Comptes Rendus
Géométrie algébrique
Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials
[Courbes et surfaces bi-algébriques non-linéaires dans les espaces de modules de différentielles abéliennes]
Bertrand Deroin1 ; Carlos Matheus2 
1 CNRS & Université de Cergy-Pontoise (UMR CNRS 8088), 95302, Cergy-Pontoise, France
2 CNRS & École Polytechnique (UMR CNRS 7640), 91128, Palaiseau, France
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1691-1698.

Les strates des espaces de modules de différentielles abéliennes sont des espaces non-homogènes possédant des structures bi-algébriques naturelles. Partiellement inspirés par le cas des espaces homogènes bi-algébriques (comme les tores, les variétés abéliennes et les variétés de Shimura), Klingler et Lerer ont récemment montré qu’une courbe bi-algébrique dans un strate d’un espace de modules de différentielles abéliennes est linéaire pourvu que la soi-disant condition (☆) est satisfaite.

Dans cette note, on construit une courbe, resp. surface, bi-algébrique non-linéaire de différentielles abéliennes de genre 7, resp. 10.

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties and Shimura varieties), Klingler and Lerer recently showed that any bi-algebraic curve in a stratum of the moduli space of Abelian differentials is linear provided that the so-called condition (☆) is fulfilled.

In this note, we construct a non-linear bi-algebraic curve, resp. surface, of Abelian differentials of genus 7, resp. 10.

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DOI : 10.5802/crmath.529
Bertrand Deroin 1 ; Carlos Matheus 2

1 CNRS & Université de Cergy-Pontoise (UMR CNRS 8088), 95302, Cergy-Pontoise, France
2 CNRS & École Polytechnique (UMR CNRS 7640), 91128, Palaiseau, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Bertrand Deroin and Carlos Matheus},
     title = {Non-linear bi-algebraic curves and surfaces in moduli spaces of {Abelian} differentials},
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     year = {2023},
     doi = {10.5802/crmath.529},
     language = {en},
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Bertrand Deroin; Carlos Matheus. Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1691-1698. doi : 10.5802/crmath.529. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.529/

[1] Artur Avila; Carlos Matheus; Jean-Christophe Yoccoz The Kontsevich-Zorich cocycle over Veech-McMullen family of symmetric translation surfaces, J. Mod. Dyn., Volume 14 (2019), pp. 21-54 | DOI | MR | Zbl

[2] Benjamin Bakker; Bruno Klingler; Jacob Tsimerman Tame topology of arithmetic quotients and algebraicity of Hodge loci, J. Am. Math. Soc., Volume 33 (2020) no. 4, pp. 917-939 | DOI | MR | Zbl

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[7] Giovanni Forni; Carlos Matheus; Anton Zorich Square-tiled cyclic covers, J. Mod. Dyn., Volume 5 (2011) no. 2, pp. 285-318 | DOI | MR | Zbl

[8] Eugene Gutkin; Chris Judge Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Volume 103 (2000) no. 2, pp. 191-213 | MR | Zbl

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[11] Carlos Matheus; Jean-Christophe Yoccoz The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., Volume 4 (2010) no. 3, pp. 453-486 | DOI | MR | Zbl

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