Semisimplicity of Frobenius operation on etale cohomology?

Let $X_0$ be a variety defined over a finite field of characteristic $p \neq l$. Is it true, that the action of the frobenius on the l-adic cohomology $H_l^*(X)$ is semisimple (say for smooth $X_0$)? If not, what would be a counter-example?

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See Emerton's comment to Bondarko's question "Morphisms between pure complexes of sheaves". He says that semi-simplicity of Frobenius is part of the Tate conjectures. – Geordie Williamson Aug 6 '12 at 12:11
This is famous open problem... It is true for abelian varieties. – Damian Rössler Aug 6 '12 at 12:11
Ok thanks this is good to know! – Jan Weidner Aug 6 '12 at 13:15
Also true for K3 surfaces, by Deligne. See Lei Fu's AMS article 'On the semisimplicity conjecture and Galois representations' for more info. For mixed cohomology groups, the weight filtration does not split in general (it would be split if Frobenius were always semisimple), and examples can be found already in dimension 1. – shenghao Aug 6 '12 at 16:02
@Jan Weidner. The connection with the standard conjectures is explained in Kleiman's article "The standard conjectures" in the first volume of the proceedings of the conference on Motives (ed. Jannsen, Kleiman, Serre), th. 5-6, p. 19. – Damian Rössler Aug 7 '12 at 11:13

Let $X$ be a smooth projective variety over a finite field $\mathbb{F}_{q}$ of caracteristic $p$ and let $l$ be a prime number different from p. We consider the following statement :

(A) The action of the Frobenius on the etale cohomology $H^{i}_{et}( X_{\overline{\mathbb{F}}_{q}}, \mathbb{Q}_{l})$ is semisimple.

How to suppress the projective hypothesis is the subject of the mathoverflow question link text

(A) is true in the following cases :

1) $X=A$ an abelian variety (and so for $X$ a curve via the jacobian). As mentionned in comment by Emerton, it is a consequence of the Weil's work on the Riemann hypothesis in this case. Fix a polarization on $A$. For $x$ an endomorphism of $X$ which gives an endomorphim on $H^{1}_{et}$, we can define an endomorphism $x'$ (' : "Rosati involution") by $x' = *x^{T}*$ where in the middle we have the transposition with respect to the intersection product and * comes from the duality theory of abelian varieties ( the polarisation gives a identification between $H^{1}_{et}(A)$ and $H^{1}_{et}(\check{A})$). Weil proved that $Tr(xx')>0$ if $x$ is non-zero. Let $F$ be the (geometric) Frobenius. For $x = q^{-1/2}F$, we have x'=$x^{-1}$. So $Tr(aa')$ is a definite positive bilinear form on the $\mathbb{Q}$ algebra generated by x and is preserved by multiplication by $x$ : so multiplication by $x$ is unitary which shows that $x$ is semi-simple (and eigenvalues of modulus one gives the Riemann hypothesis).

2) $X$ a K3 surface. As mentionned in comment by shenghao, it is a consequence of the work of Deligne : link text The result is deduced from the case of abelian varieties via the Kuga-Satake construction (of course there is a non-trivial thing to do because Kuga-Satake construction is a priori of transcendental nature but Deligne did it).

For $X$ general, (A) is conjectured. It is a consequence of standard conjectures. More precisely, things should work as in the case of abelian varieties. We can still define $x \mapsto x'$ at the cohomological level but $Tr(xx')>0$ is conjectural : a standard conjecture of Lefschetz type implies $x'$ algebraic if $x$ is, which permits to use a trace formula expressing $Tr(xx')$ as an intersection product. The positivity should then be a consequence of a standard conjecture of Hodge type. For more details, as mentionned in comment by Damian Rössler, see Kleiman "The standard conjectures" (whose some details depend on Kleiman, "Algebraic cycles and the Weil conjectures").

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Thanks, do you know a source where it is discussed how my question relates to the standard conjectures? Also is it known whether semisimplicity of cohomology fails without the assumptions smooth / projective? – Jan Weidner Aug 6 '12 at 13:12
I'm not sure if Faltings' work leads to semisimplicity of the $Gal(\bar{\mathbb Q}/\mathbb Q)$-representation, which is a different conjecture. – shenghao Aug 6 '12 at 16:07
@shenghao. It does (for abelian varieties); see the original article or the translation in Cornell-Silverman. – Damian Rössler Aug 7 '12 at 11:10
Dear unknown, The semisimplicity of Frobenius on the $\ell$-adic cohomology of a curve over a finite field is not due to Faltings; it goes back to the foundational theory of abelian varieties (due to Weil in the non-complex setting, perhaps?). Faltings's results pertain to the context of curves and abelian varieties over number fields. Regards, – Emerton Aug 14 '12 at 5:36
Dear unknown, This is a nice survey of the situation. Regards, – Emerton Aug 31 '12 at 18:41