Moduli Spaces and Formal Operads

Let M g,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations which relate the different moduli spaces identifying marked points, the family (M g,l) g,l is a modular operad of projective smooth Deligne-Mumford stacks, M. In this paper we prove that the modular operad of singular chains C * (M; Q) is formal; so it is weakly equivalent to the modular operad of its homology H * (M; Q). As a consequence, the " up to homotopy " algebras of these two operads are the same. To obtain this result we prove a formality theorem for operads analogous to Deligne-Griffiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field.