How to Push-Forward Differential Forms on a Complex Surface to a Complex Surface by a Holomorphic Function

Question

Let $M$ and $ N$ be two Compact Complex Surfaces (i.e., $M$ and $N$ are Compact Complex Manifolds of complex dimension Two).

Let $A$ be a (non-empty) subset of the complex surface $M$ (not necessarily a sub-manifold).

Let $f: M \longrightarrow N$ be a Holomorphic map such that:

1. $f(A)$ is a finite subset $N$;
2. $f: M \setminus A \longrightarrow N\setminus f(A)$ is a Diffeomorphism (or even a Bi-Holomorhpic).

Let $\omega$ be a $(1,1)$ differntial form on the complex surface $M$.

Under the assumptions above, can we push-forward the differential form $\omega$ to the complex surafce $N$ by the map $f$?