Riemann-Roch 09: Cohomological Form
Riemann-Roch notes / IX
The cohomological form
In modern language, Riemann-Roch is the Euler characteristic formula for the line bundle attached to a divisor.
The divisor \(D\) determines a line bundle \(\mathcal O(D)\). Its global sections are exactly the classical space:
\[H^0(X,\mathcal O(D))=L(D).\]Thus \(h^0(X,\mathcal O(D))=\ell(D)\).
The cohomological form of Riemann-Roch is
Curve Riemann-Roch
\(h^0(X,\mathcal O(D))-h^1(X,\mathcal O(D))=\deg(D)+1-g.\)
The left side is the Euler characteristic \(\chi(\mathcal O(D))\).
Recovering the classical formula
Serre duality gives
\[H^1(X,\mathcal O(D))^*\cong H^0(X,\mathcal O(K-D)).\]Therefore
\[h^1(X,\mathcal O(D))=\ell(K-D).\]Substitution gives
\[\ell(D)-\ell(K-D)=\deg(D)+1-g.\]This is exactly the classical theorem.
Why this version is useful
The cohomological statement explains the correction term. It is not a mysterious extra space added to fix the formula. It is the first cohomology group of the line bundle, written in dual classical language.
| Classical term | Cohomological term |
|---|---|
| \(L(D)\) | \(H^0(X,\mathcal O(D))\) |
| \(\ell(K-D)\) | \(h^1(X,\mathcal O(D))\) by duality |
| dimension formula | Euler characteristic formula |
Two quick checks make the dictionary less abstract.
On \(\mathbb P^1\), the line bundle \(\mathcal O(n)\) satisfies
\[h^0(\mathbb P^1,\mathcal O(n))=n+1, \qquad h^1(\mathbb P^1,\mathcal O(n))=0\]for \(n\ge 0\). For \(n=-3\), the global sections vanish, but
\[h^1(\mathbb P^1,\mathcal O(-3))=2,\]which is dual to \(H^0(\mathbb P^1,\mathcal O(1))\).
On an elliptic curve, a line bundle of positive degree \(d\) has
\[h^0=d, \qquad h^1=0.\]For the trivial bundle, however, \(h^0=1\) and \(h^1=1\). The Euler characteristic is zero, exactly as \(\deg+1-g=0\) predicts for degree zero and genus one.
For computations on curves, the classical notation is often quicker. For proofs, generalizations, and higher-dimensional analogues, the cohomological form is the durable one.