Lenses on very curved zones of a singular line field of ${\mathbb C}^2$ or of a singular plane field of ${\mathbb C}^3$

Rémi Langevin (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We renormalize, using suitable lenses, small domains of a singular holomorphic line field of ${\mathbb C}^2$ or plane field of ${\mathbb C}^3$ where the curvature of a plane-field is concentrated. At a proper scale the field is almost invariant by translations. When the field is integrable, the leaves are locally almost translates of a surface that we will call {\it profile}. When the singular rays of the tangent cone (a generalization to a plane-field of the tangent cone of a singular surface is defined) are isolated, we obtain more precise results. We also generalize a result of Merle (\cite{Me}) concerning the contact order of generic polar curves with the singular level $f=0$ when $\omega = df$. On the way we obtain some classical results (Lê's carousels) on the knot $K = (\{f=0\} \cap B_{\epsilon}(0,0,0))$ in dimension $2$ an a maybe less classical ones in dimension $3$ .

Mots clés: complex polynomial; complex one-form; isolated singularity; polar curve; profile; profile domains [mathAG]; [mathIT]; [mathGT]; [mathDG]

Domaines: Mathématiques [math]; Géométrie algébrique [math.AG]; Géométrie différentielle [math.DG]

Rémi Langevin. Lenses on very curved zones of a singular line field of ${\mathbb C}^2$ or of a singular plane field of ${\mathbb C}^3$. 2020. ⟨hal-02487381⟩