On the Rational Homotopy Type of Configuration Spaces

Abstract

In this article we prove that the rational homotopy type of the configuration space of n distinct points on a smooth projective variety X over C is determined by the rational cohomology ring of X together with its canonical orientation class. This strengthens a previous result of Fulton and MacPherson [FM] (see previous issue in this volume). A few words are due on the history of differential graded algebras. By a DGA we shall mean a graded-commutative differential graded algebra. Fulton and MacPherson [FM] gave a model F(n) for the space F(X, n) by resolving the singularities of the inclusion F(X, n) C Xn and using a result of Morgan [M] on the rational homotopy type of smooth quasiprojective varieties over C. Surprisingly the DGA F(n) depended not only on H*X and its orientation class, but also on the Chern classes of X. Fulton suggested that one should be able to find another explicit model of F(X, n) independent of Chern classes. The DGA used here, E(n), was found independently by the author and by Totaro [T]. Totaro's idea was to study the Leray spectral sequence of the inclusion F(X, n) C Xn. By a weight-filtration argument he showed that the only nontrivial differential in this spectral sequence is d2m. The E2 term with the differential d2m is isomorphic to E(n). Thus, by a result of Deligne [D2], H*E(n) ' H*F(X, n) as rings. Finally the relations (1)-(3) of Theorem 1.2 are much older than any of the results mentioned above. They were discovered by F. Cohen [C] (see also [CLMa]) in the study of configuration spaces of IR'. What happens could be described by saying that H*F(X, n) for a smooth projective variety X is related to H*F(lR2m, n) in the simplest possible way.