Lagrangian Subvarieties in the Chow Ring of Some Hyperkähler Varieties

Abstract

Let X be a hyperkähler variety, and let Z ⊂ X be a Lagrangian subvariety. Conjecturally, Z should have trivial intersection with certain parts of the Chow ring of X. We prove this conjecture for certain Hilbert schemes X having a Lagrangian fibration, and Z ⊂ X a general fibre of the Lagrangian fibration.

On the Chow ring of some Lagrangian fibrations

“…In dimension 2, a Lagrangian fibration is an elliptic K3 surface 1 ). As explained in [21], Conjecture 1.2 plus the Bloch-Beilinson conjectures lead in particular to the following: Conjecture 1.3. Let X be a hyperkähler variety of dimension 4.…”

“…(For a more general conjecture, which is more awkward to state, cf. [21,Conjecture 1.3].) The goal of this note is to study the conjectural injectivity property (as outlined by Conjectures 1.1, 1.2 and 1.3) for some classical examples of Lagrangian fibrations.…”