Riemann-Roch 12: Application I - Compact Riemann Surfaces Are Algebraic

Riemann-Roch notes / XII

Application I: compact Riemann surfaces are algebraic

Riemann-Roch produces enough meromorphic functions to move from an analytic surface to a projective algebraic curve.

Riemann-Roch is one of the standard tools behind the theorem that compact Riemann surfaces are not merely analytic objects. They admit enough meromorphic functions to be realized as algebraic curves.

A precise version is:

Algebraicity theorem

Every compact connected Riemann surface is biholomorphic to a smooth projective algebraic curve over \(\mathbb C\).

The theorem is deep, but the role of Riemann-Roch is concrete: it manufactures meromorphic functions with prescribed poles. With enough such functions, one obtains a map to projective space that separates points and tangent directions. The image is then a projective algebraic curve, and the original Riemann surface is its normalization.

How Riemann-Roch supplies functions

Fix a point \(P\in X\). For large \(n\), Riemann-Roch gives

\[\ell(nP)=n+1-g\]

because \(\ell(K-nP)=0\) once \(n>2g-2\). In particular, \(L(nP)\) eventually has nonconstant functions. By increasing the divisor and comparing dimensions, one can choose functions that separate two given points and distinguish tangent directions at a point.

This is the analytic-to-algebraic bridge: meromorphic functions become rational functions on the eventual algebraic model.

Two concrete models

The sphere

For \(X=\mathbb P^1\), the divisor \(n\infty\) gives

\[L(n\infty)=\operatorname{span}\{1,z,\ldots,z^n\}.\]

The single meromorphic function \(z\) already identifies the sphere with the projective line. Here algebraicity is visible before the theorem is needed, but it is the model for the general construction.

A complex torus

Let \(X=\mathbb C/\Lambda\). The Weierstrass function \(\wp(z)\) has a double pole at the origin, and \(\wp'(z)\) has a triple pole. Thus

\[\wp\in L(2O), \qquad \wp'\in L(3O).\]

They satisfy the cubic equation

\[(\wp')^2=4\wp^3-g_2\wp-g_3.\]

So the torus is realized as the smooth projective cubic

\[y^2=4x^3-g_2x-g_3.\]

Riemann-Roch predicts the dimensions behind this picture: on a genus one curve, \(\ell(2O)=2\) and \(\ell(3O)=3\).

A third familiar family is hyperelliptic. On

\[C:y^2=f(x),\qquad \deg f=2g+1,\]

the functions \(x\) and \(y\) have pole orders \(2\) and \(2g+1\) at infinity. The algebraic equation is not imposed from outside; it is the relation among meromorphic functions generated by these pole-bounded spaces.

What the application really says

The slogan “compact Riemann surface equals algebraic curve” should not be read as a coincidence between two categories. Riemann-Roch explains why compactness plus one complex dimension is rigid enough to force many meromorphic functions. Those functions provide projective coordinates, and their algebraic relations cut out the curve.