The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio–Kirchheim’s theory of metric currents. The notion of weak differentiability we intro...

We show that the moduli space M of marked cubic surfaces is biholomorphic to (B 4 ? H)=? 0 where B 4 is complex hyperbolic four-space, where ? 0 is a speciic group generated by complex reeections, and where H is the union of reeection hyperplanes for ? 0. Thus M has complex hyperbolic structure, i.e., an (incomplete) metric of constant holomorphic sectional curvature. Une structure hyperbolique...