Multiplicative models for configuration spaces of algebraic varieties

Fulton and MacPherson (Ann. Math. 139 (1994) 183) found a Sullivan dg-algebra model for the space of n-configurations of a smooth complex projective variety X. K?í? (Ann. Math. 139 (1994) 227) gave a simpler model, E_n(H), depending only on the cohomology ring, H?H^*X.We construct an even simpler and smaller model, J_n(H). We then define another new dg-algebra, E_n(H^°), and use J_n(H) to prove that E_n(H^°) is a model of the space of n-configurations of the non-compact punctured manifold X^°, when X is 1-connected. Following an idea of Drinfel’d (Leningrad Math. J. 2 (1991) 829), we put a simplicial bigraded differential algebra structure on {E_n(H^°)}_n?0.