THE HODGE POLYNOMIAL OF THE FULTON-MACPHERSON COMPACTIFICATION OF CONFIGURATION SPACES

Abstract. Let X be a smooth complex variety. In the paper “A compactification of configuration spaces, ” Fulton and MacPherson construct a smooth compactification X[n] of the configuration space of n distinct labelled points on X; the space X[n] is projective if X is projective. In this paper, we obtain a generating function which expresses the virtual Hodge polynomials of the spaces X[n] in terms of the virtual Hodge polynomial of X. This enables us to read off the Euler characteristic of X[n] in terms of that of X and in the case when X is projective, the Hodge polynomial and the Poincare ́ polynomial of X[n] in terms of those of X. Let X be a smooth complex variety. In [Ful-Mac], Fulton and MacPherson construct a smooth compactification X[n] of the configuration space of n distinct labelled points on X; the space X[n] is projective if X is projective. In this paper, we obtain a generating function which expresses the virtual Hodge polynomials of the spaces X[n] in terms of the virtual Hodge polynomial of X. This enables us to read off the Euler characteristic of X[n] in terms of that of X and in the case when X is projective, the Hodge polynomial and the Poincare ́ polynomial of X[n] in terms of those of X