Algebraic cycles on quadric sections of cubics in P^4 under the action of symplectomorphisms

TL;DR

This paper proves that a symplectomorphism induced by an involution acts as the identity on the second Chow group of certain K3 surfaces and describes its action on the Chow group of a related cubic threefold, using algebraic and geometric methods.

Contribution

It generalizes Voisin's method to non-conjecturally prove the identity action on CH^2 for specific K3 surfaces and analyzes the involution's action on the Chow group of a cubic threefold.

Findings

Proves the involution acts as the identity on CH^2 of the K3 surface.

Describes the involution's action on the Chow group of the cubic threefold.

Determines the involution acts as multiplication by -1 on a specific subgroup of CH^2.

Abstract

We consider the involution changing the sign of two coordinates in 4-dimensional projective space. The intersection S of invariant cubic and quadric hypersurfaces in P^4 is a K3-surface with the induced symplectomorphic action on its second cohomology group. The Bloch-Beilinson conjecture predicts that the induced action on the second Chow group CH^2(S) must be the identity. Generalizing the method developed by C. Voisin we non-conjecturally prove the identity action on CH^2 for K3-surfaces as above. Then we look at a smooth invariant cubic hypersurface C in P^4 and project it from the 1-dimensional onto the 2-dimensional linear spaces of the fixed locus of the involution. The discriminant curve splits into two components of degree 2 and 3, and the generalized Prymian can be described in terms of the Prymians P_2 and P_3 associated to the double covers of these components. Such an approach gives precise information about the induced action of the involution on the continuous part in the second Chow group CH^2(C) of the threefold C, as well as on the Hodge pieces of its third cohomology group. The action on the subgroup in CH^2(C) corresponding to the Prymian P_3 is the identity, and the action on the subgroup corresponding to P_2 is the multiplication by -1.