Rigidity of certain finite group actions on the complex projective plane
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 equivalence, and which provide the topological classification of the actions. A projective model for the G-action on the Chern manifold consists of a linear G- action on \(P^ 2({\mathbb{C}})\) and a G-homotopy equivalence from the Chern manifold to \(P^ 2({\mathbb{C}})\). Therefore, the examples of non-linear actions on \(P^ 2({\mathbb{C}})\) described by the authors provide further evidence to the fact that \(P^ 2({\mathbb{C}})\) admits more actions than the Chern manifold does.
locally linear action on \(P^ 2({\mathbb{C}})\)
topologically conjugate to a linear action
non-linear group actions on \(P^ 2({\mathbb{C}})\)
scientific article; zbMATH DE number 4138802