Hyperbolic geometry and moduli of real cubic surfaces

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

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 1, pp. 69-115.

On note ${ℳ}_0^{ℝ}$ 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 ${ℳ}_0^{ℝ}$, dont certains sont nouveaux. On note ${ℳ}_s^{ℝ}$ 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 ${ℳ}_0^{ℝ}$ est celle évoquée ci-dessus. Nous décrivons un domaine fondamental pour le réseau correspondant de $PO(4,1)$, qui s’avère être non arithmétique.

Let ${ℳ}_0^{ℝ}$ 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 $\mathrm{PO}(4,1)$. We also derive several new and several old results on the topology of ${ℳ}_0^{ℝ}$. Let ${ℳ}_s^{ℝ}$ 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 ${ℳ}_0^{ℝ}$ is that just mentioned. The corresponding lattice in $\mathrm{PO}(4,1)$, for which we find an explicit fundamental domain, is nonarithmetic.

Classification : 14J15, 14P99, 20F55, 22E40
Keywords: cubic surface, moduli, real agebraic geometry, hyperbolic geometry, arithmetic groups, Coxeter groups, uniformization
Mot clés : surfaces cubiques, espaces des modules, géométrie algébrique réelle, géométrie hyperbolique, groupes arithmétiques, groupes de Coxeter, uniformisation

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