Motivic HyperK\"ahler Resolution Conjecture : I. Generalized Kummer varieties

TL;DR

This paper proves a motivic version of Ruan's HyperK"ahler Resolution Conjecture for generalized Kummer varieties and Hilbert schemes of abelian surfaces, establishing isomorphisms of Chow motives and providing Chow-K"unneth decompositions.

Contribution

It introduces and proves a motivic version of the HyperK"ahler Resolution Conjecture for specific varieties, linking Chow motives of Hilbert schemes and Kummer varieties to orbifold motives.

Findings

Chow motive of $A^{[n]}$ is isomorphic to orbifold motive of $[A^{n}/\mathfrak{S}_{n}]$

Chow motive of $K_n(A)$ is isomorphic to orbifold motive of $[A_{0}^{n+1}/\mathfrak{S}_{n+1}]$

Established multiplicative Chow-K"unneth decompositions for these varieties.

Abstract

Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-G\"ottsche, 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, one can formulate a motivic version of his Cohomological HyperK\"ahler Resolution Conjecture. We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura, 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 $[A^{n}/\mathfrak{S}_{n}]$. 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 $[A_{0}^{n+1}/\mathfrak {S}_{n+1}]$, where $A_{0}^{n+1}$ is the kernel abelian variety of the summation map $A^{n+1}\to A$. As a byproduct, we prove the original Cohomological HyperK\"ahler 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 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.