On the splitting of the Bloch-Beilinson filtration

By A. Beauville. 2004

117 citations recorded by the source corpus.

For a smooth projective variety X, let CH(X) be the Chow ring (with rational coefficients) of algebraic cycles modulo rational equivalence. The conjectures of Bloch and Beilinson predict the existence of a functorial ring filtration of CH(X). We want to investigate for which varieties this filtration splits, that is, comes from a graduation on CH(X) -- this occurs for K3 surfaces and, conjecturally, for abelian varieties. We observe that, though the Bloch-Beilinson filtration is only conjectural, the fact that it splits has some simple consequences which can be tested in concrete examples. Namely, for a regular variety X, it implies that the sub-Q-algebra of CH(X) spanned by divisor classes injects into the cohomology of X . We give examples of Calabi-Yau threefolds which do not satisfy this property. On the other hand we conjecture that the property does indeed hold for (holomorphic) symplectic manifolds, and we give some (weak) evidence in favour of this conjecture.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d4ac38f557c6d84e7452fbc5494960224b71a4b9

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Citations in the corpus

THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ (2015, methodology): “Then Beauville [2], after studying intersection of divisors on certain hyperKähler varieties, asked whether the filtration, if it exists, would split for all hyperKähler varieties.”
Holomorphic symplectic geometry: a problem list (2010, background): “This would follow from a much more general (and completely out of reach) conjecture, for which we refer to the introduction of [ 6 ].”
Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties (2015, background): “In fact, we describe a parametrization of them which should make accessible for them a number of conjectures made in this paper, [5], or [28], by reduction to the case of the variety of lines of a cubic fourfold.”
On the Chow ring of certain algebraic hyper-K\ (2006, background): “For any line bundle L on S, there is an induced line bundle, which we still denote by L on S[n], which is the pull-back via the Hilbert-Chow morphism of the line bundle on S(n) corresponding to the Sn-invariant line bundle L⊠ . . .⊠L on Sn.”
Lectures on K3 Surfaces (2016, background): “This has been put forward by Beauville in [49] and strengthened and verified in a series of examples by Voisin in [620].”
The Fourier Transform for Certain HyperKähler Fourfolds (2013, methodology): “Property (2) for divisors on hyperkähler varieties F is Beauville’s weak splitting conjecture [9] ; it was subsequently strengthened in [54] to include the Chern classes of F .”
Monodromy and birational geometry of O'Grady's sixfolds (2019, methodology): “As an immediate consequence of the above proposition by using [40, Theorem 4.2], we obtain that the Weak Splitting property conjectured by Beauville [3] holds when the manifold has a square zero divisor.”
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, background): “In [9], Beauville conjectured that a result similar to Theorem 5.”