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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.

##### MSC:
 32V40 Real submanifolds in complex manifolds 32S05 Local complex singularities
##### Keywords:
complex tangent; formal normal form
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##### References:
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