Sábháilte in:
| Príomhchruthaitheoir: | |
|---|---|
| Formáid: | Preprint |
| Foilsithe / Cruthaithe: |
2026
|
| Ábhair: | |
| Rochtain ar líne: | https://arxiv.org/abs/2605.02129 |
| Clibeanna: |
Cuir clib leis
Níl clibeanna ann, Bí ar an gcéad duine le clib a chur leis an taifead seo!
|
| _version_ | 1866913085204201472 |
|---|---|
| author | Rahman, Abdul |
| author_facet | Rahman, Abdul |
| contents | 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 $π_1^*α_1\cupπ_2^*β_2\in H^3(V,\mathbb Z/2(2))$. Here \(α_1\) is the K3 double-cover class and \(β_2\) 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 $Θ_n=π_1^*α_1\cupπ_2^*β_2\cup\cdots\cupπ_n^*β_n \in H^{2n-1}(X_n,\mathbb Z/2(n))$,with Bockstein $Δ_n=δ(Θ_n)\in H^{2n}(X_n,\mathbb Z(n))$. Using external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson--Vilonen boundary, we prove unconditionally that \(Δ_n\) 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 \(Δ_n\) 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--Künneth 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02129 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Diaz's Enriques Product to an $n$-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples Rahman, Abdul Algebraic Geometry Algebraic Topology 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 $π_1^*α_1\cupπ_2^*β_2\in H^3(V,\mathbb Z/2(2))$. Here \(α_1\) is the K3 double-cover class and \(β_2\) 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 $Θ_n=π_1^*α_1\cupπ_2^*β_2\cup\cdots\cupπ_n^*β_n \in H^{2n-1}(X_n,\mathbb Z/2(n))$,with Bockstein $Δ_n=δ(Θ_n)\in H^{2n}(X_n,\mathbb Z(n))$. Using external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson--Vilonen boundary, we prove unconditionally that \(Δ_n\) 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 \(Δ_n\) 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--Künneth 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. |
| title | From Diaz's Enriques Product to an $n$-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2605.02129 |