Bloch’s conjecture for (anti-)symplectic birational automorphisms of hyper‑Kähler varieties
Prove that for a smooth projective hyper‑Kähler variety Y of dimension 2n, any birational automorphism φ satisfies φ*|_{H^{2,0}(Y)} = ± id if and only if φ_* acts on the graded pieces of Voisin’s filtration by φ_*|_{Gr_s CH_0(Y)} = (±1)^s id for all 1 ≤ s ≤ n.
References
A more approachable question to examine these conjectures (and expectations) is to consider the action of (anti)-symplectic birational automorphisms on the Chow group of a Bridgeland moduli space. More precisely, we have the following conjecture
Let $Y$ be a hyper-K\"ahler variety of dimension $2n$. Set $$\Gr_{s}\CH_{0}(Y):=S_{s}{\rm BV}\CH_0(Y)/S_{s-1}{\rm BV}\CH_0(Y)$$ to be the $s$-th graded piece. For any $\phi\in \mathrm{Bir}(Y)$, we have \begin{equation*} \phi\ast|_{H{2,0}(Y)}=\pm \id \Leftrightarrow \phi_\ast|_{\mathrm{Gr}_s\CH_0(Y)}=(\pm 1)s\id,~\forall ~1\leq s\leq n. \end{equation*}