For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. To be more specific, here are some straight questions.

- What can Dolbeault tell us that de Rham can't?
- Does there exist some simple relationship between these two cohomologies?
- When are they equal?
- Do things become simpler for the Kahler case?
- What happens for the projective spaces?
- Why does nobody talk about the holomorphic cohomology?