Classes of hyperplane sections in cohomology

Question

Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.

Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive dimension?

The reason why I ask, is that when $X$ is a complex algebraic variety then the vector space of rational Hodge classes is always nonzero because it contains a power of the class of a Kähler form.

Can one argue in a similar way using the class of a hyperplane section in $H^2(X,\mathbf{Q}_{\ell}(1))$?

Can one deduce that the group of con dimension $k$-cycles modulo homological equivalence on $X$ is always nonzero?

Answer 1

The deduction goes the other way around. We want to check that for $H$ a hyperplane class in the group of codimension $1$ cycles, the class induced by $H^k$ in $H^{2k} (X, \mathbb Q_\ell(-k))$ is nonzero for $k$ from $0$ to $\dim X$.

Because the cycle class map is compatible with the intersection product (i.e. the cup product of two cycles classes is the class of the intersection of the cycles), it suffices to check this for $k = \dim X$.

For $k = \dim X$ one has to check that the cycle class map on zero-cycles is the same as the degree, and that the degree of $H^k$ is equal to the degree of $X$ as a projective variety and thus is nonzero.