Beauville-Voisin Conjecture for Generalized Kummer Varieties

Abstract

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...

Algebraic cycles on a very special EPW sextic

“…The symmetrically distinguished cycles form a Q-subalgebra Remark 2.25. For discussion and applications of the notion of symmetrically distinguished cycles, in addition to [37] we refer to [43,Section 7], [50], [3], [20].…”

Lagrangian Subvarieties in the Chow Ring of Some Hyperkähler Varieties

“…(cf. [5], [48], [3], [10], [41], [13], [57] for cases where conjecture 1.1 is satisfied.) The "motivation" underlying conjecture 1.1 is that for a hyperkähler variety X, the Chow ring A * (X) is expected to have a bigrading A *…”

“…Suppose we know in addition that Γ b ∈ A 2 (0) (X b ) (for instance because Γ b is the fibre of a Lagrangian fibration). Then conjecturally, equality (13) implies there is a rational equivalence (14) Γ b…”

A remark on the Chow ring of some hyperkähler fourfolds

“…Remark 2.9. For discussion and applications of the theory of symmetrically distinguished cycles, in addition to [20] we refer to [24,Section 7], [26], [1], [10], [11]. Proposition 2.10.…”