Say $A_0$ is an ordinary abelian variety over ${\mathbf{F}}_q$. Call $\mathcal{A}$ the canonical lift of $A_0$ over $R := W({\mathbf{F}}_q)$. It carries a lift of the $q$-th power map on $A_0$. We call $\phi : \mathcal{A}\to\mathcal{A}$ this lift. It exists by functoriality of the canonical lift.
Call $K = \text{Frac}(R)$ and choose a field embedding $K\subset\mathbf{C}$.
Call $A$ the complex torus $(\mathcal{A}\times_K\mathbf{C})(\mathbf{C})$ and $F$ the endomorphism of $A$ induced by the Frobenius lift on $\mathcal{A}$, i.e.
$$F = (\phi \times_{\text{Spec}(R)}\text{id}_{\mathbf{\text{Spec}(C)}})^{\rm an}:A\to A.$$
$F$ acts on the Betti cohomology $H^*(A,\mathbf{C})$. By the Weil conjectures (or by an argument of Serre, in this case), its eigenvalues are of the form $q^{*/2}\zeta$ for algebraic numbers $\zeta$ of complex absolute value $1$.
Suppose $v \in H^{2m}(A,\mathbf{C})$ is an eigenvector of $F$ whose eigenvalue is of the form $q^m\zeta$ for $\zeta$ a root of unity. Is $v$ a class of type $(m,m)$?
Best Answer
Let's examine how $\phi$ acts on the algebraic Dolbeaut cohomology $$H^1(\mathcal A_K , \mathcal O_{\mathcal A})+ H^0 ( \mathcal A_K, \Omega^1_{\mathcal A}).$$
I claim its eigenvalues on $H^1(\mathcal A_K , \mathcal O_{\mathcal A})$ are units and its eigenvalues on $H^0 ( \mathcal A_K, \Omega^1_{\mathcal A})$ are $q$ times units.
For the first claim, we can use the Artin-Schreier exact sequence $$H^1 ( A_{0, \overline{\mathbb F_q}}, \mathbb Z/p) \to H^1 ( A_{0, \overline{\mathbb F_q}} , \mathcal O_{ A_0} )\to H^1 ( A_{0, \overline{\mathbb F_q}} , \mathcal O_{ A_0} ) ,$$ the image of whose first arrow is a basis for $H^1 ( A_{0, \overline{\mathbb F_q}} , \mathcal O_{ A_0} )$ on which Frobenius acts invertibly, showing that Frobenius acts invertibly on $H^1 ( A_0, \mathcal O_{A_0})$ and thus invertibly on $H^1 (\mathcal A, \mathcal O_{\mathcal A})$.
For the second claim, we can use the fact that the pullback of a polarization along $\phi$ is $q$ times that polarization, so Frobenius is a symplectic similitude with similitude character $q$ for the form on $H^1(\mathcal A_K)$ induced by that polarization, hence for each eigenvalue $\lambda$ that is a $p$-adic unit, $q/\lambda$ must also be an eigenvalue.
Using that claim and the isomorphism $$\wedge ^a H^1(\mathcal A_K , \mathcal O_{\mathcal A})\otimes \wedge^b H^0 ( \mathcal A_K, \Omega^1_{\mathcal A}) \to H^a ( \mathcal A_K, \Omega^b_{\mathcal A})$$ we conclude that all eigenvalues of $\Phi$ on $$H^a ( \mathcal A_K, \Omega^b_{\mathcal A})$$ are $q^b$ times a $p$-adic unit.
So any eigenvalues in degree $2m$ of the form $q^m$ times a unit must occur in $H^{m,m}$, as desired.