PUB - Publikationen an der Universität Bielefeld

Given a smooth projective variety M endowed with a faithful action of a finite group G, following Jarvis-Kaufmann-Kimura (Invent. Math. 168 (2007) 23-81), and Fantechi-Gottsche (Duke Math. J. 117 (2003) 197-227), we define the orbifold motive (or Chen-Ruan motive) of the quotient stack [M/G] as an algebra object in the category of Chow motives. Inspired by Ruan (Contemp. Math. 312 (2002) 187-233), one can formulate a motivic version of his cohomological hyper-Kahler resolution conjecture (CHRC). We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura in loc. cit., in two situations related to an abelian surface A and a positive integer n. Case (A) concerns Hilbert schemes of points of A: the Chow motive of A([n]) is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [A(n)/S-n]. Case (B) concerns generalized Kummer varieties: the Chow motive of the generalized Kummer variety K-n(A) is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [A(0)(n+1)/Sn+1], where A(0)(n+1) is the kernel abelian variety of the summation map A(n+1) -> A. As a by-product, we prove the original cohomological hyper-Kahler resolution conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow-Kunneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch-Beilinson-Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville (London Math. Soc. Lecture Note Ser. 344 (2007) 38-53). Finally, as another application, we prove that over a nonempty Zariski open subset of the base, there exists a decomposition isomorphism R pi(*)Q similar or equal to circle plus R-i pi(*)Q[-i] in D-c(b)(B) which is compatible with the cup products on both sides, where pi: kappa(n)(A) -> B is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces A -> B.