# Cup products and correspondences

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 $$a=p(f)$$.

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].