Holomorphic automorphisms of quadrics

Abstract

a R-valued Hermitian form. The following set is called a quadric Q = {(z, w) ∈ C : v = 〈z, z〉}. Q is presumed to be nondegenerate, i.e., i.) 〈z, b〉 = 0 for all z implies b = 0 ii.) The forms 〈·, ·〉 are linearly independent for j = 1, . . . , k. Any quadric Q is a homogeneous manifold (the transformations z∗ = p + z, w∗ = q + w + 2i〈z, p〉 with (p, q) ∈ Q act transitively on Q). Therefore, we restrict ourselves to the local automorphisms preserving a fixed point, say the origin. The subgroup of these automorphisms, being called isotropy group, is a finite dimensional Lie group iff Q is nondegenerate (see [2, 5]). The connected component of the unit of the isotropy group will be denoted by Aut0Q. Our goal is to find the explicit description for Aut0Q. This is important for the study of real submanifolds of codimension k in C. It follows from the results of Henkin, Tumanov and Forstneric [6, 11, 5] that any local CR-diffeomorphism of a nondegenerate quadric extends to a birational map of C with the degree uniformly bounded within the same (n, k). This result might be considered as a generalization of the Poincare-Alexander theorem [7, 1] about the extension of a local CR-diffeomorphism of a hyperquadric in C. However, explicit descriptions of the isotropy groups are only known for hyperquadrics (k = 1) and direct products of hyperquadrics.