I'm the beginner in complex geometry, and I'm curious about the motivation of dolbeault cohomology,I know that group$H^{q,0}(M)$ represents the holomorphic forms of the manifold $M$,but what's about the higher group?
and for kahler and compact case ,we have the hogde decomposition to found a connection of the dolbeault cohomology and de Rham cohomology .It seems that we set up a connection between the topology and the complex structure of the manifold .But we take a opposite example, the complex plane, which is kahler but not compact,its 0 th de rham cohomology group is only $\mathbb{C}$,but the $\mathbb{Z}-module$ generate by holomorphic functions on complex plane is sufficiently large.How do the kahler and compact conditions force the topology structure and complex structure coinciding to a certain extent？(to be honest, I want to know what happen with the compact but not kahler case, but I have no idea to caculate the group $H^{p,q}(M)$(here I mean the dolbeault cohomology) even for some explicit cases. )
Although I know the proof of Hodge decomposition explains the reason ,I seek for a more geometric(or visual) explaination.
Thank you for your answer!