Bloch's conjecture for (anti-)symplectic automorphisms of K3 surfaces
Abstract
We prove Bloch's conjecture for (anti-)symplectic automorphisms of smooth projective complex K3 surfaces of Picard number two. Together with the recent result for abelian surfaces and the known K3 cases, this settles the remaining case of Bloch's conjecture for (anti-)symplectic automorphisms of smooth projective complex surfaces with trivial canonical bundle. More generally, we prove Bloch's conjecture for (anti-)symplectic derived autoequivalences of projective K3 surfaces whose Picard number is at least three, or is two with anisotropic Néron--Severi lattice. The key new result is the congruence property of the subgroup generated by reflective involutions in O(U⊕ N) for anisotropic even lattices N of signature (1,1). Combined with the local generation theorem and Chow-theoretic criterion of Li--Yu--Zhang, this gives the required reflective decompositions and their consequences for zero-cycles.
Related
- Finite Symplectic Automorphisms of Hyperkähler Manifolds of K3[n]-Type Act Trivially on Discriminant Groups
- Numerically trivial automorphisms of elliptic surfaces III: the non-algebraic case
- On supersoluble like groups of automorphisms of compact Riemann surfaces
- Bound of automorphisms of fibred surfaces in positive characteristic
- On possible values of the signature of flat symplectic bundles over surfaces with boundary
BibTeX
@misc{li2026blochs,
title = {{Bloch's conjecture for (anti-)symplectic automorphisms of K3 surfaces}},
author = {Zhiyuan Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39368},
eprint = {2610.39368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39368},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bloch's conjecture for (anti-)symplectic automorphisms of K3 surfaces
AU - Zhiyuan Li
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39368
UR - https://arxiv.org/abs/2610.39368
AB - We prove Bloch's conjecture for (anti-)symplectic automorphisms of smooth projective complex K3 surfaces of Picard number two. Together with the recent result for abelian surfaces and the known K3 cases, this settles the remaining case of Bloch's conjecture for (anti-)symplectic automorphisms of smooth projective complex surfaces with trivial canonical bundle. More generally, we prove Bloch's conjecture for (anti-)symplectic derived autoequivalences of projective K3 surfaces whose Picard number is at least three, or is two with anisotropic Néron--Severi lattice. The key new result is the congruence property of the subgroup generated by reflective involutions in O(U⊕ N) for anisotropic even lattices N of signature (1,1). Combined with the local generation theorem and Chow-theoretic criterion of Li--Yu--Zhang, this gives the required reflective decompositions and their consequences for zero-cycles.
ER -