An n-fold cup-product Bockstein on products of Enriques surfaces produces non-algebraic 2-torsion integral Hodge classes in dimension 2n under the Brauer-separation hypothesis.
Generalizations of intersection homology and perverse sheaves with duality over the integers
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Computations for A1, Ak, D4, E8 and other singularities show finite discriminant torsion is a codimension-two phenomenon, not generic for nodes, with threefold ordinary double points being torsion-free.
For normal surface singularities the integral perverse obstruction E equals H^2(link, Z)_tors, the discriminant group of the exceptional lattice (|E|=|det(M)|), and for hypersurfaces the torsion cokernel of (T-id) on vanishing cohomology (|E|=|det(T-id)| under a Q-isomorphism hypothesis).
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From Diaz's Enriques Product to an $n$-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples
An n-fold cup-product Bockstein on products of Enriques surfaces produces non-algebraic 2-torsion integral Hodge classes in dimension 2n under the Brauer-separation hypothesis.
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Torsion Trajectories from Local Discriminants to Global Obstructions
Computations for A1, Ak, D4, E8 and other singularities show finite discriminant torsion is a codimension-two phenomenon, not generic for nodes, with threefold ordinary double points being torsion-free.
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Integral Perverse Obstructions for Normal Surface Singularities: Resolution Determinants and Monodromy
For normal surface singularities the integral perverse obstruction E equals H^2(link, Z)_tors, the discriminant group of the exceptional lattice (|E|=|det(M)|), and for hypersurfaces the torsion cokernel of (T-id) on vanishing cohomology (|E|=|det(T-id)| under a Q-isomorphism hypothesis).