Grothendieck's standard conjectures

Let $Y$ be a smooth projective variety of dimension $d_Y$ over a field $k$, let $H^*$ be a Weil cohomology, and let $H^{2j}_{\operatorname{alg}}(Y) \subset H^{2j}(Y)$ denote the image of the cycle class map $CH^j(Y) \to H^{2j}(Y)$. For an ample divisor $h \in \operatorname{Pic}(Y)$, write

$$L_h^j:=\cdot h^j:H^j(Y)\to H^{2d_Y-j}(Y).$$

Let $\delta_j$ denote the Künneth components of the diagonal in

$$\Delta_Y\in H^{2d_Y}(Y\times Y)\cong\bigoplus_{j=0}^{2d_Y}H^j(Y)\otimes H^{2d_Y-j}(Y).$$

Grothendieck's standard conjectures. The following assertions hold: (1) there exist correspondences $\Lambda_j\in CH^j(Y\times Y)$ such that $\Lambda_{j*}:H^{2d_Y-j}(Y)\to H^j(Y)$ is the inverse of $L_h^j$; (2) every Künneth component $\delta_j$ is induced by an algebraic cycle, equivalently $\delta_j\in H^{2d_Y}_{\operatorname{alg}}(Y\times Y)$; and (3), for every $\gamma\in H^{2j}_{\operatorname{alg}}(Y)$, if $\gamma\cdot\gamma'=0$ for all $\gamma'\in H^{2(d_Y-j)}_{\operatorname{alg}}(Y)$, then $\gamma=0$.

These are Grothendieck's standard conjectures: the Lefschetz standard conjecture, the Künneth standard conjecture, and the assertion that numerical and homological equivalence coincide on algebraic cohomology classes. The paper's abstract states that it proves them for the Fano variety of lines on a smooth cubic hypersurface, but the supplied text does not specify which of the assertions is being proved or provide a resolution status for the general formulation.