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

Lie Fu, Zhiyu Tian and Charles Vial

Geometry & Topology 23 (2019) 427–492
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 [MG] 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[n] is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [AnSn]. Case (B) concerns generalized Kummer varieties: the Chow motive of the generalized Kummer variety Kn(A) is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [A0n+1Sn+1], where A0n+1 is the kernel abelian variety of the summation map An+1 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 Rπ Riπ[i] in Dcb(B) which is compatible with the cup products on both sides, where π: Kn(A) B is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces A B.

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Keywords
hyper-Kähler varieties, symplectic resolutions, crepant resolution conjecture, Chow rings, motives, orbifold cohomology, Hilbert schemes, generalized Kummer varieties, abelian varieties
Mathematical Subject Classification 2010
Primary: 14C15, 14C25, 14C30, 14J32, 14N35
Secondary: 14K99
References
Publication
Received: 1 September 2017
Revised: 15 February 2018
Accepted: 23 April 2018
Published: 5 March 2019
Proposed: Jim Bryan
Seconded: Dan Abramovich, Gang Tian
Authors
Lie Fu
Département de Mathématiques
Université Claude Bernard Lyon 1
Institut Camille Jordan
Villeurbanne
France
Zhiyu Tian
Beijing International Center for Mathematical Research
Peking University
Beijing
China
Charles Vial
Fakultät für Mathematik
Universität Bielefeld
Bielefeld
Germany