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CR singularities of real threefolds in 4 . (English) Zbl 1120.32024

The author first classifies the defining functions of 3-dimensional real analytic submanifolds in 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 4 is a graph over the (z 1 ,x 2 ,x 3 )-subspace of the form y 2 =O(3),z 3 =z ¯ 1 2 +O(3), and z 4 =(z 1 +x 2 )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 )z ¯ 1 =0. If N 4 is defined by y 2 =O(3),z 3 =z ¯ 1 2 +O(3) and z 4 =z 1 z ¯ 1 +O(3), then the normal form of N under a formal transformation is defined by y 2 =z 3 -z ¯ 1 2 =z 4 -z 1 z ¯ 1 =0, or by y 2 =z 3 -z ¯ 1 2 =z 4 -z 1 z ¯ 1 -z ¯ 1 x 2 k =0 with k2.

The author conjectures that the normal form of M can be achieved by a convergent transformation.

MSC:
32V40Real submanifolds in complex manifolds
32S05Local singularities (analytic spaces)