The Support of Differential Forms defined by Pulling-Back on Complex Manifolds of the Same Dimension

Question

Let $M,N$ be two complex manifolds of the same (complex) dimension.

Let $\omega$ be a differential form on the manifold $N$ whose support denoted by $S$.

Let $f: M \longrightarrow N$ be a smooth map.

How to define the support of the form $f^* \omega$ in terms of the support of $\omega$?