JournalscmhVol. 79 , No. 1DOI 10.1007/s00014-003-0787-4

Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus

  • Alain Hénaut

    Université de Bordeaux I, Talence, France
Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus cover

Abstract

A dd-web \CalW(d){\Cal W}(d) is given by dd complex analytic foliations of codimension nn in (\sumbbbCN,0)({\sumbbb C}^N,0) such that the leaves are in general position. We are interested in the geometry of such configurations. A complex (\CalA,δ)({\Cal A}^{\bullet},\delta) of \sumbbbC{\sumbbb C}-vector spaces is defined in which \CalA0{\Cal A}^0 corresponds to functions and \CalAp{\Cal A}^p to pp-forms of the web \CalW(d){\Cal W}(d) for 1pn1\leq p\leq n. If N=knN=kn with k2k\geq 2, it is proved that rp:=dim\sumbbbC\CalApr_p:=\dim_{\,\sumbbb C}{\Cal A}^p is a finite analytic invariant of \CalW(d){\Cal W}(d) with an optimal upper bound πp(d,k,n)\pi_{p}(d,k,n) for 0pn0\leq p\leq n. These bounds generalize the Castelnuovos ones for genus of curves in \sumbbbPk{\sumbbb P}^{k} with degree dd. Some characterization of the the space H0(Vn,ωVnp)H^0(V_n,\omega^p_{V_n}) of abelian differentials to an algebraic variety VnV_n in \sumbbbPn+k1{\sumbbb P}^{n+k-1} of pure dimension nn with degree dd is given. Moreover, using duality and Abels theorem, we investigate how for suitable VnV_n the natural complex (H0(Vn,ωVn),d)\bigr(H^0(V_n,\omega^{\bullet}_{V_n}),d\,\bigr) and the abelian relation complex (\CalA,δ)({\Cal A}^{\bullet},\delta) of the linear web associated to VnV_n in (\sumbbbCkn,0)({\sumbbb C}^{kn},0) are related.