Given a smooth projective variety M endowed with a faithful action of a finite group G, following Jarvis-Kaufmann-Kimura [36] and Fantechi-Göttsche [26], 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 [51], one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture (CHRC). We prove this motivic version, as well as its K-theoretic analogue conjectured in [36], 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+1is the kernel abelian variety of the summation map A n+1 → A. As a byproduct, we prove the original Cohomological HyperKähler Resolution Conjecture for generalized Kummer varieties.As an application, we provide multiplicative Chow-Künneth 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 [10]. Finally, as another application, 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.Meta-conjecture 1.2 (MHRC). Let X be a smooth proper complex Deligne-Mumford stack with underlying coarse moduli space X being a (singular) symplectic variety. If there is a symplectic resolution Y → X, then we have an isomorphism h(Y) ≃ h orb (X) of commutative algebra objects in CHM C , hence in particular an isomorphism of graded C-algebras : CH * (Y) C ≃ CH * orb (X) C .See Definition 3.1 for generalities on symplectic singularities and symplectic resolutions. The reason why it is only a meta-conjecture is that the definition of orbifold Chow motive for a smooth proper Deligne-Mumford stack in general is not available in the literature and we will not develop the theory in this generality in this paper (see however Remark 2.10). From now on, let us restrict ourselves to the case where the Deligne-Mumford stack in question is of the form of a global quotient X = [M/G], where M is a smooth projective variety with a faithful action of a finite group G, in which case we will define the orbifold Chow motive h orb (X) in a very explicit way in Definition 2.5.The Motiv...

Abstract. We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of klt singularities. As an application, we verify that the local fundamental group of a three dimensional klt singularity and the fundamental group of the smooth locus of a three dimensional Fano variety with canonical singularities are always finite.

We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersections of index at least 2 is generated by lines.As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

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We discuss the Picard group of the moduli space K g of quasi-polarized K3 surfaces of genus g ≤ 12 and g = 11. In this range, K g is unirational, and a general element in K g is a complete intersection with respect to a vector bundle on a homogenous space, by the work of Mukai. In this paper, we find generators for the Picard group Pic Q (K g ) using the Noether-Lefschetz (NL) theory. This verifies the NL conjecture on the moduli of K3 surfaces in these cases.