A map $\mathbb{C}^2\rightarrow \mathbb{C}^2$ that preserves a holomorphic 2-form.

Question

Let $\mathbb{C}^2\rightarrow \mathbb{C}^2$ be a map. If $f$ preserves a non-zero holomorphic 2-form, is $f$ a holomorphic map?

Answer 1

Yes. At the points where $\omega\neq 0$ both maps $(Re\omega)^\#,(Im\omega)^\#:T\mathbb{C}^2\to T^*\mathbb{C}^2$ (given by contraction) are invertible and if you compose one with the inverse of the other you get the complex structure $J:T\mathbb{C}^2\to T\mathbb{C}^2$. Hence $J$ is preserved and $f$ is holomorphic at the points where $\omega\neq0$, which implies that $f$ is holomorphic everywhere.