Prove that every compact Riemann surface is an algebraic curve.

Question

I am currently studying Riemann surfaces and I find a theorem stating "Every compact Riemann surface is an algebraic curve." but I don't find a proof. Thanks in advance for helping by providing a proof...

Answer 1

There is a proof in Griffiths and Harris (page 215) that uses nothing other than the Kodaira vanishing theorem. But the proof of the Kodaira vanishing theorem (page 154) uses a version of the Hodge decomposition theorem for line bundles (page 152), and the proof of Hodge decomposition uses the usual tools for elliptic PDEs (Sobolev inequalities, Rellich compactness, spectral theorem, regularity lemma, pages 85-96).

The basic idea is to show that there exists a line bundle $\mathcal L$ on the Riemann surface $C$ whose global sections separate points and separate tangent vectors. The global sections of such a line bundle $\mathcal L$ will then define an embedding $C \hookrightarrow \mathbb P^{H^0(\mathcal L) - 1}$.

The statement that the global sections of $\mathcal L$ separates points $p, q \in C$ (with $p \neq q$) is equivalent to the statement that the restriction map $$r_{p,q} : \mathcal L \to \mathcal L_p \oplus \mathcal L_q$$ induces a surjection on global sections (where $\mathcal L_p$ and $\mathcal L_q$ are the stalks of $\mathcal L$ at $p$ and $q$, which can be though of as skyscraper sheaves supported at these points).

In view of the short exact sequence, $$ 0 \to \mathcal L(-p-q) \to \mathcal L \overset{r_{p,q}}{\to} \mathcal L_q \oplus \mathcal L_q \to 0, $$ the induced map $r_{p,q} : H^0(\mathcal L) \to H^0(\mathcal L_q \oplus \mathcal L_q)$ is surjective if $$ H^1(\mathcal L(-p-q)) = 0,$$ and this is true if ${\rm deg \ } \mathcal L - 2 > {\rm deg \ } \mathcal K_C$, by Kodaira vanishing. So $\mathcal L$ separates points if $${\rm deg \ } L> {\rm deg \ } \mathcal K_C + 2.$$ Under the same inequality, it can also be proved that $\mathcal L$ separates tangent vectors. (The proof is almost a special case of the previous one, but with $p = q$.)

The conclusion is that the global sections any line bundle with sufficiently high degree define an embedding of $C$ into projective space, hence $C$ must be an algebraic curve.