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I am currently learning about Riemann surfaces, and just recently learned about line bundles. I've been asked to show on an assignment that $K_X=\cup_{x \in X} \Omega_x$ is a holomorphic line bundle on X, called the canonical line bundle on X. Here, $\Omega$ is the sheaf of holomorphic 1-forms on X.

It seems natural to define the map from $K_X$ to X by sending an element of $\Omega_x$ to $x \forall x \in X$. In this case, the fiber over any $x$ is just $\Omega_x$.

I don't see why this is going to be a finite dimensional vector space over $\mathbb{C}$. I understand it will be a vector space, but I don't see why it will be finite dimensional. For example, if we take $X = \mathbb{C}$ and take $x = 0$, we can expand any holomorphic function in a Taylor series centred at $0$, and so we'd have as our basis for $\Omega_0$ the set $\{1dz, xdz, x^2dz, x^3dz, ..., x^ndz, ...\}$, no?

In that we refer to $K_X$ as a line bundle, other sources seem to suggest we should have the fibers as 1-dimensional vector spaces over the complex numbers, so I guess a basis would just be $dz$ for $z$ a local coordinate, so I'm not convinced in my above reasoning. However, if we just had a 1-dimensional vector space with basis $dz$, we would only get germs corresponding to locally constant functions, which aren't the only holomorphic functions.

I'm using Otto Forster's book "Lectures on Riemann Surfaces" for reference.

Garnet
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Given a locally free sheaf $\mathcal L$ of rank one on a Riemann surface you can associate to it at a point $x\in X$:
$\bullet $ Its stalk $\mathcal L_x$, which is a free, rank-one module of rank one over $\mathcal O_{X,x}$ but an infinite-dimensional $\mathbb C$-vector space.
$\bullet \bullet $ Its fiber $\mathcal L[x]=\mathcal L_x/\mathfrak m_x \mathcal L_x$ where $\mathfrak m_x \subset \mathcal O_{X,x}$ is the maximal ideal consisting of germs vanishing at $x$.
This fiber is a vector space of dimension $1$ over $\mathbb C$ .

If you now take for $\mathcal L$ the sheaf of holomorphic $1$-forms $\Omega_X^1$, you obtain fibers $\Omega_X^1[x]$ whose disjoint union $K_X=\sqcup_{x\in X}\Omega_X^1[x]$ is the canonical line bundle.

To sum up : the stalk of the sheaf of holomorphic one forms $\Omega^1_{X}$ can be written in a local coordinate chart $z$ centered at $x$ as the infinite-dimensional complex vector space $\Omega^1_{X,x}=\oplus_k \mathbb C\cdot (z^kdz)_x$, whereas the fiber of that sheaf is the one-dimensional complex vector space $K_X[x]=\mathbb C\cdot (dz)[x]$ .

Georges Elencwajg
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