Birational properties of tangent to the identity germs without non‐degenerate singular directions

Abstract

AbstractWe provide a family of isolated tangent to the identity germs which possess only degenerate characteristic directions, and for which the lift of to any modification (with suitable properties) has only degenerate characteristic directions. This is in sharp contrast with the situation in dimension 2, where any isolated tangent to the identity germ admits a modification where the lift of has a non‐degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of , showing that this phenomenon is not related to the non‐existence of complex separatrices for vector fields of Gomez‐Mont and Luengo. Finally, we describe the set of formal ‐invariant curves, and the associated parabolic manifolds, using the techniques recently developed by López‐Hernanz et al.