I was thinking about this lately. On a complex manifold of dimension $m$, one has a bigrading on the complexified space of differential forms, namely the forms of type $(p,q)$, say, have $p$ $dz$'s and $q$ $d\bar{z}$'s. Moreover, integration gives us a pairing between complex chains and (complex) differential forms.
My question is now this. What does the bigrading on the complexified space of differential forms correspond to on the level of complex chains?
Edit: by complex chains I mean complex linear combinations of real subvarieties, and not of complex submanifolds. So it is more general. For example, a contour integral would also fall in this category.