2023/05/17 by Zhiyuan Li, Li, Zhiyuan, Yu Xun +3
Mathematics · #14F08 #14J28 #14J42 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2305.10078
openalex publication_date 2023/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate Bloch's conjecture for autoequivalence on K3 surfaces. We introduce the notion of reflective autoequivalence of twisted K3 surfaces and prove Bloch's conjecture for such autoequivalences, thereby confirming the conjecture for all (anti-)symplectic autoequivalences of K3 surfaces with Picard number at least 3. The main idea is that we find a Cartan-Dieudonné type decomposition of (anti)-symplectic autoequivalences. Our findings have several interesting consequences. Firstly, we verify Bloch's conjecture for (anti-)symplectic birational automorphisms of Bridgeland moduli space on a K3 surface with Picard number at least 3. This notably implies that Bloch's conjecture holds for (anti)-symplectic birational automorphisms of finite order (≠ 2,4) on arbitrary hyper-Kähler varieties of K3[n]-type. Secondly, we extend Huybrechts' work in \citeHuy12 to twisted K3 surfaces. This extension enables us to affirm Bloch's conjecture for symplectic birational automorphisms on any hyper-Kähler variety of K3[n]-type preserving a birational Lagrangian fibration. Finally, we prove the constant cycle property for the fixed loci of anti-symplectic involutions on hyper-Kähler varieties of K3[n]-type, provided that n ≤ 2 or the invariant sublattice has rank greater than 1.