Holomorphic flows in $\textbf {C}^ 3,0$ with resonances

Abstract:

The topological classification, by conjugacy, of the germs of holomorphic diffeomorphisms $f: {{\mathbf {C}}^2},0 \to {{\mathbf {C}}^2},0$ with $df(0) = \operatorname {diag} ({\lambda _1},{\lambda _2})$, where ${\lambda _1}$ is a root of unity and $|{\lambda _2}| \ne 1$ is given. This type of diffeomorphism appears as holonomies of singular foliations ${\mathcal {F}_X}$ induced by holomorphic vector fields $X:{{\mathbf {C}}^3},0 \to {{\mathbf {C}}^3},0$ normally hyperbolic and resonant. An explicit example of a such vector field without holomorphic invariant center manifold is presented. We prove that there are no obstructions in the holonomies for ${\mathcal {F}_X}$ to be topologically equivalent to a product type foliation.