MOTIVIC HYPERKAHLER RESOLUTION CONJECTURE FOR GENERALIZED KUMMER VARIETIES

Given a smooth projective variety M endowed with an action of a finite group G, following Jarvis–Kaufmann–Kimura [33] and Fantechi–G ¨ ottsche [25], 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 [46], one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture (CHRC). We prove this conjecture 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) for 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 n+1 0 /S n+1 ], where A n+1 0 is the kernel abelian variety of the summation map A n+1 → A. In particular, these results give complete descriptions of the Chow motive algebras (resp. Chow rings) of A [n] and K n (A) in terms of h 1 (A) the first Chow motive of A (resp. CH * (A) the Chow ring of A). As a byproduct, we prove the Cohomological HyperKähler Resolution Conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow–K ¨ unneth 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 are 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 [10]. Finally, as another application analogous to Voisin's result in [54], we prove that over a non-empty Zariski open subset of the base, there exists a decomposition isomorphism Rπ * Q ⊕R i π * Q[−i] in D b c (B) which is compatible with the cup-products on both sides, where π : K n (A) → B is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces A → B