On the Chow ring of a K3 surface

Beauville, Arnaud; Voisin, Claire

doi:10.1090/s1056-3911-04-00341-8

Cited by 172 publications

(408 citation statements)

References 6 publications

Supporting

Mentioning

395

Contrasting

Order By: Relevance

“…However, contrary to what was claimed there, it does not suffice to show that CH 2 (X) Q is finitedimensional. Our proof relies deeply on the result of Beauville-Voisin [2].…”

Section: Introductionmentioning

confidence: 99%

“…Let S be a projective K3 surface. In [2], Beauville and Voisin studied the Chow ring CH * (S) of S. They showed that there is a canonical class c S ∈ CH 0 (S) represented by a point on a rational curve in S, which satisfies the following properties:…”

Section: Introductionmentioning

confidence: 99%

“…Further, it was shown in [3,Proposition 2] that the Chow ring CH * (G (2,3,4)) is generated by the Chern classes of V 16 , V 16 . To prove Theorem 0.2 we have to take care of the second Chern classes of both tautological bundles.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On O'Grady's Generalized Franchetta Conjecture

Pavic

Shen

Yin

2016

Int Math Res Notices

View full text Add to dashboard Cite

Abstract. We study relative zero cycles on the universal polarized K3 surface X → Fg of degree 2g − 2. It was asked by O'Grady if the restriction of any class in CH 2 (X) to a closed fiber Xs is a multiple of the Beauville-Voisin canonical class c Xs ∈ CH 0 (Xs). Using Mukai models, we give an affirmative answer to this question for g ≤ 10 and g = 12, 13, 16, 18, 20. 0. Introduction Throughout, we work over the complex numbers. Let S be a projective K3 surface. In [2], Beauville and Voisin studied the Chow ring CH * (S) of S. They showed that there is a canonical class c S ∈ CH 0 (S) represented by a point on a rational curve in S, which satisfies the following properties:(i) The intersection of two divisor classes on S always lies in Zc S ⊂ CH 0 (S).(ii) The second Chern class c 2 (T S ) equals 24c S ∈ CH 0 (S). This result is rather surprising since the Chow group CH 0 (S) is infinite-dimensional by Mumford's theorem [7].Let F g denote the moduli space of (primitively) polarized K3 surfaces of degree 2g − 2. For g ≥ 3, let F 0 g ⊂ F g be the open dense subset parametrizing polarized K3 surfaces with trivial automorphism groups, which carries a universal family X → F 0 g . Motivated by Franchetta's conjecture on the moduli spaces of curves (see [1]), O'Grady asked the following question in [12], referred to as the generalized Franchetta conjecture.Question 0.1 (Generalized Franchetta conjecture). Given a class α ∈ CH 2 (X) and a closed point s ∈ F 0 g , is it true that α| Xs ∈ Zc Xs ? The goal of this paper is to give an affirmative answer to Question 0.1 for a list of small values of g. By the work of Mukai [8,9,10,11], for these g a general polarized K3 surface can be realized in a variety with "small" Chow groups as a complete intersection with respect to a vector bundle.Theorem 0.2. The generalized Franchetta conjecture holds for g ≤ 10 and g = 12, 13, 16, 18, 20.The paper is organized as follows. In Section 1 we review Mukai's constructions and make some comments about Question 0.1. In Section 2 we prove Theorem 0.2 for all cases except g = 13, 16. Two independent proofs are presented, one using Voisin's result [17], the other via a direct calculation. The cases g = 13, 16 have a different flavor and are treated in Section 3. Acknowledgement. We are grateful to Rahul Pandharipande for his constant support and his enthusiasm in this project. We also thank Kieran O'Grady for his careful reading of a preliminary version of this paper.

show abstract

“…However, contrary to what was claimed there, it does not suffice to show that CH 2 (X) Q is finitedimensional. Our proof relies deeply on the result of Beauville-Voisin [2].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On O'Grady's Generalized Franchetta Conjecture

