On the automorphism groups of algebraic bounded domains

1994/10/25 by Zaitsev, Dmitri
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.alg-geom/9410025

Abstract

Let D be a bounded domain in Cⁿ. By the theorem of H.~Cartan, the group Aut(D) of all biholomorphic automorphisms of D has a unique structure of a real Lie group such that the action Aut(D)× D→ D is real analytic. This structure is defined by the embedding C_v\colon Aut(D)\hookrightarrow D× Gl_n(C), f↦ (f(v), f_*v), where v∈ D is arbitrary. Here we restrict our attention to the class of domains D defined by finitely many polynomial inequalities. The appropriate category for studying automorphism of such domains is the Nash category. Therefore we consider the subgroup Aut_a(D)⊂ Aut(D) of all algebraic biholomorphic automorphisms which in many cases coincides with Aut(D). Assume that n>1 and D has a boundary point where the Levi form is non-degenerate. Our main result is theat the group Aut_a(D) carries a unique structure of an affine Nash group such that the action Aut_a(D)× D→ D is Nash. This structure is defined by the embedding C_v\colon Aut_a(D)\hookrightarrow D× Gl_n(C) and is independent of the choice of v∈ D.

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