Pullback of support of differential form

Question

Let $f:X \to Y$ be a diffeomorphism between smooth complex compact manifolds. Let $\omega$ be a differential form on $Y$. It is true that the support of $f^*\omega$ is equal to $f^{-1}$ of the support of $\omega$?

Answer 1

Yes. Intuitively this is because the pullback and the preimage are the natural way to transfer forms and sets respectively across a smooth map, so you're essentially just "renaming" everything via the diffeomorphism.

If that doesn't satisfy you then the details are fairly mundane: from the definition of the pullback and the fact that every $Df_p$ is an isomorphism you get that $f^* \omega(p)$ is zero if and only if $\omega(fp)$ is. The result then follows from the fact that diffeomorphisms commute with taking closures.