Cup products and correspondences

Question

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 $a^*$.

• 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 $a=p(f)$.

Answer 1

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].

---

References.

[Kleiman] S. L. Kleiman, The standard conjectures. In: Motives. 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.