Linearity of automorphisms of standard quadrics of codimension \(m\) in \(\mathbb C^{m+n}\) (Q1810269)

Consider the quadric \(M:=\{(z,w)\in\mathbb C^{n+k}\mid (zA_1\overline z^t,\dots,zA_k\overline z^t)=\Im w\}\) where \(A_1,A_2,\dots,A_k\in\mathbb C^{n\times n}\) are linearly independent Hermitian matrices with the property that \(\bigcap_{1\leq j\leq k} \operatorname {Ker}A_j=\{0\}\). The affine transformations \[ (z,w)\mapsto (z+z_0,w+w_0+2i(zA_1\overline z_0^t,\dots,zA_k\overline z_0^t)),\quad (z_0,w_0)\in M, \] induce automorphisms of \(M\), hence \(M\) is homogeneous. The author deals with the general question whether the automorphisms of \(M\) fixing the origin are linear, i.e. restrictions of linear transformations of the ambient space \(\mathbb C^{n+k}\). The main result is the following sufficient condition: If there exist complex numbers \(a_1,\dots,a_{j_0-1}, a_{j_0+1},\dots,a_k, 1\leq j_0\leq k\), such that the polynomial \(P(x):=\det(xA_{j_0}+\sum _{j\not=j_0}a_jA_j)\) has at least \(k+1\) different roots, then every element in the isotropy group of the origin is linear. The proof is based on former results of the author [Russ. Acad. Sci., Sb., Math. 80, 137--178 (1995); translation from Mat. Sb. 184, No. 10, 3--52 (1993; Zbl 0826.32011)]. For results in the opposite direction cf. \textit{P. B. Utkin} [Math. Notes 72, 138--141 (2002); translation from Mat. Zametki 72, No. 1, 152--156 (2002; Zbl 1028.32019)].