A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on PC2

Abstract

Let d >= 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(W), we establish an effective criterion for the holomorphy of the curvature of W along D, generalizing results on decomposable webs due to Mar & iacute;n, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) 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 Leg H in the particular case where the foliation H is Galois. When the Galois group of H is cyclic, we show that Leg H is flat if and only if H is given, up to linear conjugation, by one of the two 1-forms omega(d)(1)=y(d)dx-x(d)dy, omega(d)(2)=xddx-y(d)dy. When the Galois group of H is noncyclic, we obtain that Leg H is always flat.