By Jorge Vitório Pereira, Luc Pirio
This e-book takes an in-depth examine abelian family of codimension one webs within the complicated analytic atmosphere. In its classical shape, net geometry is composed within the learn of webs as much as neighborhood diffeomorphisms. an important a part of the idea revolves round the suggestion of abelian relation, a specific type of practical relation one of the first integrals of the foliations of an internet. major focuses of the booklet contain what number abelian relatives can an internet hold and which webs are wearing the maximal attainable variety of abelian kinfolk. The booklet deals entire proofs of either Chern’s sure and Trépreau’s algebraization theorem, together with all of the invaluable necessities that transcend undemanding advanced research or easy algebraic geometry. lots of the examples recognized modern of non-algebraizable planar webs of maximal rank are mentioned intimately. A old account of the algebraization challenge for maximal rank webs of codimension one is additionally presented.
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Additional resources for An Invitation to Web Geometry
The simplest example of an asymptotic web is certainly the asymptotic web of a smooth quadric Q in P3 . As is well-known, the latter is isomorphic to P1 P1 . The fibers of both natural projections to P1 are lines in P3 . Since any line contained in it is an asymptotic curve of Q, it is clear that its asymptotic web WQ is formed by these two families of lines. Asymptotic Webs: Alternative Approach When S P3 is a smooth projective surface the definition of the asymptotic web of S is amenable to a more intrinsic formulation.
The fibers of both natural projections to P1 are lines in P3 . Since any line contained in it is an asymptotic curve of Q, it is clear that its asymptotic web WQ is formed by these two families of lines. Asymptotic Webs: Alternative Approach When S P3 is a smooth projective surface the definition of the asymptotic web of S is amenable to a more intrinsic formulation. Suppose that S is cut out by an irreducible homogenous polynomial F 2 CŒx0 ; : : : ; x3 . F / D @2 F @xi @xj Ã ; when evaluated at the tangent vectors of S , gives rise to a morphism Sym2 T S !
For condition (d) the situation is even worse. / ^ ! 2]), the integrability condition for k-symmetric 1-forms has not been treated yet in the literature. 2 Germs of Singular Webs II There is an alternative definition for germs of singular webs that is in a certain sense more geometric. The idea is to consider the (germ of) web as a meromorphic section 2 Here, and throughout, iv denotes the interior product. 3 Singular and Global Webs 19 of the projectivization of the cotangent bundle. This is a classical point of view in the theory of differential equations which has been recently explored in web geometry by Cavalier–Lehmann, see .
An Invitation to Web Geometry by Jorge Vitório Pereira, Luc Pirio