Let $X$ be a smooth projective variety over $\mathbf{C}$ of complex dimension $2n$. Let $C_1,C_2\subseteq X$ be two closed subvarieties of complex dimension $n$. Then we get two Betti homology classes $[C_1],[C_2]\in H_{2n}(X,\mathbf{Z})$.

Because the singular locus of $C_1$ and $C_2$ occur (at least) in real codimension 2 we may integrate any smooth complex valued $2n$-form of $X$ on $C_i$. **But more generally**, if
$\omega$ is a $\mathbf{C}$-valued real analytic $2n$-form with "no residue" (I'm not so sure of what this mean in complex dimension $>1$) it should make sense to integrate $\omega$ against $C_i$. Now let us look at the **algebraic** de Rham cohomology of $X$. Then $C_1$
and $C_2$ gives rise to two classes $C_1^*,C_2^*\in \mathbb{H}_{DR}^{2n}(X,\mathbf{C})^*$
(the last $*$ here means dual) which is given by integration against $C_i$. Now there is a Poincare duality which relates $\mathbb{H}_{DR}^{k}(X,\mathbf{C})$ to $\mathbb{H}_{DR}^{4n-k}(X,\mathbf{C})$. However, beware, this Poincare duality is not induced by the integration over $X$ of the wedge product of a $k$-form with a $(4n-k)$ form.
(so all the subtlety in my question is hidden in my inability to describe with enough precision this duality). Nevertheless, from this Poincare duality we get a privileged $\mathbf{C}$-vector space isomorphism:
$$
PD:\mathbb{H}_{DR}^{2n}(X,\mathbf{C})^*\rightarrow \mathbb{H}_{DR}^{2n}(X,\mathbf{C}).
$$
Let $\iota: \mathbb{H}_{DR}^{2n}(X,\mathbf{C})\rightarrow H_{DR}^{2n}(X,\mathbf{C})$
be the comparaison isomorphism between algebraic de Rham cohomology and smooth de Rham cohomology. Define $[\eta_i]:=\iota(PD(C_i^*))$. So $\eta_i$ are smooth $2k$-forms.

Now assume that $C_1$ and $C_2$ intersect at finitely many points.

Q: How do we compute the period $p$ defined by the relation
$$
(\star)\;\;\;\; p\int_X \eta_1\wedge \eta_2=[C_1]\cup[C_2]\in\mathbf{Z}_{\geq 0}
$$
Note 1: In the case where $X$ is an elliptic curve and $C_1$ and $C_2$ are the two standard loops; one finds that $p=2\pi i$ (the so-called Legendre's relation). Note here tough that
$C_1$ and $C_2$ are **not** algebraic cycles since they have real dimension $1$. In any case, from this simple example I would expect $p$ to be a power of $2\pi i$.

Note 2: In Griffiths and Harris they prove that if $(L,\nabla)$ is a line with a connexion on a compact Riemann surface then $c_1(L)=[\frac{-1}{2\pi i}\Theta_\nabla]$, where $c_1(L)$ corresponds to the first Chern class and $\Theta_\nabla$ corresponds to the connexion matrix of $\nabla$. One of the key idea of the computation is to apply Stoke's theorem to a differential form which has singularities. So I would like to adapt GH computation's in order to prove $(\star)$. However in order to do so I need to get a precise description of $\mathbb{H}_{DR}^{2n}(X,\mathbf{C})$ and of the map $PD$. I don't think that working with an explicit Cech-covering is the right strategy since it becomes quickly too complicated. There should be a better approach...