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 $S$.


1. 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. $T\_{S} \in H^{2r} (M)$.
2. 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 $\eta\_{S}$?



Integration currents vs Poincaré dual
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.


