# Triviality of canonical line bundle of $\mathbb{C}^n$

complex-geometrydifferential-formsdifferential-geometry

This is a very basic question but I'm learning differential geometry and I'm currently thinking about the following canonical line bundle, $$K_{\mathbb{C}^n}=\bigwedge^nT^*\mathbb{C}^n$$. We know that this line bundle is trivial if and only if there exists a nowhere vanishing holomorphic $$n$$ form. My feeling is that this line bundle is trivial since we can just take the form $$dz_1\wedge\cdots \wedge dz_n$$. Is this correct?

Does this same form then also work for the canonical line bundle of $$(\mathbb{C}^*)^n$$? I'm sure I've read somewhere that the appropriate volume form to take for $$(\mathbb{C}^*)^n$$ is $$\frac{dz_1}{z_1}\wedge \cdots \wedge \frac{dz_n}{z_n}$$ but wouldn't the form $$dz_1\wedge \cdots \wedge dz_n$$ also show triviality of the canonical bundle in this situation?

Any help is much appreciated!

Any nowhere-zero multiple of a nowhere-zero $$n$$-form is another nowhere-zero $$n$$-form. So both work. The reason you've seen $$\dfrac{dz_1}{z_1}\wedge\dots\wedge\dfrac{dz_n}{z_n}$$ is that it is naturally $$(\Bbb C^*)^n$$-invariant.