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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.

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