Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the de Rham cohomology of $X(\mathbb{C})$. By Poincare duality, there is a form $[Z] \in \mathrm{H}_{\mathrm{dR}}^{2n-2p}(X,\mathbb{C})$, such that $$\int_{X} [Z] \wedge \alpha = \int_{Z} \alpha$$ for all $\alpha \in \mathrm{H}_{\mathrm{dR}}^{2p}(X,\mathbb{C})$.
Denote $\mathrm{H}^{i}(X,\mathbb{Q})$, the singular cohomology with rational coefficients. Recall that $\mathrm{H}^{i}(X,\mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C} \simeq \mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$.
Why is $[Z]$ in $\mathrm{H}^{2n-2p}(X,\mathbb{Q})$?