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

TL;DR

This paper constructs examples of tangent to the identity germs in three complex dimensions with only degenerate characteristic directions, contrasting with the two-dimensional case, and analyzes their invariant structures and singularity resolutions.

Contribution

It introduces a family of three-dimensional tangent to the identity germs with exclusively degenerate characteristic directions, expanding understanding of their birational properties.

Findings

Existence of three-dimensional germs with only degenerate characteristic directions.

Contrast with the two-dimensional case where non-degenerate directions always appear after modifications.

Description of invariant curves and parabolic manifolds for these germs.

Abstract

We provide a family of isolated tangent to the identity germs $f:(\mathbb{C}^3,0) \to (\mathbb{C}^3,0)$ which possess only degenerate characteristic directions, and for which the lift of $f$ 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 $f$ admits a modification where the lift of $f$ has a non-degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of $f$, 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 $f$-invariant curves, and the associated parabolic manifolds, using the techniques recently developed by L\'opez-Hernanz, Raissy, Rib\'on, Sanz S\'anchez, Vivas.