Equivalence of real submanifolds under volume-preserving holomorphic automorphisms of \(\mathbb{C}^ n\)
Let \(G \subset \Aut \mathbb{C}^n\) be a group of holomorphic automorphisms of \(\mathbb{C}^n\). The author defines continuous maps \(f_0, f_1 : M \to \mathbb{C}^n\) of a compact topological space \(M\) to be \(G\)-equivalent if there is a biholomorphic map \(F\) of an open neighborhood \(U \supset f_0 (M)\) to a neighborhood \(F(U) \supset f_1 (M)\) such that \(F \circ f_0 = f_1\) and such that \(F\) is the uniform limit in \(U\) of a sequence of automorphisms in \(G\). The author's main results apply to the subgroup \(G = \Aut_1 \mathbb{C}^n\) of automorphisms that preserve the standard volume form \(\Omega\) on \(\mathbb{C}^n\), or equivalently, that have complex Jacobian constantly equal to 1. Let \(M\) be a compact real-analytic manifold of real dimension \(m\) and let \(f_0, f_1 : M \to \mathbb{C}^n\) be real-analytic embeddings of \(M\) such that \(f_i (M)\) is totally real and polynomially convex, \(i = 0,1\). Suppose one of the following holds: (1) \(m \leq n - 2\) or (2) \(m = n - 1\) and \(H^{n - 1} (M, \mathbb{R}) = 0\) or (3) \(m = n - 1\), \(M\) is closed and orientable, and \(\int_M f^*_0 \beta = \int_M f^*_1 \beta\) where \(\beta\) is an \((n - 1)\)-form with \(d \beta = \Omega\). The author proves that, under these conditions, if \(f_0\) and \(f_1\) are \(\Aut \mathbb{C}^n\)-equivalent, then they are also \(\Aut_1 \mathbb{C}^n\)-equivalent. Using previous results with \textit{J.-P. Rosay} [Invent. Math. 112, No. 2, 323-349 (1993; Zbl 0792.32011)] the author also shows that if \(M\) is a compact connected real analytic manifold of dimension \(m\) and if \(n \geq \max \{m + 2,3 m/2\}\), then every two real-analytic embeddings \(f_0, f_1 : M \to \mathbb{C}^n\) whose images are totally real and polynomially convex are \(\Aut_1 \mathbb{C}^n\)-equivalent. The author gives corresponding results for curves in \(\mathbb{C}^2\) and surfaces in \(\mathbb{C}^3\).
scientific article; zbMATH DE number 784710
scientific article