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!
Best Answer
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.