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On the algebraization of webs of codimension \(n\) in \(\mathbb{C}^{2n}\). (Sur l’algébrisation des tissus de codimension \(n\) de \(\mathbb{C}^{2n}\).) (French) Zbl 0906.53007
The rank problems for webs \(W (d, k, n)\) of codimension \(n\), given on a \((kn)\)-dimensional differentiable manifold by \(d\) foliations of codimension \(n\) in general position, are: (1) to find an upper bound for the \(n\)-rank, and (2) to describe webs of maximum \(n\)-rank. The author describes the \(d\)-webs \(W(d, 2, n)\) of maximum \(n\)-rank.
The main result of the paper is the following. For \(d \geq n+3 \geq 5\), any web \(W(d, 2, n)\) of maximum \(n\)-rank is algebraizable, i.e., it is equivalent to an algebraic web \(AW (d, 2, n)\) generated by an algebraic hypersurface \(V^n\) of degree \(n\) in the projective space \(P^{n+1}\). For \(n = 2\), this result matches the reviewer’s result for webs \(W(d, 2, 2)\) of maximum 2-rank [see V. V. Goldberg, C. R. Acad. Sci., Paris, Sér. I 297, 339-342 (1983; Zbl 0539.53012); Colloq. Math. Soc. János Bolyai 56, 317-357 (1992; Zbl 0789.53008); or Section 8.3 of the reviewer’s book “Theory of multicodimensional \((n+1)\)-webs” (Mathematics and its Applications 44, Kluwer, Dordrecht) (1988; Zbl 0668.53001)].
MSC:
53A60 Differential geometry of webs
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References:
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