By Jorge Vitório Pereira, Luc Pirio
This publication takes an in-depth examine abelian kin of codimension one webs within the advanced analytic environment. In its classical shape, internet geometry is composed within the research of webs as much as neighborhood diffeomorphisms. an important a part of the speculation revolves round the suggestion of abelian relation, a selected form of practical relation one of the first integrals of the foliations of an online. major focuses of the ebook contain what number abelian kinfolk can an online hold and which webs are wearing the maximal attainable variety of abelian family members. The e-book bargains entire proofs of either Chern’s certain and Trépreau’s algebraization theorem, together with the entire helpful necessities that transcend easy advanced research or easy algebraic geometry. lots of the examples identified modern of non-algebraizable planar webs of maximal rank are mentioned intimately. A historic account of the algebraization challenge for maximal rank webs of codimension one is usually presented.
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Additional info for An Invitation to Web Geometry
First notice that Eq. u/ ; iD1 then dividing by fk , setting hi D fi =fk and deriving repeatedly with respect to the hamiltonian vector field of u until arriving to an equation depending only on u. 1). k2 1/ then Eq. 1) has to have at least k 1 non-constant solutions vanishing at zero. Consequently k D `. u// D 0. k 1/. t u Webs such as W F of the proposition above, which are obtained from the superposition of a parallel web and one non-linear foliation, will be called quasiparallel webs. 4. C2 ; 0/ !
The real graph of f , which is not differentiable since every point of f is tangent to some leaf of the foliation without being a leaf itself. To summarize: the real trace of a holomorphic (or even polynomial) web can be a non-differentiable, although continuous, foliation (Fig. 9). Fig. Pn ; Symk 1Pn ˝ N / be a k-web on Pn . The degree of W is defined as the number of tangencies, counted with multiplicities, of W with a line not everywhere tangent to W. More precisely, if i W P1 ! P1 ; Symk 1P1 ˝ i N /.
I ^ Ái D 0. Cn ; 0/ ! C; 0/ ! C; 0/. Condition (c) translates into k X gi ı ui D 0 iD1 which is the functional equation among the first integrals of W mentioned at the beginning of the discussion. Cn ; 0/k . It is said to be non-trivial if at least one of the Ái ’s is not identically zero. If none of the 1-forms Ái is identically zero, then the abelian relation P k iD1 Ái D 0 is called complete. W/ Á 0 ” W has a non-trivial abelian relation: To some extent, the main results presented in this book can be seen as generalizations of this equivalence to arbitrary webs of codimension one.