Cancelling complex points in codimension two (Q254252)

A generically embedded real submanifold \(Y\) of codimension 2 in a complex manifold \(X\) has isolated complex points -- either elliptic or hyperbolic. The author proves that a pair consisting of one elliptic and one hyperbolic point of the same sign can be cancelled by a \({\mathcal C}^0\)-small isotopy of embeddings. The main idea of the proof is to connect the elliptic and the hyperbolic point with an arc \(\gamma\), to find an appropriate complex basis of \(TX\) in a neighbourhood \(U\) of \(\gamma\) and to use it to construct a diffeomorphism \(F : U \rightarrow \mathbb{C}^n\) so that \(F(U \cap Y)\) is a graph over a domain \(D \subset \mathbb{C}^{n-1}\). The map \(F|_{U \cap Y}\) can thus be viewed as an embedding \(F :D \rightarrow \mathbb{C}^n\). The h-principle of Gromov then yields an embedding \(F_1: U \rightarrow \mathbb{C}^n\) that is \({\mathcal C}^0\)-close to \(F\) and agrees with \(F\) near the boundary of \(D\).