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

Abstract

Let $d\ge 3$ be an integer. For a holomorphic $d$-web $W$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $\Delta \left(W\right)$, we establish an effective criterion for the holomorphy of the curvature of $W$ 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) $\mathrm{Leg}H$ of a homogeneous foliation $H$ of degree $d$ on $P_C^2$, generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{Leg}H$ in the particular case where the foliation $H$ is Galois. When the Galois group of $H$ is cyclic, we show that $\mathrm{Leg}H$ is flat if and only if $H$ is given, up to linear conjugation, by one of the two 1-forms ${\omega }_1^d=y^{d}dx-x^{d}dy$, ${\omega }_2^d=x^{d}dx-y^{d}dy$. When the Galois group of $H$ is noncyclic, we obtain that $\mathrm{Leg}H$ is always flat.