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Une structure hyperbolique complexe pour les modules des surfaces cubiques. (A complex hyperbolic structure for moduli of cubic surfaces.) (English. Abridged French version) Zbl 0959.32035
From the paper: Main results: To a (marked) cubic surface corresponds a (marked) cubic threefold defined as the triple cover of \(\mathbb{P}^3\) ramified along the surface. The period map \(f\) for these threefolds is defined on the moduli space \(M\) of marked cubic surfaces and takes its values in the quotient of the unit ball in \(\mathbb{C}^4\) by the action of the projective monodromy group. This group \(\Gamma\) is generated by complex reflections in a set of hyperplanes whose union we denote by \({\mathcal H}\). Then we have the following result:
Theorem. The period map defines a biholomorphism: \(f:M \to(B^4-{\mathcal H})/ \Gamma\), where \(B^4\) is the complex hyperbolic four-space.
From this identification we obtain results on the metric structure and the fundamental group:
Corollary. (1) The moduli space of marked cubic surfaces carries a complex hyperbolic structure: an (incomplete) metric of constant negative holomorphic sectional curvature.
(2) The fundamental group of the space of marked cubic surfaces contains a normal subgroup which is not finitely generated.
(3) The fundamental group of the space of marked cubic surfaces is not a lattice in a semisimple Lie group.

MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32G20 Period matrices, variation of Hodge structure; degenerations
14J10 Families, moduli, classification: algebraic theory
14F30 \(p\)-adic cohomology, crystalline cohomology
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