Tangent to the Identity Germs and Affine Surfaces

Formal transversality of the infinitesimal generator along the fixed point set

11 févr. 2025, 14:00

50m

Amphithéatre Schwartz (Institut de Mathématiques de Toulouse)

Orateur

Javier Ribon

Description

Given a tangent to the identity germ of holomorphic diffeomorphism $\varphi$, we consider the map $P\mapsto {\varphi }_P$ that associates to any fixed point $P$ of $\varphi$ near the origin the germ ${\varphi }_P$ of $\varphi$ at $P$. Such germs are not in general tangent to the identity. Given the infinitesimal generator $X$ of $\varphi$, a formal vector field, it is natural to ask whether we can define ``infinitesimal generators" $X_P$ of ${\varphi }_P$ for $P\in \mathrm{Fix}(\varphi )$ and if the dependence of $X_P$ on $P\in \mathrm{Fix}(\varphi )$ is analytic. In other works, we are asking whether $X$ is formally transversal along $\mathrm{Fix}(\varphi )$.

We introduce a weak and a strong concept of formal transversality for $X$. On the one hand, we will see that $X$ is always formally transversal along $\mathrm{Fix}(\varphi )$ in the weak sense. On the other hand, there are examples where $X$ is not formally transversal along $\mathrm{Fix}(\varphi )$ in the strong sense. We will discuss the gap between the weak and the strong concepts and provide a characterization of strong formal transversality. This is a joint work with Rudy Rosas.