Multiplicative Models for Configuration Spaces of Algebraic Varieties

W. Fulton–R. MacPherson [15] found a Sullivan dg-algebra model for the space of n-configurations of a smooth compact complex algebraic variety X . I. Kř́ıž [16] gave a simpler model, En(H), depending only on the cohomology ring, H := H X . We construct an even simpler and smaller model, Jn(H). We then define another new dg-algebra, En( o H), and use Jn(H) to prove that En( o H) is a model of the space of nconfigurations of the non-compact punctured manifold o X , when X is 1-connected. Following an idea of V.G. Drinfel’d [10], we put a simplicial bigraded differential algebra structure on {En( o H)}n≥0.