Let $M$ be a complex manifold, $A^{p,q}(M)$ be $C^{\infty}$ $(p,q)$ form.
Dolbeault cohomology $H^{p,q}_{\bar{\partial}}(M)$ is defined as the cohomology with boundary map $\bar{\partial}$, but why not define $H^{p,q}_{\partial}(M)$, with boundary map $\partial$?