Notes on symplectic action on $(2,1)$-cycles on $K3$ surfaces

Abstract: In this paper, we propose and study a conjecture that symplectic automorphisms of a $K3$ surface $X$ act trivially on the indecomposable part $\mathrm{CH}^2(X,1)_{\mathrm{ind}}\otimes \mathbb{Q}$ of Bloch's higher Chow group. This is a higher Chow analogue of Huybrechts' conjecture on the symplectic action on $0$-cycles. We give several partial results verifying our conjecture, some conditional and some unconditional. Our unconditional results include the full proof for Kummer surfaces of product type.