Riemann-Roch 04: Canonical Divisors

Riemann-Roch notes / IV

Canonical divisors and differentials

The correction term in Riemann-Roch comes from differentials. Canonical divisors are the divisor language for those differentials.

A meromorphic differential is locally written as

\[\omega=f(z)\,dz.\]

Just as a meromorphic function has a divisor of zeros and poles, a nonzero meromorphic differential has a divisor

\[(\omega)=\sum_P \operatorname{ord}_P(\omega)P.\]

Any divisor of this form is called canonical. It is denoted by \(K\).

Independence of the choice

If \(\omega_1\) and \(\omega_2\) are two nonzero meromorphic differentials, then \(\omega_1/\omega_2\) is a meromorphic function. Therefore

\[(\omega_1)-(\omega_2)=(\omega_1/\omega_2).\]

So different choices of differential give linearly equivalent canonical divisors. The canonical class is well defined.

Canonical degree

For a compact Riemann surface of genus \(g\),

\(\deg(K)=2g-2.\)

On \(\mathbb P^1\), the differential \(dz\) has a double pole at infinity, so \(K\sim -2\infty\). On an elliptic curve, the invariant differential has no zeros and no poles, so \(K\sim 0\).

Why differentials enter Riemann-Roch

The theorem contains \(\ell(K-D)\), not merely \(\ell(D)\). This is the dual term. Classically it is a space of meromorphic differentials constrained by \(D\). Cohomologically it is Serre dual to \(H^1(X,\mathcal O(D))\).

A first consequence comes from setting \(D=0\):

\[\ell(0)-\ell(K)=1-g.\]

Since the only holomorphic functions on a compact Riemann surface are constants, \(\ell(0)=1\). Hence

\[\ell(K)=g.\]

Interpretation

The dimension of the space of holomorphic one-forms is the genus. This is one of the cleanest places where topology appears as a vector-space dimension.