*Published Paper*

**Inserted:** 24 jan 2019

**Last Updated:** 30 apr 2021

**Journal:** Ann. Mat. Pura Appl. (4)

**Volume:** 199

**Number:** 3

**Pages:** 985–995

**Year:** 2020

**Abstract:**

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.

**Tags:**
SIR2014-AnHyC