The Lefschetz standard conjecture

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, let $\alpha\in H^{1,1}(X,\mathbb{Q})$ be a polarization, and let $L_\alpha$ denote cup product with $\alpha$. By the Hard Lefschetz theorem, for each $k\leq n$ there is an isomorphism

$$L_\alpha^{n-k}:H^k(X,\mathbb{Q})\xrightarrow{\sim}H^{2n-k}(X,\mathbb{Q}).$$

The Lefschetz standard conjecture. For each $k\leq n$, there exists an algebraic self-correspondence

$$[\mathcal{Z}]\in H^{2k}(X\times X,\mathbb{Q}),$$

arising from a codimension-$k$ cycle $\mathcal{Z}\in \operatorname{CH}^k(X\times X)_\mathbb{Q}$, such that

$$[\mathcal{Z}]^*:H^{2n-k}(X,\mathbb{Q})\xrightarrow{\sim}H^k(X,\mathbb{Q})$$

is the inverse of $L_\alpha^{n-k}$. This is one of Grothendieck's standard conjectures; its validity would provide algebraic correspondences realizing the inverse Lefschetz operators and has important consequences for algebraic cycles and the structure of cohomology.