Hyperbolic geometry and moduli of real cubic surfaces

[Géométrie hyperbolique et espace des modules des surfaces cubiques réelles]

Allcock, Daniel ; Carlson, James A. ; Toledo, Domingo

Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010), p. 69-115 / Harvested from Numdam

Résumé

On note $\mathcal {M}_0^\mathbb {R}$ l’espace des modules des surfaces cubiques réelles lisses. Nous montrons que chacune de ses composantes admet une structure hyperbolique réelle. Plus précisément, en enlevant de l’espace hyperbolique réel $H^4$ certaines sous-variétés totalement géodésiques de dimension inférieure, puis en prenant le quotient par un groupe arithmétique, on obtient une orbifold isomorphe à une composante de l’espace des modules. Il y a cinq composantes. Nous décrivons le réseau de $PO(4,1)$ qui correspond à chacune d’entre elles. Nous démontrons également quelques résultats sur la topologie de $\mathcal {M}_0^\mathbb {R}$, dont certains sont nouveaux. On note $\mathcal {M}_s^\mathbb {R}$ l’espace des modules des surfaces cubiques réelles qui sont stables au sens de la théorie géométrique des invariants. Nous montrons que cet espace admet une structure hyperbolique dont la restriction à $\mathcal {M}_0^\mathbb {R}$ est celle évoquée ci-dessus. Nous décrivons un domaine fondamental pour le réseau correspondant de $PO(4,1)$, qui s’avère être non arithmétique.

Abstract

Let $\mathcal {M}_0^\mathbb {R}$ be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space $H^4$ and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in ${\rm PO}(4,1)$. We also derive several new and several old results on the topology of $\mathcal {M}_0^\mathbb {R}$. Let $\mathcal {M}_s^\mathbb {R}$ be the moduli space of real cubic surfaces that are stable in the sense of geometric invariant theory. We show that this space carries a hyperbolic structure whose restriction to $\mathcal {M}_0^\mathbb {R}$ is that just mentioned. The corresponding lattice in ${\rm PO}(4,1)$, for which we find an explicit fundamental domain, is nonarithmetic.

Détail

Publié le : 2010-01-01

MR 2583265 | Zbl 1187.14043

DOI : https://doi.org/10.24033/asens.2116
Classification:  14J15,  14P99,  20F55,  22E40
Mots clés: surfaces cubiques, espaces des modules, géométrie algébrique réelle, géométrie hyperbolique, groupes arithmétiques, groupes de Coxeter, uniformisation

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