From Diaz's Enriques Product to an $n$-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples

Abstract

We reinterpret Diaz's construction of Chow-trivial smooth projective varieties violating the integral Hodge conjecture as the level-two case of an \(n\)-fold cup-product Bockstein mechanism. Diaz's dimension-four example is \(V=S_1\times S_2\), where \(S_1,S_2\) are Enriques surfaces, and its obstruction is the Bockstein of $\pi_1^*\alpha_1\cup\pi_2^*\beta_2\in H^3(V,\mathbb Z/2(2))$. Here \(\alpha_1\) is the K3 double-cover class and \(\beta_2\) is an Enriques Brauer-detecting class. We extend the finite-coefficient source construction to \(X_n=S_1\times\cdots\times S_n\) by forming $\Theta_n=\pi_1^*\alpha_1\cup\pi_2^*\beta_2\cup\cdots\cup\pi_n^*\beta_n \in H^{2n-1}(X_n,\mathbb Z/2(n))$,with Bockstein $\Delta_n=\delta(\Theta_n)\in H^{2n}(X_n,\mathbb Z(n))$. Using external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson--Vilonen boundary, we prove unconditionally that \(\Delta_n\) has nonzero image in a distinguished Enriques--Brauer component of the MV obstruction channel. Under the Brauer-separation hypothesis, which asserts that algebraic codimension-\(n\) cycle classes have zero image in this same component, the class \(\Delta_n\) is a non-algebraic \(2\)-torsion integral Hodge class. We verify this separation for decomposable algebraic cycles and reduce the remaining non-decomposable case, via integral even Chow--K\"unneth projectors on the Enriques factors, to a single coefficient-level algebraic-control problem involving the \(H^1(S_1,\mathbb Z/2(1))\) Enriques double-cover direction. We also record a motivic finite-coefficient lift of the tower via the finite-coefficient cone \(\mathbf 1_X(n)/2:=\operatorname{Cone}(\mathbf 1_X(n)\xrightarrow{\times 2} \mathbf 1_X(n))\), and explain which formal part of the MacPherson--Vilonen zig-zag construction lifts motivically under Betti realization.