Bloch's conjecture for equivalences between twisted abelian surfaces and applications

The Beauville–Voisin conjecture predicts a canonical descending filtration on the Chow group of zero-cycles of a hyperkähler variety, opposite to the conjectural Bloch–Beilinson filtration. A basic test for such filtrations is a Bloch-type principle: the action on zero-cycles should be governed by the action on the holomorphic symplectic form. While this principle has been verified in several cases of hyperkähler varieties of ${\mathrm{K}3}^{[n]}$-type, the ${\mathrm{K}\mathrm{u}\mathrm{m}}_n$-type case remains much less understood. In this paper, we study this problem through twisted abelian surfaces and their associated ${\mathrm{K}\mathrm{u}\mathrm{m}}_n$-type varieties. We first construct a natural action of autoequivalences of twisted abelian surfaces on the Albanese kernel and prove Bloch’s conjecture for all (anti-)symplectic autoequivalences. As an application, we prove the corresponding Bloch conjecture for symplectic birational automorphisms of twisted modular ${\mathrm{K}\mathrm{u}\mathrm{m}}_n$-type varieties; in particular, this applies to those admitting a birational Lagrangian fibration. Finally, we introduce and study a Shen–Yin–Zhao type filtration on twisted modular varieties and compare it with Voisin’s filtration in the sixfold case. We also establish the anti-symplectic Bloch conjecture for twisted modular ${\mathrm{K}\mathrm{u}\mathrm{m}}_3$-type varieties.