Birational Chow–Künneth decompositions

Shen, Mingmin

doi:10.4310/pamq.2016.v12.n1.a4

Cited by 3 publications

(2 citation statements)

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“…In addition, a rational map f : X Y induces a well-defined morphism f * : h • (X) → h • (Y ), obtained by restricting to the generic point of X the graph of f | U , where U ⊆ X is a dense open subset over which f is defined. We refer to [She16,§2] for further explicit calculations involving morphisms of birational motives ; of particular relevance is the fact that a generically finite rational map f : X Y induces a well-defined morphism…”

Section: On Co-algebra Objects In a ⊗-Categorymentioning

confidence: 99%

“…Birational Chow-Künneth decompositions. The following definitions are borrowed from Shen [She16,§3]. Fix a Weil cohomology theory H • for smooth projective varieties defined over K ; e.g., ℓ-adic cohomology for ℓ = char(K), or Betti cohomology if K ⊆ C. For a smooth projective variety X over K, we then define its transcendental cohomology to be the quotient…”

Section: Co-multiplicative Birational Chow-künneth Decompositionsmentioning

confidence: 99%

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On the birational motive of hyper-Kähler varieties

Vial¹

2020

Preprint

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We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for certain moduli spaces of stable sheaves on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with respect to the diagonal embedding and to show in the aforementioned cases the existence of a so-called strict grading. Furthermore, we upgrade to rational equivalence Voisin's notion of "surface decomposition" and use this to show that the birational motive of some projective hyper-Kähler manifolds is determined, as a co-algebra object, by the birational motive of a surface. In an appendix, we relate our co-radical filtration on the Chow groups of abelian varieties to Beauville's eigenspace decomposition.

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Section: On Co-algebra Objects In a ⊗-Categorymentioning

confidence: 99%

Section: Co-multiplicative Birational Chow-künneth Decompositionsmentioning

confidence: 99%