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CR singularities of real threefolds in \(\mathbb C^4\). (English) Zbl 1120.32024
The author first classifies the defining functions of 3-dimensional real analytic submanifolds in \(\mathbb C^4\), up to third order terms. Excluding the trivial totally real case, there are 13 types of quadratic terms. The author then finds normal forms under a formal change of coordinates for two types of quadratic terms.
More precisely, if \(M\subset\mathbb C^4\) is a graph over the \((z_1,x_2,x_3)\)-subspace of the form \(y_2=O(3), z_3=\overline z_1^2+O(3)\), and \( z_4=(z_1+x_2)\overline z_1+O(3)\), then the normal form of \(M\) under formal holomorphic transformation is defined by \(y_2=z_3-z_1^2=z_4-(z_1+x_2)\overline z_1=0\). If \(N\subset\mathbb C^4\) is defined by \(y_2=O(3), z_3=\overline z_1^2+O(3)\) and \(z_4=z_1\overline z_1+O(3)\), then the normal form of \(N\) under a formal transformation is defined by \(y_2=z_3-\overline z_1^2=z_4-z_1\overline z_1=0\), or by \(y_2=z_3-\overline z_1^2=z_4-z_1\overline z_1-\overline z_1x_2^k=0\) with \(k\geq 2\).
The author conjectures that the normal form of \(M\) can be achieved by a convergent transformation.

32V40 Real submanifolds in complex manifolds
32S05 Local complex singularities
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