Motivic hyper-Kähler resolution conjecture, I: Generalized Kummer varieties

Abstract

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–Göttsche (Duke Math. J. 117 (2003) 197–227), we define the orbifold motive (or Chen–Ruan motive) of the quotient stack $\left[M∕G\right]$ 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-Kähler 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 $A^{\left[n\right]}$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $\left[A^n∕{\mathfrak{S}}_n\right]$. Case (B) concerns generalized Kummer varieties: the Chow motive of the generalized Kummer variety $K_n\left(A\right)$ is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack $\left[A_0^{n+1}∕{\mathfrak{S}}_{n+1}\right]$, where $A_0^{n+1}$ is the kernel abelian variety of the summation map $A^{n+1}\to A$. As a by-product, we prove the original cohomological hyper-Kähler 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 (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 $R{\pi }_{\ast }\mathbb{Q}\simeq \oplus R^i{\pi }_{\ast }\mathbb{Q}\left[-i\right]$ in $D_c^b\left(B\right)$ which is compatible with the cup products on both sides, where $\pi :{\mathcal{K}}_n\left(\mathcal{A}\right)\to B$ is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces $\mathcal{A}\to B$.