Assume now we have a holomorphic vector bundles $E$ over a complex manifold $X$, $E=E_1\oplus E_2$, i.e, $E_1, E_2$ are subvectorbundles of $E$, then if we have map $f:A^{p,q}(E)\longrightarrow A^{p,q}(E_1)$, more precisely, $f$ is the identity on $A^{p,q}$, tensored with the projection of $E$ onto $E_1$ along $E_2$, then my question is: how to understand this map on local representation?
For a section $\alpha$ of $A^{p,q}(E)$, locally, it can be written as $\alpha=\sum_{i=1}^{r}\alpha_i\otimes e_i$, where $\left\{e_i\right\}_{i=1}^{r}$ is the local holomorphic frame of $E$.
Then what is the local representation of $f(\alpha)$? How about $f(\alpha)=\sum_{i=1}^{r_1}\alpha_i\otimes e_i$ where $\left\{e_i\right\}_{i=1}^{r_1}$ is the local holomorphic frame of $E_1$?
Is my method right? Any suggestion will be appreciated. Thanks a lot!
If your choice of local basis of sections $\{e_1, \dots, e_{r_1+r_2}\}$ for $E$ is such that $\{e_1, \dots, e_{r_1}\}$ is a local basis of sections for $E_1$, then your projection formula is correct. Likewise, if $\{e_1, \dots, e_{r_1+r_2}\}$ is such that $\{e_{r_1+1}, \dots, e_{r_1+r_2}\}$ is a local basis of sections for $E_2$, then there is a similar formula for the projection onto $E_2$. Not every choice of $\{e_1, \dots, e_{r_1+r_2}\}$ will have either of these properties, but one can always choose a local basis of sections for $E$ which satisfies both: take a local bases of sections for $E_1$ and $E_2$, then together they form a local basis of sections for $E$.
This is just the bundle analogue of choosing a basis $\{v_1, \dots, v_{r_1+r_2}\}$ for a vector space $V = V_1\oplus V_2$ such that the projection map $p_1 : V \to V_1$ satisfies
$$p_1(v_i) = \begin{cases} v_i & i = 1, \dots, r_1\\ 0 & i = r_1 + 1, \dots, r_1 + r_2. \end{cases}$$
Likewise for the projection $p_2 : V \to V_2$.