Analytic web geometry (Q2782004)

In the paper under review, the author presents some fundamental results of web geometry. Some of these results are classical, and others related to the author's research. In the course of his presentation, he poses a few open problems for webs. NEWLINENEWLINENEWLINEIn Part 2, the author considers \(d\)-webs of curves in \(\mathbb{C}^2\). He introduces abelian relations, the web rank, gives several proofs for the optimal inequality for the rank, defines algebraizable and linearizable webs and discusses relations between webs of maximum rank, algebraizable webs and linearizable webs. In particular, the author presents here his result on the necessary and sufficient condition of linearizability of a \(d\)-web [see \textit{A. Hénaut}, Topology 32, No.~3, 531-542 (1993; Zbl 0799.32010)].NEWLINENEWLINENEWLINEIn Part 3, the author discusses the general properties of \(d\)-webs \(W (d, k, n)\) of codimension \(n\) in \(\mathbb{C}^{kn}\), defines abelian relations of degree \(n\) and the \(n\)-rank for them, states the theorem giving an optimal upper bound for their rank, indicates a possibility to define abelian relations of degree \(p\), \(1 \leq p \leq n,\) and the \(p\)-rank of a web \(W (d, k, n)\), discusses relations between webs of maximum rank and algebraizable webs \(W(d, k, n)\), and considers the topics indicated above in more detail, for the webs \(W (d, 2, n)\). He pays a special attention to the exceptional \(d\)-webs \(W (d, 2, n)\) that have maximum rank but are not algebraizable. There are known only five examples of such webs: a 5-web \(W (5, 2, 1)\) constructed by Bol [see, for example, p. 31 of the book of \textit{W. Blaschke} and \textit{G. Bol}, Geometrie der Gewebe, Springer, Berlin (1938; Zbl 0020.06701)], three explicit examples of webs \(W (4, 2, 2)\) constructed by the reviewer [see, for example, \textit{V. V. Goldberg}, Theory of multicodimensional \((n+1)\)-webs, Kluwer Academic Publishers, Dordrecht (1988; Zbl 0668.53001), Section 8.4], and an implicit example of a web \(W (4, 2, 2)\) constructed by \textit{J. B. Little} [see Geom. Dedicata 31, No. 1, 19-35 (1989; Zbl 0677.53017)].NEWLINENEWLINENEWLINEIn connection with exceptional webs, the author presents a brief observation of polylogarithmic webs in \(\mathbb{C}^2\). The author concludes Part 3 by introducing the complex of abelian relations of a web \(W (d, 2, n)\), the subject which he investigates in detail in his forthcoming paper.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00024].