On the geometry of Poincaré's problem for one-dimensional projective foliations

Abstract

We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation.

Characteristic numbers and invariant subvarieties for projective webs

“…Remark 2.1. This corollary generalize the bound obtained by M. Soares for one dimensional foliations in [12]. We also remark that the bound d ≤ deg(W) + (n − p) + 1 is proved in [2] for normal crossing hypersurfaces invariant by a p-dimensional foliation.…”

“…The question of bounding the degree of an algebraic curve which is a solution of a foliation on P 2 in terms of the degree of the foliation was treated by H. Poincaré in [9]. Versions of this problem have been considered in a number of recent works, see for example [12] and references therein. In that paper, M. Soares considered one dimensional projective foliations and their tangency locus with a pencil of hyperplanes.…”

“…Polar classes was also considered by R. Mol in [6] for holomorphic distributions of arbitrary dimension. He expresses these classes in terms of the Chern-Mather classes of the tangent sheaf of the distribution, moreover, Theorem 1 of [12] is generalizated.…”

Polar Varieties Revisited

“…More recently, polar varieties have been applied to study the topology of real affine varieties and to find real solutions of polynomial equations by Bank et al, Safey El Din-Schost, and Mork-Piene [1,2,3,4,18,13,14], to complexity questions by Bürgisser-Lotz [5], to foliations by Soares [19] and others, to focal loci and caustics of reflection by Catanese and Trifogli [6,22] and Josse-Pène [8], to Euclidean distance degree by Draisma et al [7].…”

Degré d'une solution algébrique d'un tissu sur le plan projectif complexe

“…Etant donné un feuilletage algébrique G de degré n sur le plan projectif complexe P 2 admettant une solution algébrique γ de degré δ, il existe plusieurs formules liant n et δ à des indices dont le support est contenu dans le lieu singulier de G (voir [2,3,7,8,10,4]). Nous voulons donner une formule analogue dans le cas des tissus.…”