Yesterday I saw the definition of Dolbeaut cohomology: let $(M,J)$ be a complex manifold, write the exterior derivative as ${\rm d} = \partial +\overline{\partial}$, where $\partial: \Omega^{\ell,m}(M) \to \Omega^{\ell+1,m}(M)$ and $\overline{\partial}: \Omega^{\ell,m}(M) \to \Omega^{\ell,m+1}(M)$, look at the complex $$\cdots \xrightarrow{\hspace{.4cm}\overline{\partial}\hspace{.4cm}} \Omega^{\ell,m-1}(M)\xrightarrow{\hspace{.4cm}\overline{\partial}\hspace{.4cm}} \Omega^{\ell,m}(M) \xrightarrow{\hspace{.4cm}\overline{\partial}\hspace{.4cm}} \Omega^{\ell,m+1}(M)\xrightarrow{\hspace{.4cm}\overline{\partial}\hspace{.4cm}} \cdots$$and put $$H^{\ell,m}_{\rm Dolbeaut}(M) \doteq \frac{\ker(\overline{\partial}: \Omega^{\ell,m}(M) \to \Omega^{\ell,m+1}(M))}{{\rm Im}(\overline{\partial}: \Omega^{\ell,m-1}(M) \to \Omega^{\ell,m}(M))}.$$Great.
Question: Why did we took the above complex instead of $$\cdots \xrightarrow{\hspace{.4cm}\partial\hspace{.4cm}} \Omega^{\ell-1,m}(M)\xrightarrow{\hspace{.4cm}\partial\hspace{.4cm}} \Omega^{\ell,m}(M) \xrightarrow{\hspace{.4cm}\partial\hspace{.4cm}} \Omega^{\ell+1,m}(M)\xrightarrow{\hspace{.4cm}\partial\hspace{.4cm}} \cdots?$$
I get that $f: M \to \Bbb C$ is $J$-holomorphic if and only if $\overline{\partial}f = 0$ so that $\overline{\partial}$ sort of measures how stuff in general would be away from being holomorphic. But I'd like a more solid explanation.
Both $\partial$ and $\bar{\partial}$ are well defined locally. However, when we are looking at a global section of a holomorphic vector bundle, then $\partial$ is in general not well defined! (When both are well-defined, indeed I have seen people who consider both complexes and cohomologies.)
The point is that to describe a section, we usually pass to local trivialization, and patch them together with transition functions. When the bundle is holomorphic, by definition there is an atlas such that the transition functions are holomorphic. So by product rule and that $\bar{\partial}f=0$ for any holomorphic $f$, $\bar{\partial}$ can be extended from local to global.
Usually people care about holomorphic bundles more (I wonder why). If you are dealing with anti-holomorphic bundles, then indeed $\partial$ is well-defined.