I am currently trying to explicitly calculate the pullback of a (p,q) form. So let's say that we have $\omega \in \Omega_X^{p,q}(U)$ and it is $\omega=\sum_{I \in I(p,n),J\in I(q,n)}\omega_{I,J}dz^I \wedge d\overline{z}^J$.
How can I now calculate the pullback $f^*\omega$ with some (holomorphic) $f: V \rightarrow U$ in local coordinates? I would really appreciate it if someone could show me how this can be done.
Note: I already tried it myself but for some reason I can't handle the expression $df^I \wedge d\bar{f}^J$ that appears when I try to use $f^*(dz\wedge d\overline z) = df\wedge d\overline{f}$ with multi indices.