## 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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### References:

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