On Murre’s conjectures for blow-ups: a constructive approach

We show, by an explicit construction of the relevant Chow-Kunneth projectors, that if X is the quotient of a smooth projective variety by a finite group action and Y is obtained from X by blowing up a finite set of points, then (under appropriate hypotheses) each of Murre’s conjectures holds for Y if and only if it holds for X. The novelty of our approach is that, finite dimensionality of the corresponding motives is never used or needed and that our construction explicitly provides the ChowKunneth projectors for X in terms of the Chow-Kunneth projectors for Y and vice-versa. This is applied to two classes of examples: one where the group action is trivial and the other to Kummer manifolds over algebraically closed fields of characteristic different from 2, which are obtained by blowing up the corresponding Kummer varieties along the 2-torsion points. In particular, our results imply that the Kummer manifolds satisfy part of Murre’s vanishing conjecture (B) in all dimensions and the full vanishing conjecture in dimensions 3 and 4.