Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties

Abstract

This paper proposes a conjectural picture for the structure of the Chow ring CH * (X) of a (projective) hyper-Kähler variety X, that seems to emerge from the recent papers [9], [23], [24], [25], with emphasis on the Chow group CH 0 (X) of 0-cycles (in this paper, Chow groups will be taken with Q-coefficients). Our motivation is Beauville's conjecture (see [5]) that for such an X, the Bloch-Beilinson filtration has a natural, multiplicative, splitting. This statement is hard to make precise since the Bloch-Beilinson filtration is not known to exist, but for 0-cycles, this means thatwhere the decomposition is given by the action of self-correspondences Γ i of X, and where the group CH 0 (X) i depends only on (i, 0)-forms on X (the correspondence Γ i should act as 0 on H j,0 for j = i, and Id for i = j). We refer to the paragraph 0.1 at the end of this introduction for the axioms of the Bloch-Beilinson filtration and we will refer to it in Section 3 when providing some evidence for our conjectures. Note that a hyper-Kähler variety X has H i,0 (X) = 0 for odd i, so the Bloch-Beilinson filtration F BB has to satisfy F i BB CH 0 (X) = F i+1 BB CH 0 (X) when i is odd. Hence we are only interested in the F 2i BBlevels, which we denote by F ′ i BB . Note also that there are concrete consequences of the Beauville conjecture that can be attacked directly, namely, the 0-th piece CH(X) 0 should map isomorphically via the cycle class map to its image in H * (X, Q) which should be the subalgebra H * (X, Q) alg of algebraic cycle classes, since the Bloch-Beilinson filtration is conjectured to have F 1 BB CH * (X) = CH * (X) hom . Hence there should be a subalgebra of CH * (X, Q) which is isomorphic to the subalgebra H 2 * (X, Q) alg ⊂ H 2 * (X, Q) of algebraic classes. Furthermore, this subalgebra has to contain NS(X) = Pic(X). Thus a concrete subconjecture is the following prediction (cf. This conjecture has been enlarged in [28] to include the Chow-theoretic Chern classes of X, c i (X) := c i (T X ) which should thus be thought as being contained in the 0-th piece of the conjectural Beauville decomposition. Our purpose in this paper is to introduce a new set of classes which should also be put in this 0-th piece, for example, the constant cycles subvarieties of maximal dimension (namely n, with dim X = 2n, because they have to be isotropic, see Section 1) and their higher dimensional generalization, which are algebraically coisotropic. Let us explain the motivation for this, starting from the study of 0-cycles.Based on the case of S [n] where we have the results of [4], [22], [25] that concern the CH 0 group of a K3 surface but will be reinterpreted in a slightly different form in Section 2, we introduce the following decreasing filtration S · on CH 0 (X) for any hyper-Kähler manifold *

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