Suppose $X$ is a smooth projective complex variety, connected of dimension $n$.


Let $a$ be an algebraic correspondence in $A^n(X\times X)$, the group of cycles modulo homological equivalence in $H^{2n}((X\times X)(\mathbf{C}),\mathbf{Q})$. This is a ring with the usual composition of correspondences.


$a$ acts on the cohomology of $X$ by the Künneth formula, as the graded linear map of degree zero $a^\*$.



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> * Is $a^\*$ compatible with cup products?
> * Do we have, for another algebraic correspondence $b$, $(a\circ b)^\*=b^\*\circ a^\*$?
> 
> 
> 


I expect the answer to the first question to be no, and the answer to the second question to be yes. It would be great to collect a few examples.


For instance, I’m thinking if $X$ is an abelian variety and $f$ is the graph of multiplication by $N>1$, then $H^2\times H^i\to H^{i+2}$ should give counterexamples to the first question for $i\le n-1$ by taking the characteristic polynomial $p(T)$ of $f^\*$ on $H^{i+2}$ and choosing $a=p(f)$.



Cup products and correspondences
It is not true that $a^\*(\alpha \smile \beta) = a^\*\alpha \smile a^\*\beta$:


**Example.** Let $X = \mathbf P^2$, and let $a$ be the Künneth projector onto $H^2(\mathbf P^2) \subseteq H^\*(\mathbf P^2)$. Explicitly, $a$ can be represented by the algebraic cycle $\Delta\_{\mathbf P^2} - \mathbf P^2 \times x - x \times \mathbf P^2$ for any point $x \in \mathbf P^2$. If $\alpha \in H^2(\mathbf P^2)$ is the class of a line, then $a^\*(\alpha \smile \alpha) = a^\*(x) = 0$, but $a^\*\alpha \smile a^\*\alpha = \alpha \smile \alpha$ is the class of a point.


The answer to the second question is positive, and this is in fact purely a linear algebra statement. Indeed, the construction of $a \circ b$ in $A^n(X \times X)$ is in terms of pullbacks, pushforwards, and intersection products, all of which are preserved by the cycle class map $\operatorname{cl} \colon A^n(X \times X) \to H^{2n}(X \times X)$. This is explained for instance in §3 of Kleiman's chapter *The standard conjectures* in the 1991 Seattle Motives proceedings [Kleiman].




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**References.**


[Kleiman] S. L. Kleiman, *The standard conjectures*. In: [Motives](http://dx.doi.org/10.1090/pspum/055.1). Proceedings of the summer research conference on motives, Seattle WA, July 20-August 2, 1991. American Mathematical Society. Proc. Symp. Pure Math. **55**.1, p. 3-20 (1994). [ZBL0820.14006](https://zbmath.org/?q=an:0820.14006).


