Abstract
The main result of this chapter is a converse to Abel’s addition Theorem stated in Sect. 4.1. It ensures the algebraicity of local datum satisfying the hypotheses of Abel’s addition Theorem. Its first version was established by Sophus Lie in the context of double-translation surfaces. Lie’s arguments consisted in a tour-de-force analysis of an overdetermined system of PDEs. Later Poincaré introduced a geometrical method to handle the problem solved analytically by Lie. Poincaré’s approach was later revisited, and made more precise by Darboux, to whom the approach presented in Sect. 4.2 can be traced back. By the way, those willing to take for granted the validity of the converse of Abel’s Theorem can safely skip Sect. 4.2.
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Notes
- 1.
Here and throughout in the proof of the converse of Abel’s Theorem, the notation (X, Y ) means the germ of the variety X along (or at) Y. One should think of open subsets of X, arbitrarily small among the ones containing Y.
- 2.
Here and throughout, the convention about germs made in Sect. 1.1.1 is in use. If one wants to be more precise, then \(\mathcal{W}\) has to be thought as a web defined on an open subset U of \(\mathbb{C}^{n}\) containing the origin and \(F_{x}^{1}\mathcal{A}(\mathcal{W})\) is the filtration of the germ of \(\mathcal{W}\) at x.
- 3.
This is the case if and only if \(\mathcal{W}\) has an abelian relation whose ith component is non-trivial.
- 4.
Recall that the canonical map of the projective curve C is defined as
$$\displaystyle\begin{array}{rcl} \kappa _{C}:\, C& --\rightarrow & \mathbb{P}H^{0}(C,\omega _{ C})^{{\ast}}\simeq \mathbb{P}^{g_{a}(C)-1} {}\\ x& \longmapsto & \mathbb{P}(\ker \{\omega \mapsto \omega (x)\})\,. {}\\ \end{array}$$ - 5.
The Riemann’s theta function θ A of a polarized abelian variety \(A = \mathbb{C}^{g}/\Delta \) with \(\Delta = (I_{g},Z)\) (where \(Z \in M_{g}(\mathbb{C})\) is such that \(Z =\! ^{t}\!Z\) and Im Z > 0) is defined by \(\theta _{A}(z) =\sum _{m\in \mathbb{Z}^{g}}\exp \big(i\pi \langle m,Zm\rangle + 2i\pi \langle m,z\rangle \big)\) for all \(z \in \mathbb{C}^{g}\) (see [7, Chap. I]).
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Pereira, J.V., Pirio, L. (2015). The Converse to Abel’s Theorem. In: An Invitation to Web Geometry. IMPA Monographs, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-14562-4_4
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