No abstract

The moduli space of cubic threefolds in CP 4 , with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.In the second-to-last step we used the relation σ −2 +σ 2 = σ+σ −1 = −1. That ·|· is a C-valued Hermitian form now follows from α|β = β| α ; to prove this it suffices to check that the imaginary part of (2.2) changes sign when α and β are exchanged, i.e.,

L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

No abstract

Let D be bounded symmetric domain of rank p, and let o9 be the KRhler form of its Bergmann metric, where the metric is normalized so that the minimum holomorphic sectional curvature is -1. IfX is a compact manifold whose universal cover is D, to defines a cohomology class [to] e H 2 (X, R). The purpose of this paper is to compute its sup norm in the sense of Gromov [5]. We show II [to] II co = pn. Strictly speaking, we only prove this for three of the four classes of classical domains (cf. Sect. 2).This theorem has the following topological corollary. Let S be a Riemann surface of genus 0 > 1 and f : S~X a continuous map. Then ! f'to < 4p(#-1)n.This was proved in [8] for the case p= 1, i.e., D=unit ball in C", by different methods: harmonic mappings and Bochner's formula. The original motivation for this paper was to extend these methods to higher rank. This turned out to be impossible, in the sense that the pointwise inequality needed to apply Boehner's formula is actually false for p> 1. We thus find it quite interesting that Gromov's methods can be used to derive the topological corollary.In [8] the case of equality was also settled: it can only hold for mappings homotopic to a totally geodesic holomorphic (or anti-holomorphic) immersion. For higher rank we find that certain complex geodesics give equality, but that (at least in certain "reducible" situations) these need not be the only extremal homotopy classes.We point out the relation of this paper to bounded cohomology. One would like to prove that ifD = G/K is a symmetric space of non-compact type and to is a G-invariant form on D, then for any compact manifold X covered by D, the Corresponding cohomology class [to]