# Continuity of Intersection Pairing on Chow monoids

Let $$X$$ be a smooth irreducible complex projective variety. As we know, if $$\alpha,\beta$$ are two cycles intersecting properly in $$X$$, we can define, via Serre's Intersection Formula, their intersection $$\alpha\cdot\beta$$. Let

$$\Omega:=\lbrace (\alpha,\beta)\in\mathcal{C}_p(X)\times\mathcal{C}_{p'}(X) \: : \: \alpha \textit{ and } \beta \textit{ intersect properly}\rbrace.$$ One can easly check that $$\Omega$$ is open. I would like to prove that the intersection pairing $$\Omega\longrightarrow \mathcal{C}_s(X), (\alpha,\beta)\longmapsto \alpha\cdot\beta$$ is continuous. In the case $$X=\mathbb{P}^N$$ this can be seen as follows.

First, one uses the reduction to diagonal: $$\alpha\cdot\beta=\Delta\cdot(\alpha\times\beta)$$, then one applies the following fact.

Proposition 7.1 [LR01] Let $$V\subseteq \mathbb{P}^N$$ be a dimension $$t$$-irreducible projective variety and $$\ell_1,\ldots,\ell_k$$ be linear forms such that one can define the intersection $$V'=V\cdot\mathsf{div}(\ell_1)\cdots\mathsf{div}(\ell_k)$$. Then $$F_{V'}(\theta_0,\ldots,\theta_t)=F_V(\theta_0,\ldots,\theta_{t-k},\ell_1,\ldots,\ell_k).$$

As the map $$\mathcal{C}_p(\mathbb P^N)\times\mathcal{C}_{p'}(\mathbb{P}^N)\longrightarrow \mathcal{C}_{p+p'}(\mathbb{P}^N\times\mathbb P^N)$$ taking $$(\alpha,\beta)\longmapsto \alpha\times\beta$$ is continuos, one obtains the continuity of the intesection pairing in the case $$X=\mathbb{P}^{N}$$.

I do not know how to prove the continuity of the intersection pairing for any $$X$$. Friedlander and Lawson in p. 371 of [FL92] claim that the result follows by [6.1,Ful84] and using the reduction to the diagonal. Anyway, I cannot see this. I would appreciate very much any help to prove this.

EDIT: Thinking more on the problem, I realized that my strategy to check the case $$X=\mathbb P^N$$ was wrong. Indeed, the correct way to proceed is considering the join $$\mathbb{P}^N\#\mathbb P^N=\mathbb P^{2N+1}$$ of $$\mathbb{P}^N$$ (see below) with itself and taking the intersection of the cycle $$\alpha\#\beta$$ with the diagonal $$\Delta=\lbrace x_0=y_0,\ldots , x_{N}=y_N\rbrace$$, where $$x_0,\ldots,x_N,y_0,\ldots , y_N$$ are the homogeneous coordinates of $$\mathbb{P}^N\#\mathbb P^N=\mathbb P^{2N+1}$$. One easly see, by using the properties of the $$\mathsf{Tor}$$-functor, that $$\alpha\cdot \beta=\Delta\cdot \alpha\#\beta.$$ Now, as it is very well known, the join map $$\#\colon \mathcal{C}_p(\mathbb{P}^N)\times\mathcal{C}_{p'}(\mathbb{P}^N)\longrightarrow \mathcal{C}_{p+p'+1}(\mathbb{P}^N)$$ is continuous, thus by Prp. 7.1 we are done.

NOTATION

$$\mathcal{C}_{p,d}(X)$$ is the Chow variety of $$p$$-cycles of degree $$d$$ with support contained in $$X$$.

$$\mathcal{C}_p(X)$$ is the disjoint union on $$d$$ of the Chow varieties $$\mathcal{C}_{p,d}(X)$$ endowed with the analytic topology.

The join of $$V\subseteq \mathbb{P}^n$$ and $$W\subset \mathbb P^m$$ is the projectivization of the product of their affine cones: $$\mathbb{P}\big(C(V)\times C(W)\big)\subset \mathbb P^n\#\mathbb P^m=\mathbb P^{n+m+1}$$.

REFERENCES

[LR01] Laurent, M., & Roy, D. (2001). Criteria of algebraic independence with multiplicities and approximation by hypersurfaces. J. Reine Angew. Math. 536, 65-114; DOI: 10.1515/crll.2001.053, author's website.

[FL92] Friedlander, E. M., & Lawson, H. B. (1992). A theory of algebraic cocycles. Ann. of Math. 136, 361-428; DOI: 10.2307/2946609.

[Ful84] Fulton, W. (1984). Intersection theory. New York: Springer-Verlag.