Chow Motive of Fulton-macpherson Configuration Spaces and Wonderful Compactifications

The purpose of this article is to study the Chow groups and Chow motives of the so-called wonderful compactifications of an arrangement of subvarieties, in particular the Fulton-MacPherson configuration spaces. All the varieties in the paper are over an algebraically closed field. Let Y be a nonsingular quasi-projective variety. Let S be an arrangement of subvarieties of Y (cf. Definition 2.2). Let G be a building set of S, i.e., a finite set of nonsingular subvarieties in S satisfying Definition 2.3. The wonderful compactification YG is constructed by blowing up Y along subvarieties in G successively (cf. Definition 2.5). There are different orders in which the blow-ups can be carried out, for example we can blow up along the centers in any order that is compatible with the inclusion relation. There are many important examples of such compactifications: De Concini and Procesi’s wonderful model of a subspace arrangement, the Fulton-MacPherson configuration spaces, the moduli space M0,n of stable rational curves with n marked points, etc. These spaces have many properties in common. Studying them by a uniform method gives us better understanding of these spaces. In this article, we study their Chow groups and Chow motives. If we assume that Y is projective, then the Chow motive of YG , denoted by h(YG), can be decomposed canonically into a direct sum of the motive of Y and the twisted motives of the subvarieties in the arrangement (cf. §2.1 for a review of Chow motives). We will prove the following theorem, where the precise definition of the set MT and the subvarieties Y0T of Y are in §3.