Standard conjectures on algebraic cycles

Statement

Let $k$ be an algebraically closed field, and let $\ell$ be a prime different from the characteristic of $k$. Let $X$ be a smooth irreducible projective variety of dimension $n$ over $k$. Write $H^i(X) = H^i(X, \Q_\ell)$ for ℓ-adic cohomology, with the cup product. Let $\mathrm{cl} \colon Z^j(X) \to H^{2j}(X)$ be the cycle class map on algebraic cycles of codimension $j$, and let $A^j(X) \subset H^{2j}(X)$ be the $\Q$-span of its image. A class in $H^\bullet(X)$ is algebraic if it lies in the span of the $A^j(X)$. Let $L \colon H^i(X) \to H^{i+2}(X)$ be the cup product with the class of a hyperplane section. By the $\ell$-adic form of the hard Lefschetz theorem, due to Deligne, $L^{n-i} \colon H^i(X) \to H^{2n-i}(X)$ is an isomorphism for $i \le n$. The standard conjectures are the following.

1. Lefschetz type (conjecture B): the operator $\Lambda$ on $H^\bullet(X)$ that inverts $L$ in the Lefschetz decomposition is induced by an algebraic class in $H^\bullet(X \times X)$.
2. Künneth type (conjecture C): the projectors $\pi_i$ of $H^\bullet(X)$ onto $H^i(X)$ are induced by algebraic classes in $H^{2n}(X \times X)$.
3. Conjecture D: for $Z \in Z^j(X) \otimes \Q$, one has $\mathrm{cl}(Z) = 0$ if and only if $\deg(Z \cdot W) = 0$ for every $W \in Z^{n-j}(X)$; that is, homological and numerical equivalence agree.
4. Hodge type (conjecture I): for $2j \le n$, the pairing $(x, y) \mapsto (-1)^j \deg\big(L^{n-2j} x \cdot y\big)$ is positive definite on the primitive part $\set{x \in A^j(X) : L^{n-2j+1} x = 0}$.