Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Abstract

Let $X\subset {\P }^n$ be a possibly singular hypersurface of degree $d\le n$, defined over a finite field $k$. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its $\ell$-adic cohomology $H^i(\bar{X}, \Q_\ell)$, for $\ell \ne \mathrm{char}(k)$, are divisible by $q$, without using the result on the existence of rational points by Ax and Katz.

16 pages

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• Cohomological divisibility and point count divisibility