  We define the relative Dolbeault homology of a complex manifold with currents
via a \v{C}ech approach and we prove its equivalence with the relative
\v{C}ech-Dolbeault cohomology as defined in [T. Suwa, \v{C}ech-Dolbeault
cohomology and the $\overline\partial$-Thom class, {\em
Singularities---Niigata---Toyama 2007}, 321--340, Adv. Stud. Pure Math.,
\textbf{56}, Math. Soc. Japan, Tokyo, 2009. ]. This definition is then used to
compare the relative Dolbeault cohomology groups of two complex manifolds of
the same dimension related by a suitable proper surjective holomorphic map.
Finally, an application to blow-ups is considered.

Differential Geometry;Complex Variables