On local automorphisms of certain quadrics of codimension 2
Quadrics of the form \(M=\{(z,\zeta)\in\mathbb{C}^{n+k}|\text{Im} \zeta_ j=(\omega^ jz,z)\), \(j=1,\ldots,k\}\), where \((\omega^ jz,z)=\sum_{\mu,\nu=1}^ n\omega_{\mu\nu}^ jz_ \mu\overline{z_ \nu}\) are Hermitian linear forms were considered by \textit{A. E. Tumanov} and \textit{G. M. Khenkin} in Funct. Anal. Appl. 17, 285-294 (1983); translation from Funkts. Anal. Prilozh. 17, No. 4, 49-61 (1983; Zbl 0572.32018). Let \(\Aut_{w_ 0}M\) denote the group of local automorphism of \(M\) fixing a given point \(w_ 0=(z_ 0,\zeta_ 0)\). In the above paper [loc. cit.] it was conjectured that \(\Aut_{w_ 0}M\) consists of rational automorphisms. This was later proved by \textit{A. E. Tumanov} [Math. USSSR, Izv. 32, No. 3, 655-662 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 3, 651-659 (1988; Zbl 0655.32026)]. In the present paper the case \(k=2\) is considered and the following refinement is proved. Suppose \(\omega^ j:\mathbb{C}^ n\to\mathbb{C}^ n\) are hermitian linear operators for \(j=1,2\). Assume \(\omega^ 1\) is invertible and \(A:=(\omega^ 1)^{-1}\omega^ 2\) has more than two distinct eigenvalues. Then every automorphism in \(\Aut_{w_ 0}M\) is linear. Two examples are also given which show that the result is false if the operator \(A\) does not have more than two distinct eigenvalues.
scientific article; zbMATH DE number 148607