Ordinary holomorphic webs of codimension one
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 197-214.

To any $d$-web of codimension one on a holomorphic $n$-dimensional manifold $M$ ($d>n$), we associate an analytic subset $S$ of $M$. We call ordinary the webs for which $S$ has a dimension at most $n-1$ or is empty. This condition is generically satisfied, at least at the level of germs.

We prove that the rank of an ordinary $d$-web has an upper-bound ${\pi }^{\text{'}}\left(n,d\right)$ which, for $n\ge 3$, is strictly smaller than the bound $\pi \left(n,d\right)$ proved by Chern, $\pi \left(n,d\right)$ denoting the Castelnuovo’s number. This bound is optimal. Setting $c\left(n,h\right)=\left(\begin{array}{c}n-1+hh\end{array}\right)$, let ${k}_{0}$ be the integer such that $c\left(n,{k}_{0}\right)\le d. The number ${\pi }^{\text{'}}\left(n,d\right)$ is then equal

• to 0 for $d,
• and to ${\sum }_{h=1}^{{k}_{0}}\left(d-c\left(n,h\right)\right)$ for $d\ge c\left(n,2\right)$.

Moreover, if $d$ is precisely equal to $c\left(n,{k}_{0}\right)$, we define off $S$ a holomorphic connection on a holomorphic bundle $ℰ$ of rank ${\pi }^{\text{'}}\left(n,d\right)$, such that the set of Abelian relations off $S$ is isomorphic to the set of holomorphic sections of $ℰ$ with vanishing covariant derivative: the curvature of this connection, which generalizes the Blaschke curvature, is then an obstruction for the rank of the web to reach the value ${\pi }^{\text{'}}\left(n,d\right)$.

When $n=2$, $S$ is always empty so that any web is ordinary, ${\pi }^{\text{'}}\left(2,d\right)=\pi \left(2,d\right)$, and any $d$ may be written $c\left(2,{k}_{0}\right)$: we recover the results given in .

Publié le :
Classification : 53A60,  14C21,  32S65
@article{ASNSP_2012_5_11_1_197_0,
author = {Cavalier, Vincent and Lehmann, Daniel},
title = {Ordinary holomorphic webs of codimension one},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {197--214},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {1},
year = {2012},
zbl = {1244.53014},
mrnumber = {2953049},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2012_5_11_1_197_0/}
}
Cavalier, Vincent; Lehmann, Daniel. Ordinary holomorphic webs of codimension one. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 197-214. http://www.numdam.org/item/ASNSP_2012_5_11_1_197_0/

 W. Blaschke and G. Bol, “Geometrie der Gewebe, Die Grundlehren der Mathematik”, Vol. 49, Springer, Berlin, 1938. | EuDML 203712 | JFM 64.0727.03 | MR 43502 | Zbl 0020.06701

 S. S. Chern, Abzählungen für Gewebe, Abh. Hamburg 11 (1936), 163–170. | MR 3069650 | Zbl 0011.13202

 S. S. Chern and P. A. Griffiths, Abel’s theorem and webs, Jahresber. Deutsch. Math.-Verein. 80 (1978), 13–110 and 83 (1981), 78–83. | EuDML 146681 | MR 494957 | Zbl 0386.14002

 V. Cavalier and D. Lehmann, Ordinary holomorphic webs of codimension one, preprint, arXiv: math 0703596 v2[mathDS] 13 Oct. 2008. | Numdam | MR 2953049 | Zbl 1244.53014

 V. Cavalier and D. Lehmann, Rang et courbure de Blaschke des tissus holomorphes réguliers de codimension un, C.R. Math. Acad. Sci. Paris 346 (2008), 1283–1288. | MR 2473309 | Zbl 1163.53007

 V. Cavalier and D. Lehmann, Global structure of webs in codimension one, preprint, arXiv: math v1 [mathDS] March. 2008.

 V. Cavalier and D. Lehmann, Introduction à l’étude globale des tissus sur une surface holomorphe, Ann. Inst. Fourier (Grenoble) 57 (2007), 1095–1133. | EuDML 10252 | Numdam | MR 2339328 | Zbl 1129.14015

 P. A. Griffiths and J. Harris, “Principles of Algebraic Geometry”, John Wiley & Sons, New York, 1978. | MR 507725 | Zbl 0836.14001

 A. Hénaut, On planar web geometry through Abelian relations and connections, Ann. of Math. 159 (2004), 425–445. | MR 2052360 | Zbl 1069.53020

 A. Hénaut, Systèmes différentiels, nombre de Castelnuovo et rang des tissus de ${ℂ}^{n}$, Publ. Res. Inst. Math. Sci. 31 (1995), 703–720. | MR 1371791 | Zbl 0911.53008

 A. Hénaut, Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus, Comment. Math. Helv. 79 (2004), 25–57. | MR 2031299 | Zbl 1064.53011

 A. Hénaut, Planar web geometry through abelian relations and singularities, In: “Nankai Tracts in Mathematics” P. Griffiths (ed.), Vol. 11, World Scientific and Imperial College Press, 2006. | MR 2313337 | Zbl 1133.53012

 A. Pantazi, Sur la détermination du rang d’un tissu plan, C.R. Inst. Sci. Roum. 2 (1938), 108–111. | Zbl 0018.17103

 J. V. Pereira, Algebrization of codimension one webs [after Trépreau, Hénaut, Pirio, Robert], Séminaire Bourbaki 2006-2007, n. 974, Mars 2007. | Numdam | MR 2487736 | Zbl 1184.14014

 J. V. Pereira, Resonance webs of hyperplane arrangements, Adv. Stud. Pure Math. 99 (2010), 1–30. | MR 2933800 | Zbl 1261.52015

 J. V. Pereira and L. Pirio, “An invitation to Web Geometry”, 27 Colóquio Brasileiro de Matemática, Publicaões matemáticas do IMPA, Instituto de matemática pura e aplicada, Rio de Janeiro, 2009. | MR 2536234 | Zbl 1184.53002

 L. Pirio, Sur la linéarisation des tissus, preprint, arXiv: math 0811.1810v2 [mathDG] 23 janv. 2009. | MR 2583781 | Zbl 1189.35051

 D. C. Spencer, “Selecta”, Vol. 3, World Scientific Publishing Co. Philadelphia, 1985. | MR 842668 | Zbl 0657.01016

 J. M. Trépreau, Algébrisation des tissus de codimension 1. La généralisation d’un théorème de Bol, Inspired by S. S. Chern, In: “Nankai Tracts in Mathematics”, P. Griffiths (ed.), Vol. 11, World Scientific and Imperial College Press, 2006. | MR 2313344 | Zbl 1136.53010