Lefschetz Classes on Abelian Varieties

Then C∼(X) df = ⊕sC ∼(X) becomes a graded Q-algebra under the intersection product, and we define D∼(X) to be the Q-subalgebra of C∼(X) generated by the divisor classes: D∼(X) = Q[C ∼(X)]. The elements of D∼(X) will be called the Lefschetz classes on X (for the relation ∼). They are the algebraic classes on X expressible as linear combinations of intersections of divisor classes (including the empty intersection, X). Our main theorem states that, for any Weil cohomology theory X → H∗(X) and any abelian variety A over an algebraically closed field, there is a reductive algebraic group L(A) (not necessarily connected) such that the cycle class map induces an isomorphism D hom(A)⊗Q k → H(A)(s) for all integers r, s ≥ 0; moreover, D num(A) = D hom(A). Here A = A× · · · × A (r copies), k is the coefficient field for the cohomology theory, and “(s)” denotes a Tate