[Paper Review] Motivic HyperK\"ahler Resolution Conjecture : I. Generalized Kummer varieties

Lie Fu, Zhiyu Tian+1 more|arXiv (Cornell University)|Aug 17, 2016

Algebraic structures and combinatorial models8 citations

TL;DR

This paper proves the motivic Hyperk"ahler Resolution Conjecture for generalized Kummer varieties and Hilbert schemes of abelian surfaces, establishing an isomorphism between the Chow motive of the hyperk"ahler resolution and the orbifold motive of the corresponding quotient stack. The key result is a multiplicative Chow--K"unneth decomposition for these varieties, confirming a conjecture of Beauville and extending Voisin's multiplicative decomposition theorem to relative families of generalized Kummer varieties over a Zariski open subset of the base.

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 $A^{n+1} o A$. 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.

Motivation & Objective

• To formulate and prove a motivic version of Ruan's Cohomological HyperK"ahler Resolution Conjecture for holomorphic symplectic varieties arising as symplectic resolutions of global quotient orbifolds.
• To establish a Chow motive isomorphism between the generalized Kummer variety $K_n(A)$ and the orbifold motive of $[A^{n+1}_0 / S_{n+1}]$, up to a sign twist.
• To prove the existence of multiplicative Chow--K"unneth decompositions for Hilbert schemes of abelian surfaces and generalized Kummer varieties, supporting Beauville's splitting conjecture.
• To extend Voisin's multiplicative decomposition theorem for rational cohomology to relative families of generalized Kummer varieties over a non-empty Zariski open subset of the base.

Proposed method

• Define the orbifold motive of a global quotient stack $[M/G]$ as the $G$-invariant subalgebra of a direct sum of twisted Chow motives of fixed loci, using age shifts and orbifold product via normal bundle Chern classes.
• Construct the orbifold product $\star_{\text{orb}}$ on $\bigoplus_{g \in G} CH^{*-\text{age}(g)}(M_g)$ via pushforward, restriction, and top Chern class of obstruction bundles.
• Prove the motivic Hyperk"ahler Resolution Conjecture in two cases: (A) Hilbert schemes $A^{[n]}$ and (B) generalized Kummer varieties $K_n(A)$, using base change and functoriality in Chow theory.
• Apply Voisin's result on symmetrically distinguished cycles to show that Chern classes of $K_n(A)$ are preserved under the relevant correspondences, ensuring compatibility with Chow--K"unneth projectors.
• Use spreading out of multiplicative Chow--K"unneth decompositions to extend the multiplicative decomposition theorem to relative families over a Zariski open subset of the base.
• Leverage the fact that the generic fiber of the relative family admits a multiplicative Chow--K"unneth decomposition to construct a multiplicative decomposition in the derived category of sheaves.

Experimental results

Research questions

• RQ1 Does the motivic Hyperk"ahler Resolution Conjecture hold for generalized Kummer varieties as symplectic resolutions of global quotient orbifolds?
• RQ2 Can the Chow motive of a generalized Kummer variety $K_n(A)$ be isomorphic to the orbifold motive of $[A^{n+1}_0 / S_{n+1}]$ as algebra objects in the category of Chow motives with complex coefficients?
• RQ3 Do Hilbert schemes of abelian surfaces and generalized Kummer varieties admit multiplicative Chow--K"unneth decompositions, as predicted by Beauville's splitting conjecture?
• RQ4 Is there a multiplicative decomposition of the rational cohomology of a relative generalized Kummer variety over a non-empty Zariski open subset of the base?
• RQ5 Can the multiplicative structure of the rational cohomology of relative families of generalized Kummer varieties be preserved under the derived decomposition theorem?

Key findings

• The Chow motive of the generalized Kummer variety $K_n(A)$ is isomorphic as a commutative algebra object in the category of Chow motives with complex coefficients to the orbifold motive of $[A^{n+1}_0 / S_{n+1}]$, up to a sign twist.
• The Chow motive of the Hilbert scheme $A^{[n]}$ is isomorphic to the orbifold motive of $[A^n / S_n]$, confirming the motivic Hyperk"ahler Resolution Conjecture in case (A).
• The generalized Kummer variety $K_n(A)$ admits a multiplicative Chow--K"unneth decomposition, providing a multiplicative direct sum decomposition of its Chow ring with rational coefficients.
• The original Cohomological HyperK"ahler Resolution Conjecture is proven for generalized Kummer varieties as a byproduct of the motivic result.
• For any smooth family of abelian surfaces $\pi: A \to B$, there exists a non-empty Zariski open subset $U \subset B$ such that the relative generalized Kummer variety $\pi|_U: K_n(A)|_U \to U$ admits a multiplicative decomposition of rational cohomology in the derived category.
• The Chern classes of $K_n(A)$ are symmetrically distinguished on each component of the fixed loci of the symmetric group action, ensuring compatibility with the Chow--K"unneth projectors.