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Perhaps adding that on the construction of the
Dolbeault cohomology one has to use the complex structure of the manifold and then remarking if the cohomology groups obtained give information about the complex structure (or if it is only a topological invariant) would enrich the entry.
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here. Is it better to create a new article on the topic and just link it here? The proofs are quite long, but the
Dolbeault cohomology and the Dolbeault-Grothendieck lemma are strictly related; what's more without this paragraph the article seemed really poor.
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in particular, if the
Dolbeault cohomology is an invariant of the complex structure, then would be nice to have an example of two complex structures on the same differentiable manifold yielding different Dolbeault cohomology groups.
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