Modeling complex points up to isotopy (Q375726)

Let \(Y\) be a 4-dimensional compact real manifold and let \(\tau: Y\hookrightarrow X\) be a smooth embedding into a complex 3-manifold \(X\). The author would like to construct a tubular neighborhood \(U\) of \(Y\) which is 2-complete in the sense of Andreotti and Grauert. If \(\tau\) is a holomorphic embedding and \(Y\) is a 2-dimensional compact complex manifold, such a construction is not possible, in view of the maximum principle for plurisubharmonic functions. This leads the author to study, modulo deformation by smooth isotopy, complex points of elliptic (resp. hyperbolic) type, namely: Theorem 1. Let \(p\in Y\) be a complex point and let \(z_1,z_2\in\mathbb{C}^2\) be local coordinates. Then \(p\) can be deformed by a smooth isotopy to a manifold with only complex points of types (i) \(w=|z_1|^2+ |z_2|^2\) or (ii) \(w= |z_1|^2+\overline z_2^2\), where \(w\) is the local coordinate of the submanifold \(Y\) in \(X\). Also it is noted that complex points of elliptic (resp. hyperbolic) type can be deformed to be of type (i) (resp. (ii)). Also for elliptic complex points, they can be taken in the form \(w= |z_1|^2- |z_2|^2\) or \(w= \overline z_1^2+ \overline z_2^2\).