Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus

  • Alain Hénaut

    Université de Bordeaux I, Talence, France

Abstract

A -web \( {\Cal W}(d) \) is given by complex analytic foliations of codimension in \( ({\sumbbb C}^N,0) \) such that the leaves are in general position. We are interested in the geometry of such configurations. A complex \( ({\Cal A}^{\bullet},\delta) \) of \( {\sumbbb C} \)-vector spaces is defined in which \( {\Cal A}^0 \) corresponds to functions and \( {\Cal A}^p \) to -forms of the web \( {\Cal W}(d) \) for . If with , it is proved that \( r_p:=\dim_{\,\sumbbb C}{\Cal A}^p \) is a finite analytic invariant of \( {\Cal W}(d) \) with an optimal upper bound for . These bounds generalize the Castelnuovos ones for genus of curves in \( {\sumbbb P}^{k} \) with degree . Some characterization of the the space of abelian differentials to an algebraic variety in \( {\sumbbb P}^{n+k-1} \) of pure dimension with degree is given. Moreover, using duality and Abels theorem, we investigate how for suitable the natural complex and the abelian relation complex \( ({\Cal A}^{\bullet},\delta) \) of the linear web associated to in \( ({\sumbbb C}^{kn},0) \) are related.

Cite this article

Alain Hénaut, Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus. Comment. Math. Helv. 79 (2004), no. 1, pp. 25–57

DOI 10.1007/S00014-003-0787-4