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

Fu, Lie; Tian, Zhiyu; Vial, Charles

doi:10.2140/gt.2019.23.427

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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

DOI: 10.2140/gt.2019.23.427

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 $[M ∕ G]$ 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-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 $[A^{n} ∕ S_{n}]$ . 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 $[A_{0}^{n + 1} ∕ S_{n + 1}]$ , 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 $R π_{*} ℚ ≃ \oplus R^{i} π_{*} ℚ [- i]$ in $D_{c}^{b} (B)$ which is compatible with the cup products on both sides, where $π : K_{n} (A) \to B$ is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces $A \to B$ .

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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

Volume 23, issue 1 (2019)