Classification of compact transformation groups on complex quadrics with codimension one orbits
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 \(G_2\times_{\text{SU}(3)}P_3(\mathbb C)\) denotes the quotient of \(G_2\times P_3(\mathbb C)\) by the diagonal \(\text{SU}(3)\)-action, where \(\text{SU}(3)\) acts on \(G_2\) canonically and on \(P_3(\mathbb C)\) by \(A([z_0:z_1])=[z_0:Az_1]\), for \([z_0:z_1]\in P(\mathbb C\oplus \mathbb C^3)=P_3(\mathbb C)\) and \(A\in \text{SU}(3)\). Also, \(S^7\times_{\text{Sp}(1)}P_2(\mathbb C)\) denotes the quotient of \(S^7\times P_2(\mathbb C)\) by the diagonal \(\text{Sp}(1)\)-action, where \(\text{Sp}(1)\) acts on \(S^7\) canonically and on \(P_2(\mathbb C)\) through a double covering \(\text{Sp}(1)\to \text{SO}(3)\). The author also shows that the underlying manifold \(M=G_2\times_{\text{SU}(3)}P_3(\mathbb C)\) is diffeomorphic to the genuine quadric \(Q_6\), while \(M=S^7\times_{\text{Sp}(1)}P_2(\mathbb C)\) is not diffeomorphic to \(Q_4\). When \(r\) is odd the quadric \(Q_r\) is a rational cohomology complex projective space. Compact Lie group actions with codimension one principal orbits on rational cohomology complex projective spaces were classified by \textit{F. Uchida} [Japan J. Math. (N.S.) 3, 141--189 (1977; Zbl 0374.57009)]; the cases of rational cohomology quaternionic projective spaces and of rational cohomology Cayley projective planes were treated by \textit{K. Iwata} [Osaka J. Math. 15, 475--508 (1978; Zbl 0415.57026) and Tôhoku J. Math. (2) 33, 429--442 (1981; Zbl 0506.57024)].
scientific article; zbMATH DE number 5543710
scientific article