On webs of maximum rank
Let W(d,n,r) be a d-web given on a differentiable manifold of dimension nr by d foliations of codimension r which are in general position. If the web W(d,n,r) is of maximum r-rank and \(d>r(n-1)+2\), then the web normals are (r-1)-dimensional generators of a rational normal scroll in the projectivized tangent space to the web domain. This theorem was formulated by \textit{S. S. Chern} and \textit{P. Griffiths} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 539-557 (1978; Zbl 0402.57001)] and proved by them in the case \(r=2\). In the paper under review the author, using the same method, gives the proof of this theorem in the cases \(r\geq 2\). This theorem allows to prove that webs of maximum rank, as in the theorem, are almost Grassmannizable. The latter webs were introduced by \textit{M. A. Akivis} in [Sov. Math., Dokl. 21, 707-709; translation from Dokl. Akad. Nauk SSSR 252, 267-270 (1980; Zbl 0479.53015)]. The last result (partially) gives an affirmative answer on a problem posed by the reviewer, whether every web W(d,n,r) of maximum r-rank is almost Grassmannizable. \textit{S. S. Chern} and \textit{P. Griffiths} [see Jahresber. Dtsch. Math.- Ver. 80, 13-110 (1978; Zbl 0386.14002)] also proved that so-called normal webs W(d,n,1) of maximum 1-rank are algebraizable, if \(n\geq 3\), \(d\geq 2n+1\), i.e. they are equivalent to the algebraic webs W(d,n,1) arising naturally from projective algebraic varieties. \textit{S. S. Chern} [see his paper ``Wilhelm Blaschke and web geometry'' in Wilhelm Blaschke Gesammelte Werke, Vol. 5 (1985; Zbl 0656.53003), pp. 21-24] wrote that ``the determination of all webs of maximum rank will remain a fundamental problem in web geometry and the non-algebraic ones, if there are any, will be most interesting''. It was known, that in the plane there exists at least one exceptional (non-algebraizable) web W(d,2,1) of maximum 1- rank - the famous Bol 5-web of rank 6 [see \textit{G. Bol}, Abh. Math. Semin. Univ. Hamb. 11, 387-393 (1936; Zbl 0014.23005)]. Until recently, it was not known whether any exceptional webs of codimension \(r>1\) exist. The reviewer in three recent papers [see C. R. Acad. Sci., Paris, Sér. I 301, 593-596 (1985; Zbl 0579.53015); Differential geometry, Proc. 2nd Int. Symp., Peniscola/Spain 1985, Lect. Notes Math. 1209, 168-183 (1986; Zbl 0607.53008); and Proc. Am. Math. Soc. 100, No.4, 701-708 (1987; Zbl 0628.53018)] has exhibited examples of exceptional webs W(4,2,2) of maximum 2-rank one, which are not algebraizable. In the final section of the paper under review the author proves the theorem, according to which there exist non-algebraizable webs W(2n,2,n) of maximum 2-rank one, for all \(n\geq 2\). This gives (in principle) a way of constructing further exceptional webs W(2n,2,n). (The author remarks that his construction of exceptional webs is much less explicit than the reviewer's construction.) The key idea of the proof of this theorem is that such webs arise naturally in considering families of zero-cycles on algebraic K3 surfaces [see \textit{D. Mumford}, J. Math. Kyoto Univ. 9, 195-204 (1969; Zbl 0184.466), and \textit{A. A. Rojtman}, Math. USSR, Sb. 18(1972), 571-588 (1974); translation from Mat. Sb., Nov. Ser. 89(131), 569-585 (1972; Zbl 0259.14003)].
scientific article; zbMATH DE number 4109521