Riemann-Roch Theorem

Question

Could somebody give a simple plain English explanation as to what the Riemann-Roch theorem is about to somebody who knows only standard one-variable complex analysis. Thanks.

Answer 1

Unfortunately, it's a bit hard to state with only one-variable complex analysis, since the natural setting (at least for the basic version of the theorem) is a compact Riemann surface $X$. (If you're not familiar with Riemann surfaces, keep the example $X = S^2 = \mathbb{C} \cup \{\infty\}$, the Riemann sphere, in mind.) Modulo that, here's a brief explanation: For a finite set $S\subset X$ and numbers $n(s)\in \mathbb{Z}^{\not= 0}$ for each $s\in S$, we can consider the space $V$ of meromorphic functions $f:X\to \mathbb{C}$ that are "no worse" than the function $z\to 1/z^{n(s)}$ at each $s\in S$. That is, we require that:

1) $f$ is holomorphic off $S$;

2) If $s\in S$ has $n(s) > 0$, then $f$ has a pole of order at most $n(s)$.

3) If $s\in S$ has $n(s) < 0$, then $f$ has a zero of order at least $|n(s)|$.

The Riemann-Roch theorem gives an estimate of, and often an exact value for, the dimension of $V$. (The theorem itself gives an exact value for $\dim V$, but one of the terms in it is often difficult to compute in practice.) For some motivation, consider what happens when $S$ is empty, so that $V$ is then the space of holomorphic functions on the compact surface $X$. The theorem generalizes to quite a few other objects besides meromorphic functions on Riemann surfaces.