We define the relative Dolbeault homology of a complex manifold with currents via a Čech approach, and we prove its equivalence with the relative Čech–Dolbeault cohomology as defined by Suwa (in: Singularities–Niigata–Toyama 2007. Advanced studies in pure mathematics, vol 56. Mathematical Society of Japan, Tokyo, pp 321–340, 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 and a blow-up formula for the Dolbeault cohomology in terms of relative cohomology is presented.

我们通过Čech方法定义了具有电流的复杂流形的相对Dolbeault同源性，并证明了其与Suwa定义的Čech–Dolbeault相对同调性（在：Singularities–Niigata–Toyama2007。纯数学的高级研究，第56.日本数学学会，东京，第321–340页，2009年。然后使用该定义来比较通过适当的适当的射影全同图而关联的，具有相同维数的两个复杂流形的相对Dolbeault同调群。最后，考虑了爆破的应用，并提出了基于相对同调的Dolbeault同调的爆破公式。