Abstract: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 $\mathrm{K3}^{[n]}$-type, the $\mathrm{Kum}_n$-type case remains much less understood.
In this paper, we study this problem through twisted abelian surfaces and their associated $\mathrm{Kum}_n$-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 $\mathrm{Kum}_n$-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 $\mathrm{Kum}_3$-type varieties.
From: Zaiyuan Chen [view email]
[v1]
Sun, 21 Jun 2026 03:48:22 UTC (51 KB)
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