C}}).\) The authors of the paper under review deal with the rigidity conjecture asserting that each locally linear action of G on \(P^ 2({\mathbb{C}})\), trivial on homology, is topologically conjugate to a linear action. Their main result asserts that the rigidity conjecture holds if the action has a fixed point at which the local representation is free and the action fulfills some assumptions on the singular set. The authors point out that it follows from the work of \textit{A. L. Edmonds} and \textit{J. H. Ewing} [Topology 28, 211-223 (1989)] that the rigidity conjecture is true just by assuming that the action has a fixed point at which the local representation is free. Also, using the existence of a knotted 2-sphere in \(S^ 4\), invariant under cyclic group actions, described by \textit{C. H. Giffen} [Am. J. Math. 88, 187-198 (1966; Zbl 0141.212)], the authors present a method for constructing non-linear group actions on \(P^ 2({\mathbb{C}})\). Wilczyński [loc. cit.] has characterized all finite groups acting locally linearly on the Chern manifold, the only topological four- manifold homotopy equivalent, but not homeomorphic, to \(P^ 2({\mathbb{C}})\). It turns out that fewer groups can act locally linearly on the Chern manifold than on \(P^ 2({\mathbb{C}})\). Moreover, according to \textit{D. M. Wilczyński} [Math. Ann. 281, 333-339 (1988; Zbl 0627.57025)], the actions on the Chern manifold are rigid in the sense that they have, so called, projective models which are unique up to linear equival…
reviewed by: Krzysztof Pawałowski
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 4138802
zbMATH Keywords: isolated fixed point
zbMATH Keywords: weakly complex actions
zbMATH Keywords: locally linear action on \(P^ 2({\mathbb{C}})\)
zbMATH Keywords: topologically conjugate to a linear action
zbMATH Keywords: rigidity conjecture
zbMATH Keywords: non-linear group actions on \(P^ 2({\mathbb{C}})\)
zbMATH Keywords: actions on the Chern manifold
MaRDI profile type: Publication
OpenAlex ID: W1995385808
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description: scientific article; zbMATH DE number 4138802