Chow motive of Fulton-MacPherson configuration spaces and wonderful compactifications
The Michigan mathematical journal
of the Dissertation Chow Motive of Fulton-MacPherson configuration spaces and wonderful compactifications by Li Li Doctor of Philosophy in Mathematics Stony Brook University 2006 Advisor: Mark Andrea de Cataldo We study the Chow groups and the Chow motives of the so-called wonderful compactifications of arrangements of subvarieties. Given a variety Y and a "building set" G associated to an arrangement of subvarieties of Y , the wonderful compactification Y G can be constructed by a sequence of
... d by a sequence of blow-ups of Y along the subvarieties of the arrangement. Our main result is that the Chow motive of Y G can be decomposed into a direct sum of the motive associated with Y and the twisted motives associated with the subvarieties of the arrangement. The decomposition obtained is canonical, for we iii prove it to be independent of the order of the blow-ups. Moreover, the correspondences that give the motivic decomposition are explicitly expressed in terms of the exceptional divisors in Y G and of the Chern classes of the normal bundles of the subvarieties of the arrangement. In the special case of the Fulton-MacPherson configuration space X[n], we prove a stronger result expressing the Chow group and the Chow motive in terms of X and n only. We provide a generating function for the Chow groups and for the Chow motive of X[n]. In the last chapter, we prove that the cobordism class of X[n] depends only on n and on the cobordism class of X.