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Let $k$ be a finite field, $\ell \neq \mathrm{char} k$ be prime, $X/k$ be a smooth projective geometrically integral variety of dimension $d$, and $\mathcal{A}/X$ be an Abelian scheme. Let $\eta \in H^2(X,\mathbf{Q}_\ell(1))$ be the first Chern class of $\mathcal{O}_X(1)$.

I want to prove that the hard Lefschetz morphism $\eta^{d-1}: H^1(X,V_\ell\mathcal{A}) \to H^{2d-1}(X,V_\ell\mathcal{A}(d-1))$ is an isomorphism, $V_\ell\mathcal{A}$ the rationalised Tate module of $\mathcal{A}/X$. Using the Hochschild-Serre spectral sequence $H^p(G_k,H^q(\bar{X},V_\ell\mathcal{A})) \Rightarrow H^{p+q}(X,V_\ell\mathcal{A})$, I have reduced this to $k$ the algebraic closure of a finite field.

I want to apply [Deligne, Weil II], p. 250, Théorème (6.2.13) http://www.math.harvard.edu/~gaitsgde/grad_2009/Weil-II.pdf. For this I have to check:

  1. $V_\ell\mathcal{A}$ is potentially pure (see [Deligne, Weil II], p. 248, (6.2.8): This means for $(X/S,V_\ell\mathcal{A}), f: X \to S$, $S$ the spectrum of the Henselisation of $k[T]$ in $(T)$:

a) $\mathrm{Spec} k$ is a geometric generic point of an integral scheme $A_0$ of finite type over $\mathbf{Z}[1/\ell]$. [I think we can take $A_0 = \mathrm{Spec}\mathbf{F}_p$ for $p = \mathrm{char} k$.]

b) $S$ comes from scalar extension of $k$, and Henselisation, from a smooth curve $S_0$ over $A_0$ equipped with a section.

c) $f$ comes from a proper morphism $f_0: X_0 \to S_0$. [This should follow from $X/k$ being projective.]

d) $V_\ell\mathcal{A}$ comes from a pure complex $K_0 \in D^b_c(X_0)$. [This should follow since $V_\ell\mathcal{A} = R^1\pi_*\mathbf{Q}_\ell(1)$ for $\pi: \mathcal{A}^t \to X$ the dual Abelian scheme is pure of weight $-1$ by Weil II.]

  1. $V_\ell\mathcal{A}$ and $DV_\ell\mathcal{A}[-2d]$ verify the following condition: For every $i$, the dimension of the support of the sheaf $\mathcal{H}^i$ is $\leq d-i$ (set $\mathrm{dim} \emptyset = -\infty$).

Can someone please help me proving these conditions?

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Since the question is “Can someone help... “, then “Sure, Beilinson-Bernstein-Deligne can!” seems to be a legitimate answer. Let me write a few words.

The statement you give is a variant of the relative Hard Lefschetz theorem for pure perverse sheaves — Théorème 5.4.10 in Beilinson-Bernstein-Deligne's paper Faisceaux pervers (Astérisque 100, 1982, Société mathématique de France).

Conditions d) and 2 of Deligne's theorem are equivalent to the hypothesis that $V_\ell(A)[d]$ is pure and perverse, respectively.

Since $\pi\colon\mathcal A\to X$ is an abelian scheme, it is proper and smooth, hence $R^1\pi_*\mathbf Q_\ell$ is a lisse sheaf (proper and smooth base changes). By the Weil conjectures (in this case, Weil's theorem), it is pure of weight $1$. Then $V_\ell\mathcal A$ is its dual; it is lisse, and pure of weight $-1$.

If $L$ is a lisse sheaf on a smooth purely $d$-dimensional variety, then $K=L[d]$ (aka, the complex where $L$ is placed in degree $-d$) is a perverse sheaf. The condition on $\mathcal H^i K$ holds since all groups are zero, except $\mathcal H^{-d}K = L$. The condition on the dual holds as well, because $K^\vee = L^\vee[d]$ and $L^\vee$ is lisse. (See [BBD] 4.3.1 for a general description of simple perverse sheaves along these lines.)

So, to answer specifically your question, replacing $L$ by $V_\ell\mathcal A$ in the preceding paragraph checks condition 2).

(Edited, twice, to remove inaccuracies and add details)

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  • $\begingroup$ Thank you very much for your response! Can one find also a proof of condition 2. there (is this the perversity)? $\endgroup$
    – user19475
    Nov 20 '16 at 9:23
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    $\begingroup$ Yes it is precisely the perversity condition, but it is “hidden” in the language of perverse t-structures - see the statement of 4.2.4 where they study “cohomological amplitude properties” of a morphism. $\endgroup$
    – ACL
    Nov 20 '16 at 9:48
  • $\begingroup$ Thank you again! I will give you the reputation once I have checked the reference tomorrow! $\endgroup$
    – user19475
    Nov 20 '16 at 10:56
  • $\begingroup$ I am sorry that I have to ask you again: How exactly does it follow from 4.2.4 that $\pi_!\mathbf{Q}_\ell$ is pure and perverse? $\endgroup$
    – user19475
    Nov 21 '16 at 9:00
  • $\begingroup$ Part of my answer was nonsense. I corrected it and added details. $\endgroup$
    – ACL
    Nov 21 '16 at 20:12

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