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- W3100811823 abstract "Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-Gottsche, 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 HyperKahler 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 HyperKahler Resolution Conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow-Kunneth 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." @default.
- W3100811823 created "2020-11-23" @default.
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- W3100811823 date "2019-03-05" @default.
- W3100811823 modified "2023-10-18" @default.
- W3100811823 title "Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties" @default.
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- W3100811823 doi "https://doi.org/10.2140/gt.2019.23.427" @default.
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