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.]


2. $V\_\ell\mathcal{A}$ 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?



hard Lefschetz isomorphism for rational Tate module
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)


