Let $M,N$ be two complex manifolds of the same (complex) dimension.
Let $A$ be a subset of $M$ (not necessarily a sub-manitold).
Suppose $f: M \longrightarrow N$ is a map such that $f: M \setminus A \longrightarrow N\setminus f(A)$ is a diffeomorphism (or even a bi-holomorhpic).
Let $\omega$ be a differntial form on the manifold $N$.
Under the assumption above, can I bull-pack the differential form $\omega$ to the manifold $M$ by the restriction of the map $f$ on $M \setminus A$?