On a conjectural filtration on the Chow groups of an algebraic variety

Murre, Jacob P.

doi:10.1016/0019-3577(93)90039-2

Cited by 62 publications

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“…Therefore the information necessary to study the above conjecture for a surface X is concentrated in the transcendental part of the motive t 2 (X ). More precisely, according to Murre's Conjecture (see [13]), or equivalently to Bloch-Beilinson's conjecture (see [8]) and to Kimura's Conjecture the following results should hold for a surface X (a) The motive t 2 (X ) is evenly finite dimensional; (b) h(X ) satisfies the Nilpotency conjecture, i.e. every homologically trivial endomorphism of h(X ) is nilpotent ; (c) Every homologically trivial correspondence in C H 2 (X × X ) Q acts trivially on the Albanese kernel T (X ); (d) The endomorphism group of t 2 (X ) (tensored with Q) has finite rank (over a field of characteristic 0).…”

Section: Introductionmentioning

confidence: 99%

On the finite dimensionality of a K3 surface

Pedrini

2011

manuscripta math.

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For a smooth projective surface X the finite dimensionality of the Chow motive h(X ), as conjectured by Kimura, has several geometric consequences. For a complex surface of general type with p g = 0 it is equivalent to Bloch's conjecture. The conjecture is still open for a K3 surface X which is not a Kummer surface. In this paper we prove some results on Kimura's conjecture for complex K3 surfaces. If X has a large Picard number ρ = ρ(X ), i.e. ρ = 19, 20, then the motive of X is finite dimensional. If X has a non-symplectic group acting trivially on algebraic cycles then the motive of X is finite dimensional. If X has a symplectic involution i, i.e. a Nikulin involution, then the finite dimensionality of h(X ) implies h(X ) h(Y ), where Y is a desingularization of the quotient surface X/ i . We give several examples of K3 surfaces with a Nikulin involution such that the isomorphism h(X ) h(Y ) holds, so giving some evidence to Kimura's conjecture in this case.

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Section: Introductionmentioning

confidence: 99%

On the finite dimensionality of a K3 surface

Pedrini

2011

manuscripta math.

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“…More precisely, we prove that Beauville's conjecture is equivalent to Conjecture 1 for abelian varieties or to Conjecture 1 for symmetric products (C (k) , z k ) of curves, where z k is the ample divisor C (k−1) + pt. Finally we prove in the last section that for abelian varieties, Murre's conjectures (see [13]) are equivalent to Conjecture 2. In particular, this proves Conjecture 1 (resp., Conjecture 2) for (C (k) , z k ) with g(C) 4 (resp., g(C) 3).…”

Section: Conjecturementioning

confidence: 90%

“…In [2], it was conjectured that c 0 is always injective. The following result is well-known and is easily deduced from [13]. Proof.…”

Section: Lemmamentioning

confidence: 94%

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Remarks on Hard Lefschetz conjectures on Chow groups

2010

Sci. China Ser. A-Math.

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Algebraic cycles and EPW cubes

Laterveer

2018

Mathematische Nachrichten

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Let $X$ be a hyperk\"ahler variety with an anti-symplectic involution $\iota$. According to Beauville's conjectural "splitting property", the Chow groups of $X$ should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch-Beilinson conjectures predict how $\iota$ should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a $19$-dimensional family of hyperk\"ahler sixfolds that are "double EPW cubes" (in the sense of Iliev-Kapustka-Kapustka-Ranestad). This has interesting consequences for the Chow ring of the quotient $X/\iota$, which is an "EPW cube" (in the sense of Iliev-Kapustka-Kapustka-Ranestad).Comment: 32 pages, to appear in Math. Nachrichten, feedback welcom

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On a conjectural filtration on the Chow groups of an algebraic variety

Cited by 62 publications

References 1 publication

On the finite dimensionality of a K3 surface

On the finite dimensionality of a K3 surface

Remarks on Hard Lefschetz conjectures on Chow groups

Algebraic cycles and EPW cubes

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