Bloch's conjecture for Enriques varieties (Q1667185)

The goal here is to show that the Chow group of zero-cycles of a large class of Enriques varieties is trivial and thus to provide further evidence for Bloch's conjecture. More precisely it is proved that \(A_0(X)=\mathbb Z \) when \(X\) is an Enriques variety of dimension \(\leq 6\) with the requirement that \(X\) is a quotient \( X=K/G \) where \(K=K_n(A)\) is the generalized Kummer variety of dimension \(2n-2\) associated with an abelian surface \(A\) and \(G\) is a group of automorphisms acting freely and induced by a finite order automorphism \(\phi\) of \(A\). The proof is based on Kimura's theory [Math. Ann. 331, No. 1, 173--201 (2005; Zbl 1067.14006)] and it depends on the properties of the Chow motive of \(K\) as have been determined in the work of \textit{Z. Xu} [Int. Math. Res. Not. 2018, No. 3, 932--948 (2018; Zbl 1423.14057)] and \textit{H.-Y. Lin} [Adv. Math. 298, 448--472 (2016; Zbl 1343.14005)]. The crucial fact used here is the existence of a split injection \( A_0(K)_{\mathbb Q}\;\to\;A_0(A^{(n)})_{\mathbb Q}\), where \(A^{(n)}\) is the \(n-\)th symmetric product, which injection is compatible with the Chow-Künneth decompositions of both varieties. The idea of the present paper is to consider the surface \( A^\prime:= A/<\phi_0>\), where \(\phi_0\) is the group authomorphism determined by \(\phi\). The first fact established is that one has again a split injection \( A_0(X)_{\mathbb Q}\;\to\;A_0((A^\prime) ^{(n)} )_{\mathbb Q}\), compatible with the Chow--Künneth decompositions. Now the motive of \( A^\prime\) has no transcendental part since \(p_g(A^\prime)=0\); a careful and detailed analysis of the decomposition of \((A^\prime) ^{(n)}\) exploits this fact to reach the required vanishing for \(n\leq 4\).