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CR singularities of real threefolds in $\Bbb C^4$. (English) Zbl 1120.32024
The author first classifies the defining functions of 3-dimensional real analytic submanifolds in $\Bbb C^4$, 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 $M\subset\Bbb C^4$ 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 $N\subset\Bbb C^4$ 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.

##### MSC:
 32V40 Real submanifolds in complex manifolds 32S05 Local singularities (analytic spaces)
##### Keywords:
complex tangent; formal normal form
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