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

TL;DR

This paper provides an effective criterion for the holomorphy of curvature in smooth planar webs and applies it to characterize the flatness of dual webs of homogeneous foliations on complex projective planes, especially under Galois conditions.

Contribution

It generalizes previous results by establishing a criterion for curvature holomorphy and characterizes flatness of dual webs for Galois homogeneous foliations.

Findings

Criterion for curvature holomorphy along discriminant components.

Complete characterization of flat dual webs for Galois homogeneous foliations.

Dual webs are always flat when the Galois group is non-cyclic.

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 $\Delta(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results on decomposable webs due to Mar\'{\i}n, 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 $\omega_1^{\hspace{0.2mm}d}=y^d\mathrm{d}x-x^d\mathrm{d}y$, $\omega_2^{\hspace{0.2mm}d}=x^d\mathrm{d}x-y^d\mathrm{d}y.$ When the Galois group of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{Leg}\mathcal{H}$ is always flat.