Integration currents vs Poincaré dual Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$.

*

*We have the integration current $T_{S}$ associated to $S$ defined by
$$ \langle T_{S}, \omega \rangle = \int_{S} \omega $$
for any $2(n-r)$-form $\omega$ defined on $M$. The current $T_{S}$ on $M$ has order $2r$. We can look at this current as a $2r$-cohomology class, i.e. $T_{S} \in H^{2r} (M)$.


*On the other hand, associate to $S$ the Poincaré dual $\eta_{S} \in H^{2r}(M)$.
Question: What is the relation between $T_{S}$ and $\eta_{S}$?
 A: Here is briefly the story. More details can be found  in DeRham's monograph Variétés differentiables.
Let  $M$ be a smooth, compact, oriented, $m$-dimensional manifold. Denote by $\Omega^k(M)$ the space of  smooth degree $k$-forms on $M$ and by $\Omega_k(M)$ its dual space, namely the space of $k$-dimensional currents.   Let $\newcommand{\bR}{\mathbb{R}}$
$$ \langle-,-\rangle :\Omega^k(M)\times \Omega_k(M)\to\bR,\;\; (\eta,C)\mapsto\langle \eta,C\rangle,
$$
denote the natural pairing between     topological vector space and its dual.
We have a natural map
$$\cap [M]: \Omega^k(M)\to \Omega_{m-k},\;\;\Omega^k(M)\ni \alpha \mapsto \alpha\cap [M] \in \Omega_{m-k}(M), $$
determined by $\newcommand{\lan}{\langle}$ $\newcommand{\ran}{\rangle}$
$$\lan \eta, \alpha\cap [M]\ran :=\int_M \alpha\wedge \eta ,\;\;\forall \eta\in\Omega^{m-k}(M).  $$
If we denote by $\newcommand{\pa}{\partial}$ $\pa:\Omega_k(M)\to\Omega_{k-1}(M)$ the boundary operator  on $\Omega_\bullet(M)$ defined by
$$ \lan\alpha ,\pa C\ran:= \lan d\alpha, C\ran,\;\;\forall \alpha\in\Omega^{k-1}(M),\;\;C\in \Omega_k(M), $$
then we obtain a  cochan complex
$$(\Omega_{m-\bullet}(M), \pa):\;\; \cdots \stackrel{\pa}{\to} \Omega_{m-k}(M)\stackrel{\pa}{\to}\Omega_{m-(k+1)}(M)\stackrel{\pa}\cdots . $$
We then have  a morphism of cochain complexes
$$PD_M :(\Omega^\bullet(M), d) \to (\Omega_{m-\bullet}(M),\pa),\;\;  \alpha \mapsto \alpha\cap [M]. $$
Georges DeRham proved that this  morphism induces isomorphism in cohomology.   This is one incarnation of Poincare duality.  The cohomology  of $(\Omega_{m-\bullet}(M),\pa)$  with the homology of $M$ with real coefficients.   In your question you have identified $T_S$  with a $2r$-cohomology class using    using $PD_M^{-1}$.  This identification implicitly uses Poincare duality.
