The Classification of Exceptional CDQL Webs on Compact Complex Surfaces

Abstract

Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number π(n, k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental problem in web geometry is the classification of exceptional webs, that is, webs of maximal rank not equivalent to the dual of a projective curve. Recently, J.-M. Trépreau proved that there are no exceptional k-webs for n ≥ 3 and k ≥ 2n. In dimension two there are examples for arbitrary k and the classification problem is wide open.In this paper, we classify the exceptional Completely Decomposable Quasi-Linear (CDQL) webs globally defined on compact complex surfaces. By definition, the CDQL (k + 1)-webs are formed by the superposition of k linear foliations and one non-linear foliation. For instance, we show that up to projective transformations there are exactly four countable families and thirteen sporadic exceptional CDQL webs on P 2 .

Classification des tissus exceptionnels quasilinéaires complètement décomposables

“…Leurs relations abéliennes peuvent toutes être explicitement déterminées en utilisant les résultats de [6]. Les composantes des relations abéliennes non-élémentaires des tissus B k et H l (pour k = 5, 6, 7, 8 et l = 5, 10) sont toutes des intégrales itérées de poids plus petit que 2 (voir [10] pour des détails).…”

“…Les résultats de [10] permettent en fait de classifier les tissus CDQL plats sur P 2 de degré 3 strictement plus grand que 1. Outre les treize exemples sporadiques exceptionnels décris dans le Théorème 3.1, il n'y a que trois autres tels tissus (à équivalence projective près), à savoir…”

“…Up to isogenies, there are three sporadic examples and one continuous family of exceptional CDQL webs on complex tori. These are the webs listed in Théorème 4.2 below.Detailed proofs of the two above theorems are given in [10]. …”

Flat 3-webs of degree one on the projective plane

“…The general philosophy behind our approach (considered for the first time in [7] as far as webs are concerned) is that flatness is characterized by the vanishing of a meromorphic curvature 2-form having poles along the discriminant of the 3web. In [8,5] it is shown that generic invariant components of the discriminant do not produce poles in the curvature.…”

“…In fact, in [7] the authors introduce the notion of F -barycenter β F (W) of a completely decomposable web W with respect to a foliation F , which was extended in [5] to include the case F is a web. The general statement of Theorem 1 in [5] is that the holomorphy of K(W 2 ⊠ W ′ ) is equivalent to the fact that ∆(W 2 ) be invariant by either W 2 or β W2 (W ′ ).…”