[Courbes isorésiduelles]
Given a partition $\mu $ of $-2$, the stratum $\mathcal{H}(\mu )$ parametrizes meromorphic differential one-forms on the Riemann sphere $\mathbb{CP}^{1}$ with $n$ zeros and $p$ poles of orders prescribed by $\mu $. The isoresidual fibration is defined by assigning to each differential in $\mathcal{H}(\mu )$ its configuration of residues at the poles. In the case of differentials with $n=2$ zeros, generic isoresidual fibers are complex curves endowed with a canonical translation structure, which we describe extensively in this paper. Quantitative characteristics of the translation structure on isoresidual fiber curves provide rich discrete invariants for these fibers. We determine the Euler characteristic of generic isoresidual fiber curves from intersection-theoretic computations, we describe a wall and chamber structure for the Euler characteristic of generic isoresidual fiber curves in terms of the partition $\mu $, and we classify the connected components of generic isoresidual fibers for strata in genus zero with an arbitrary number of zeros.
Étant donnée une partition $\mu $ de $-2$, la strate $\mathcal{H}(\mu )$ paramétrise les formes différentielles méromorphes de degré $1$ sur la sphère de Riemann $\mathbb{CP}^{1}$, possédant $n$ zéros et $p$ pôles dont les ordres sont prescrits par $\mu $. La fibration isorésiduelle est définie en associant à chaque différentielle de $\mathcal{H}(\mu )$ sa configuration de résidus aux pôles. Dans le cas des différentielles possédant $n=2$ zéros, les fibres isorésiduelles génériques sont des courbes complexes munies d’une structure de translation canonique, que nous décrivons en détail dans cet article. Les caractéristiques quantitatives de la structure de translation sur les courbes isorésiduelles fournissent de riches invariants discrets de ces fibres. Nous déterminons la caractéristique d’Euler des courbes fibres isorésiduelles génériques à partir de calculs de nombres d’intersection, nous décrivons une structure en murs et chambres pour la caractéristique d’Euler des courbes isorésiduelles génériques en fonction de la partition $\mu $, et nous classifions les composantes connexes des fibres isorésiduelles génériques pour les strates de genre $0$ possédant un nombre arbitraire de zéros.
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Keywords: Isoresidual fibration, translation surfaces, multi-scale compactification, resonance arrangement, Gauss–Manin connection
Mots-clés : Fibration isorésiduelle, surfaces de translation, compactification multi-échelle, arrangement de résonance, connexion de Gauss-Manin
Dawei Chen 1 ; Quentin Gendron 2 ; Miguel Prado 3 ; Guillaume Tahar 4
CC-BY 4.0
Dawei Chen; Quentin Gendron; Miguel Prado; Guillaume Tahar. Isoresidual curves. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 891-963. doi: 10.5802/jep.339
@article{JEP_2026__13__891_0,
author = {Dawei Chen and Quentin Gendron and Miguel Prado and Guillaume Tahar},
title = {Isoresidual curves},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {891--963},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.339},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.339/}
}
TY - JOUR AU - Dawei Chen AU - Quentin Gendron AU - Miguel Prado AU - Guillaume Tahar TI - Isoresidual curves JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 891 EP - 963 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.339/ DO - 10.5802/jep.339 LA - en ID - JEP_2026__13__891_0 ER -
%0 Journal Article %A Dawei Chen %A Quentin Gendron %A Miguel Prado %A Guillaume Tahar %T Isoresidual curves %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 891-963 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.339/ %R 10.5802/jep.339 %G en %F JEP_2026__13__891_0
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