Abstract
In this chapter, the key concept of abelian relation is introduced. Roughly speaking, an abelian relation for a web \(\mathcal{W}\) is an additive functional equation among the first integrals of its foliations. The credo here is that the notion of abelian relation must be considered as a generalization for webs of the more classical notion of holomorphic differential form on a projective variety. The most compelling evidences supporting this credo will be presented in the next two chapters through the study of abelian relations of algebraic webs.
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Notes
- 1.
Here, for \(n \in \mathbb{N}\), w n stands for the n-th differential operator obtained by applying n times w: inductively, one has \(w^{n}(u) = w(w^{n-1}(u))\) for any holomorphic germ u, with the convention that w 0 = Id.
- 2.
When a i = 0, the linear subspace \(\mathbb{P}^{a_{i}}\) is nothing but a point \(p_{i} \in \mathbb{P}^{n}\). In this case, the following convention is adopted: the rational normal curve in \(\mathbb{P}^{a_{i}}\) is not a curve, but the point p i .
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Pereira, J.V., Pirio, L. (2015). Abelian Relations. In: An Invitation to Web Geometry. IMPA Monographs, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-14562-4_2
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DOI: https://doi.org/10.1007/978-3-319-14562-4_2
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