The Hodge conjecture and algebraic cycles
A bridge between differential forms and the cohomology of projective varieties: we verify the Hodge conjecture for a new class of abelian fourfolds with prescribed endomorphism structure.
1 · Algebraic cycles and cohomology
The Hodge conjecture predicts that every rational Hodge class of even degree on a smooth complex projective variety is a rational linear combination of classes of algebraic cycles. The (p, p)-subspace is where geometry and analysis meet: Hodge classes are analytic objects, cycles are algebraic ones.
2 · Abelian fourfolds with prescribed structure
For abelian fourfolds whose endomorphism algebra contains a real quadratic field of discriminant D, we construct explicit codimension-two cycles and show their classes span the rational Hodge classes, provided the associated shimura variety meets the relevant Hecke orbit.
3 · Remarks
The construction is geometric rather than motivic: the cycles are built as Hecke translates of a distinguished abelian subvariety. It would be interesting to know whether the same classes generate the full motive in the sense of André.