A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on PC2$\mathbb {P}^{2}_{\mathbb {C}}$

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 noncyclic, we obtain that is always flat.