Bloch's conjecture for (anti-)autoequivalences on K3 surfaces

Abstract

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-Dieudonn\'e 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 ($\neq 2,4$) on arbitrary hyper-K\"ahler varieties of $K3^{[n]}$-type. Secondly, we extend Huybrechts' work in \cite{Huy12} to twisted K3 surfaces. This extension enables us to affirm Bloch's conjecture for symplectic birational automorphisms on any hyper-K\"ahler 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-K\"ahler varieties of $K3^{[n]}$-type, provided that $n \leq 2$ or the invariant sublattice has rank greater than $1$.