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

Ken Sato

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arXiv · 2509.13491

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

Ken Sato

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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.

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Ken Sato. 2025-09-16. Notes on symplectic action on $(2,1)$-cycles on $K3$ surfaces. https://arxiv.org/abs/2509.13491

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