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The Tate conjecture for powers of ordinary cubic fourfolds over finite fields. (English) Zbl 1056.14031

Let \(X\) be a smooth projective variety over a finite field \(k\). The Tate conjecture claims that the subspace of \(\text{H}^{2m}(\overline{X},\mathbb{Q}_{\ell})(m)\) fixed under the Galois action is spanned by cohomology classes of codimension \(m\) algebraic cycles on \(X\). Recently, N. Levin [Compos. Math. 127, 1–21 (2001; Zbl 1077.14526)] proved the Tate conjecture when \(X=Y\) is an ordinary cubic fourfold. The aim of the paper under review is to prove the Tate conjecture for self-products \(X=Y^r\). The proof is based on results and ideas of previous work of the author [J. Algebr. Geom. 5, 151–172 (1996; Zbl 0863.14004)], where he proved the Tate conjecture for self-products of ordinary \(K3\) surfaces.

MSC:

14G15 Finite ground fields in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties

Keywords:

self-products
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References:

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