Rigidification of holomorphic germs with non-invertible differential

We study holomorphic germs $f:(\mathbb{C}^2, 0) \rightarrow (\mathbb{C}^2,0) with non-invertible differential$df_0$. In order to do this, we search for a modification$π:X \rightarrow (\mathbb{C}^2,0)$(i.e., a composition of point blow-ups over the origin), and an infinitely near point$p \in π^{-1}(0)$, such that the germ$f$lifts to a holomorphic germ$\hat{f}:(X,p) \rightarrow (X,p)$which is rigid (i.e., the generalized critical set of$\hat{f}$is totally invariant and has normal crossings at$p$). We extend a previous result for superattracting germs to the general case, and deal with the uniqueness of this process in the semi-superattracting case ($\operatorname{Spec}(df_0)=\{0, λ\}$with$λ\neq 0$). We specify holomorphic normal forms for the nilpotent case and for the type$(0,\mathbb{D})$, that is$\operatorname{Spec}(df_0)=\{0, λ\}$with$λ$in the unitary disk$\mathbb{D} \subset \mathbb{C}$, and formal normal forms for the type$(0, \mathbb{C} \setminus \mathbb{D})$.

No reviews yet.

No comments yet.