Riemann-Roch theorem

Statement

Let $S$ be a compact Riemann surface. Let $K$ be a canonical divisor for $S$, and $D$ be any divisor. Let $L(D)$ denotes the L-space of $D$, i.e., the vector space of meromorphic functions on $S$ that are either zero or have divisor greater than or equal to $D$. Then:

$\mathrm{d}\mathrm{i}\mathrm{m}L(-D)=\mathrm{d}\mathrm{e}\mathrm{g}(D)-g+1+\mathrm{d}\mathrm{i}\mathrm{m}L(D-K)$.

Related facts

Applications

• Riemann's inequality
• Genus zero Riemann surface is conformally equivalent to Riemann sphere
• Every compact Riemann surface is a branched cover of the Riemann sphere