Preimages of CR submanifolds with generic holomorphic maps (Q1042663)

Let \(M \subset \mathbb{C}^{m}\) be a smooth \(CR\) manifold of codimension \(d\) and of \(CR\) dimension \(d_{1}\), let \(f :\mathbb{C}^{n} \rightarrow \mathbb{C}^{m}\) be a holomorphic map, let \(M' = f^{-1}(M)\) and denote by \(\Sigma'\) the set of points \(x \in M'\) where (the germ of) \(f\) is not \(CR\) transversal to \(M\). If \(f\) is generic, the author investigates the structure of \(\Sigma '\) and shows that it is a smooth subvariety of \(\mathbb{C}^{n}\) of real codimension \( 2(n-m+d_{1} +1)+d\) if \(n+d_{1} \geq m\) or it equals \(M'\) if \(n+d_{1} < m\). Moreover, he describes an interesting Whitney stratification of \(\Sigma '\). Further, he describes the structure of \(M'\) by analyzing what happens in \(M' \setminus \Sigma'\) and in each subset of the stratification of \(\Sigma '\). He also shows that if \(n+d_{1} \geq m\) and \( 2(n-m+d_{1} +1)+d>n\) then a generic holomorphic map is \(CR\) transversal to \(M\). Preliminary statements, necessary for the understanding of the subject are contained in the paper. Interesting examples and comments are also presented.