Pavic

Shen

Yin

2016

Int Math Res Notices

View full text Add to dashboard Cite

show abstract

“…In [7], Beauville and Voisin observe the following property of the Chow rings of projective K3 surfaces. is simply connected and H 2,0 (X) is 1-dimensional and generated by a holomorphic 2-form which is non-degenerate at each point of X.…”

Section: Introductionmentioning

confidence: 99%

“…In [7], Beauville and Voisin observe the following property of the Chow rings of projective K3 surfaces.…”

Section: Introductionmentioning

confidence: 99%

Beauville-Voisin Conjecture for Generalized Kummer Varieties

Fu¹

2014

International Mathematics Research Notices

View full text Add to dashboard Cite

Abstract. Inspired by their results on the Chow rings of projective K3 surfaces, Beauville and Voisin made the following conjecture: given a projective hyperkähler manifold, for any algebraic cycle which is a polynomial with rational coefficients of Chern classes of the tangent bundle and line bundles, it is rationally equivalent to zero if and only if it is numerically equivalent to zero. In this paper, we prove the Beauville-Voisin conjecture for generalized Kummer varieties. IntroductionIn [7], Beauville and Voisin observe the following property of the Chow rings of projective K3 surfaces. is simply connected and H 2,0 (X) is 1-dimensional and generated by a holomorphic 2-form which is non-degenerate at each point of X. In particular, a hyperkähler variety has trivial canoncial bundle. Examples 1.3. Let us give some basic examples of projective hyperkähler manifolds:• (Beauville [3]) Let S be a projective K3 surface and n ∈ N, then S [n] , which is the Hilbert scheme of subschemes of dimension 0 and length n, is hyperkähler of dimension 2n. [17]) for certain varieties with trivial canonical bundle. He suggests to verify the following down-to-earth consequence of this conjectural splitting of the conjectural filtration on Chow groups of hyperkähler varieties. As a first evidence, the special cases when X = S [2] or S [3] for a projective K3 surface S are verified in his paper loc.cit. Conjecture 1.4 (Beauville). Let X be a projective hyperkähler manifold, and z ∈ CH(X) Q be a polynomial with Q-coefficients of the first Chern classes of line bundles on X. Then z is homologically trivial if and only if z is (rationally equivalent to) zero.Voisin pursues the work of Beauville and makes in [25] the following stronger version of Conjecture 1.4, by involving also the Chern classes of the tangent bundle: Conjecture 1.5 (Beauville-Voisin). Let X be a projective hyperkähler manifold, and z ∈ CH(X) Q be a polynomial with Q-coefficients of the first Chern classes of line bundles on X and the Chern classes of the tangent bundle of X. Then z is numerically trivial if and only if z is (rationally equivalent to) zero.Here we replaced 'homologically trivial' in the original statement in Voisin's paper [25] by 'numerically trivial'. But according to the standard conjecture [18], the homological equivalence and the numerical equivalence are expected to coincide. We prefer to state the Beauville-Voisin conjecture in the above slightly stronger form since our proof for generalized Kummer varieties also works in this generality.In [25], Voisin proves Conjecture 1.5 for the Fano varieties of lines of cubic fourfolds, and for S [n] if S is a projective K3 surface and n ≤ 2b 2,tr + 4, where b 2,tr is the second Betti number of S minus its Picard number. We remark that here we indeed can replace the homological equivalence by the numerical equivalence since the standard conjecture in these two cases has been verified by Charles and Markman [8].The main result of this paper is to prove the Beauville-Voisin conjecture 1.5 for gene...

show abstract

Algebraic cycles and EPW cubes

Laterveer

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

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

show abstract

On the Chow ring of a K3 surface

Abstract: We show that the Chow group of 0-cycles on a K3 surface contains a class of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class c 2 .

Cited by 172 publications

References 6 publications

On O'Grady's Generalized Franchetta Conjecture

On O'Grady's Generalized Franchetta Conjecture

Beauville-Voisin Conjecture for Generalized Kummer Varieties

Algebraic cycles and EPW cubes

Contact Info

Product

Resources

About