Let $M$ be a compact complex manifold of complex dimension $m$.
It is well-known that, since every exact current is closed, we can define $$D_{current}^p(M)=\frac{\{Closed p-currents\}}{\{Exact p-currents\}}$$
Since every closed current is harmonic ($\partial \overline{\partial} -$closed), I would like enquire whether it is possible to define the following quotients spaces $$H_{current}^p(M)=\frac{\{Harmonic p-currents\}}{\{Closed p-currents\}}$$