Classification of compact transformation groups on complex quadrics with codimension one orbits
type: Item
instance of: scholarly article
title: Classification of compact transformation groups on complex quadrics with codimension one orbits
zbMATH Open document ID: 1170.57028
author: Shintaro Kuroki
published in: Osaka Journal of Mathematics
publication date: 2009-04-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1235574038
review text: A rational cohomology complex quadric is a simply connected closed manifold \(M\) of dimension \(2r\) which has the same cohomology ring with \(\mathbb Q\) coefficients as the compact quadric \(Q_r=\{ z=[z_0:z_1:\dots:z_{r+1}]\in{\mathbf P}_{r+1}(\mathbb C)\mid z_0^2+z_1^2+\dots+z_{r+1}^2=0\}\). A pair \((G,M)\) denotes a smooth \(G\)-action on \(M\). Two pairs \((G,M)\) and \((G',M')\) are essentially isomorphic if the induced effective actions are equivariantly diffeomorphic. In this paper the author obtains the classification of compact Lie group actions with codimension one principal orbits on a rational cohomology quadric of real dimension \(4n\), for~\(n>1\): Theorem. Let \(M\) be a rational cohomology complex quadric of real dimension \(4n\) (\(n\geq 2\)) and let \(G\) be a compact connected Lie group. If \((G,M)\) has codimension one principal orbits, then \((G,M)\) is essentially isomorphic to one of the pairs in the following list: \begin{tabular}{|c|c|c|c|} \hline \(n\) & \(G\) & \(M\) & action \\ \hline \(n\geq 2\) & \(\mathrm{SO}(2n+1)\) & \(Q_{2n}\) & \(\mathrm{SO}(2n+1) \to \mathrm{SO}(2n+2)\)\\ \(n\geq 2\) & \(\mathrm{SU}(n+1)\) & \(Q_{2n}\) & \(\mathrm{SU}(n+1) \to \mathrm{SO}(2n+2)\)\\ \(n\geq 2\) & \(U(n+1)\) &\(Q_{2n}\) & \(U(n+1)\to \mathrm{SO}(2n+2)\)\\ \(n=2m-1\geq 3\) & \(\mathrm{Sp}(1)\times \mathrm{Sp}(m)\) & \(Q_{4m-2}\) & \(\mathrm{Sp}(1)\times \mathrm{Sp}(m)\to \mathrm{SO}(4m)\)\\ 7 & \(\mathrm{Spin}(9)\) & \(Q_{14}\) & \(\mathrm{Spin}(9)\to \mathrm{SO}(16)\)\\ 3 & \(G_2\) & \(Q_{6}\)& \(G_2 \to \mathrm{SO}(7)\to \mathrm{SO}(8)\)\\ 3 & \(G_2\times T^1\) & \(G_2\times_{\mathrm{SU}(3)}P_3(\mathbb C)\) & $ \begin{cases} G_2\text{ acts on }G_2\text{ canonically} \\ \text{and }T^1\text{ acts on the}\\ \text{first coordinate of}\\ P(\mathbb C\oplus \mathbb C^3)=P_3(\mathbb C) \end{cases} $ \\ 2 & \(\mathrm{Sp}(2)\) & \(S^7\times_{\mathrm{Sp}(1)}P_2(\mathbb C)\) & \(\mathrm{Sp}(2)\text{ acts transitively on }S^7\) \\ \hline \end{tabular} Here \…
reviewed by: Laura Geatti
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57R22
zbMATH DE Number: 5543710
zbMATH Keywords: rational cohomology complex quadric
zbMATH Keywords: classification
zbMATH Keywords: compact Lie group actions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5543710