By Jorge Vitório Pereira, Luc Pirio

This ebook takes an in-depth examine abelian family of codimension one webs within the complicated analytic surroundings. In its classical shape, internet geometry is composed within the examine of webs as much as neighborhood diffeomorphisms. an important a part of the idea revolves round the thought of abelian relation, a specific form of sensible relation one of the first integrals of the foliations of an internet. major focuses of the booklet contain what number abelian kinfolk can an online hold and which webs are sporting the maximal attainable variety of abelian kin. The ebook bargains whole proofs of either Chern’s certain and Trépreau’s algebraization theorem, together with all of the important must haves that transcend uncomplicated complicated research or easy algebraic geometry. lots of the examples recognized brand new 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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Example text

It is tempting to claim that a planar k-web W on C2 defined by a real k-symmetric 1-form with only one leaf through each point of a given domain U R2 is nothing more than an analytic foliation on U . Although trivially true if the discriminant of W does not intersect U this claim is far from being true in general. Perhaps the simplest example comes from a variation on the classical Tait– Kneser Theorem presented in [57, 126]. 3 Singular and Global Webs 25 variable of degree k. t // up to order n.

More precisely, if i W P1 ! P1 ; Symk 1P1 ˝ i N /. Notice that i ! vanishes identically if and only if the image of i is everywhere tangent to W. L//. 1/ holds true. Putting these two facts together with the identity Symk 1P1 D OP1 . N / 2k : Characteristic Numbers of Projective Webs Let X Pn be an irreducible projective subvariety. 4 Examples 27 1. X // D X, where W PT Pn ! Pn is the natural projection; 1 2. X / over any smooth point x of X is PTx X The conormal variety of X PT Pn . X / D PN Xsm ; with Xsm denoting the smooth part of X and N Xsm its conormal bundle.

For an elementary proof and a comprehensive account on Chasles’ Theorem including its distinguished lineage and recent—rather non-elementary—developments, the reader is urged to consult [48]. `0 /. To choose a point x1 2 L1 is therefore the same as choosing a line through p1 2 C1 PL 2 . If such a line is sufficiently close to `0 , then it cuts C3 in a unique point still denoted x1 . In this way the leaf L1 of F1 can be identified with the curve C3 . It will also be useful to identify through the same process L2 , the leaf of F2 through `0 2 PL 2 , with C1 and L3 with C2 (Fig.

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