The semiregularity step in Markman's approach to Weil classes for quartic CM fields

Markman reduced the algebraicity of Weil classes on abelian varieties of Weil type to the semiregularity of objects built from two sheaves with Chern characters in a secant space B on an abelian variety with real multiplication. For imaginary quadratic fields he proved it on Jacobians of genus three through two properties of a sheaf equivariant for a finite group of translations, and asked for such sheaves for quartic CM fields. We show that the method does not transfer to quartic CM fields in two cases. On an abelian fourfold with real multiplication by a real quadratic field, no such sheaf with scalar invariant endomorphisms and nonzero Chern character in B has the two properties, whatever the lattice; the same holds on locally split quotient lattices for the conjugates of these actions by line-bundle twists, the kind of action for which Markman's genus-three sheaves are equivariant. The proof sets a count of invariant Ext groups against integrality: for translations the two properties force Euler characteristic at most six for the descended sheaf, while a divisibility of integral classes of degree four, valid on every lattice, gives every nonzero integral secant class Euler characteristic at least eight; a quartic invariant controls the twisted classes. In dimension six the count excludes the two properties only where the Euler form is positive, and integrality excludes their all-degree form off a rank-one cone, which is isomorphic over the reals to Markman's genus-three situation but separated from it over the rationals. At Jacobians of genus four with real multiplication that are not mixed, Markman's sheaves are obstructed along a deformation preserving the real multiplication. Groups containing -1, conjugated actions on lattices that are not locally split, other mixed or projectively twisted actions, and the semiregularity question remain open; none of the results concerns the Hodge conjecture. Code and data are at github.com/sallerk/HC-rm-semiregularity.