Stable maps and Chow groups

TL;DR

This paper investigates the Bloch-Beilinson conjecture for K3 surfaces, proving it in some cases by using stable maps and elliptic fibrations to analyze automorphisms acting trivially on the transcendental lattice.

Contribution

It demonstrates the conjecture for symplectic involutions on K3 surfaces in one third of cases using stable maps and elliptic surface techniques.

Findings

Proves the conjecture for certain symplectic involutions

Uses stable maps to construct invariant elliptic curves

Identifies conditions under which automorphisms act trivially on CH^2(X)

Abstract

According to the Bloch-Beilinson conjectures, an automorphism of a K3 surface X that acts as the identity on the transcendental lattice should act trivially on CH^2(X). We discuss this conjecture for symplectic involutions and prove it in one third of all cases. The main point is to use special elliptic K3 surfaces and stable maps to produce covering families of elliptic curves on the generic K3 surface that are invariant under the involution.