I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression
\begin{equation*} \int_M d \alpha = \int_{\partial M} \alpha \ . \end{equation*} I believe this can be done solely using the notion of currents and the support of $d \alpha$ without reference to Stokes' theorem (which relies on a definition of $\partial$), but cannot find a suitable (rigorous) reference anywhere that expands the Wikipedia entry (http://en.wikipedia.org/wiki/Current_%28mathematics%29#Homological_theory).
I am particularly interested whether, on complex manifolds, the boundary operator can admit a Dolbeault decomposition as is claimed in the "Current" entry on Encyclopedia of Mathematics https://www.encyclopediaofmath.org/index.php/Current.