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 . Kříž (Ann. Math. 139 (1994) 227) gave a simpler model, E n ( H ) , depending only on the cohomology ring, H ≔ H * X . struct 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 .