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CR singular images of generic submanifolds under holomorphic maps. (English) Zbl 1310.32040

Let \(N\) be a generic real-analytic CR manifold in \(\mathbb C^n\) and let \(f:N\longrightarrow\mathbb C^n\) be a real-analytic CR map which is a diffeomorphism onto its CR singular image \(M=f(N)\). It is well known that \(f\) extends to a holomorphic map \(F\) from a neighborhood of \(N\) in \(\mathbb C^n\) into a neighborhood of \(M\) in \(\mathbb C^n\).
The authors of the paper under review consider the following matters:
1. What can one say on the holomorphic extension \(F\)? The authors provide a necessary (and sufficient in dimension 2) condition for \(F\) to be finite.
2. Characterize the structure of the set of CR singular points of \(M\). It is shown that if \(M\) is a CR singular image with a CR singular set \(S\), and \(M\) contains a complex subvariety \(L\) of complex dimension \(j\) that intersects \(S\), then \(S\cap L\) is a complex subvariety of complex dimension \(j\) or \(j-1\). A corollary to that theorem shows that a Levi-flat CR singular image necessarily has a CR singular set of large dimension.
3. Does every real-analytic CR function on \(M\) extend holomorphically to a neighborhood of \(M\) in \(\mathbb C^n\)? Here the authors show that this is the case when \(M\) is generic at every point. In contrast, they also show that if \(M\) is a CR singular image, then there is a real-analytic function satisfying all tangential CR conditions, yet fails to extend to a holomorphic function on a neighborhood of \(M\).

MSC:

32V25 Extension of functions and other analytic objects from CR manifolds
32V40 Real submanifolds in complex manifolds
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