Lefschetz classes on abelian varieties
For a smooth projective variety \(X\), \(Z^s(X)\) denotes the group of algebraic cycles on \(X\) of codimension \(s\), and \(C^s(X)= (Z^s(X)/ \sim) \otimes \mathbb{Q}\) for any adequate equivalence relation \(\sim\) on algebraic cycles. Let \({\mathcal D}(X)\) denote the \(\mathbb{Q}\)-subalgebra of \(\bigoplus C^s(X)\) generated by the divisor classes. The main theorem of this paper states that, for any Weil cohomology theory \(X\mapsto H^*(X)\) and any abelian variety \(A\) over an algebraically closed field, there is a reductive algebraic group \(L(A)\), called the Lefschetz group, such that the cycle class map induces an isomorphism \({\mathcal D}_{\text{hom}}^s(A^r)\otimes_\mathbb{Q} k\to H^{2s} (A^r) (s)^{L(A)}\) for all integers \(r,s\geq 0\), with \(k\) the coefficient field for the cohomology theory. The Lefschetz group \(L(A)\) is defined to be the largest algebraic subgroup of \(GL(V(A)) \times {\mathbb{G}}_{m/k}\) fixing the elements of \({\mathcal D}_{\text{hom}}^s(A^r) \otimes_\mathbb{Q} k\subset H^{2s}(A^r)(s)\), thus it is a natural generalization of the corresponding notion introduced by \textit{V. Kumar Murty} [Math. Ann. 268, 197-206 (1984; Zbl 0543.14028)].
scientific article; zbMATH DE number 1425189
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