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Deformation of generic submanifolds in a complex manifold. (English) Zbl 1129.32019

A real submanifold \(M\) in a complex manifold \(X\) is generic if, at every \(p\in M\), we have \(T_pM+JT_pM=T_pX\), where \(J\) denotes the complex structure of \(M\). It is called of finite type if the vector fields with values in \(T_pM\cap JT_pM\) and their commutators span \(T_pM\) at every \(p\in M\). It is finitely nondegenerate if some higher order Levi form is nondegenerate (see the paper for a precise definition).
The authors show that a (smooth or analytic) generic real submanifold \(M\) of a complex manifold \(X\), with \(\dim _{\mathbb{C}}X<\dim M<2\dim _{\mathbb{C}}X\), can be approximated, in the Whitney topology, by (smooth or analytic, resp.) generic real submanifolds that are finitely nondegenerate and of finite type. In the smooth case, the result is refined to show that there exist a \(1\)-parameter smooth family of generic embeddings \(\phi:M\times (-1,1)\to X\) such that \(\phi(M,0)=M\) and \(\phi(M,t)\) is finitely nondegenerate and of finite type for all \(t\neq 0\). The authors also provide bounds, that depend on the dimensions of \(M\) and \(X\), for the number of commutators in the definition of finite type and for the lowest order nondegenerate Levi form for the approximating submanifolds.

MSC:

32V40 Real submanifolds in complex manifolds
32H35 Proper holomorphic mappings, finiteness theorems
58A35 Stratified sets
58A20 Jets in global analysis
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References:

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