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A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$

Samir Bedrouni, David Marín

Abstract

Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $Δ(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results on decomposable webs due to Marín, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) $\mathrm{Leg}\mathcal{H}$ of a homogeneous foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}},$ generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{Leg}\mathcal{H}$ in the particular case where the foliation $\mathcal{H}$ is Galois. When the Galois group of $\mathcal{H}$ is cyclic, we show that $\mathrm{Leg}\mathcal{H}$ is flat if and only if $\mathcal{H}$ is given, up to linear conjugation, by one of the two 1-forms $ω_1^{\hspace{0.2mm}d}=y^d\mathrm{d}x-x^d\mathrm{d}y$, $ω_2^{\hspace{0.2mm}d}=x^d\mathrm{d}x-y^d\mathrm{d}y.$ When the Galois group of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{Leg}\mathcal{H}$ is always flat.

A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$

Abstract

Let be an integer. For a holomorphic -web on a complex surface , smooth along an irreducible component of its discriminant we establish an effective criterion for the holomorphy of the curvature of along generalizing results on decomposable webs due to Marín, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) of a homogeneous foliation of degree on generalizing some of our previous results. This then allows us to study the flatness of the -web in the particular case where the foliation is Galois. When the Galois group of is cyclic, we show that is flat if and only if is given, up to linear conjugation, by one of the two 1-forms , When the Galois group of is non-cyclic, we obtain that is always flat.
Paper Structure (7 sections, 10 theorems, 68 equations)

This paper contains 7 sections, 10 theorems, 68 equations.

Table of Contents

  1. Preliminaries
  2. Webs
  3. Characteristic surface of a web
  4. Fundamental form, curvature and flatness of a web
  5. Criterion for the holomorphy of the curvature of smooth webs
  6. Holomorphy of the curvature of the dual web of a homogeneous foliation on $\mathbb{P}^{2}_{\mathbb{C}}$
  7. Galois homogeneous foliations having a flat Legendre transform

Key Result

Theorem 2.1

Let $\mathcal{W}$ be a holomorphic $d$-web on a complex surface $M$ and let $D$ be an irreducible component of the discriminant $\Delta(\mathcal{W}).$ Assume that $\mathcal{W}$ is smooth along $D$. Then the fundamental form $\eta(\mathcal{W})$ has simple poles along $D$. More precisely, choose a loc is holomorphic along $D=\{y=0\}$, where $\psi_{\alpha}$ is a function of the coordinate $x$ defined

Theorems & Definitions (29)

  • Definition 1.1
  • Example 1.2
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Corollary 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 19 more