Past exam of the geometry course of the University of Oxford / 2007 / B / 8

A Riemann surface is a connected Hausdorff second-countable topological surface equipped with coordinate charts to open subsets of $\mathbb{C}$ whose transition maps are holomorphic. This is a complex manifold of complex dimension one. The atlas makes holomorphic functions, meromorphic functions and holomorphic differentials coordinate-independent notions.

Choose $r>0$ so the closed disk centered at $a$ of radius $r$ lies in $U$. The Cauchy integral formula givesParametrize $z=a+r{e}^{i\theta }$; then $dz=ir{e}^{i\theta }d\theta$ and the denominator cancels. Thus

Choose a disk on which $|f(z)|\le M=|f(a)|$. If $M=0$ the assertion is immediate. Otherwise multiply $f$ by a fixed unit scalar so that its value at $a$ is the positive real number $M$. The mean-value identity shows the average of $\mathrm{Re}f$ on every sufficiently small circle is $M$. Since $\mathrm{Re}f\le |f|\le M$ there, continuity forces equality at every point on each circle. Equality of real part to $M$ together with modulus at most $M$ forces the imaginary part to vanish. Varying the radius covers a neighborhood of $a$. Thus both $|f|$ and $f$ are constant near $a$, proving the local maximum-modulus assertion.

Compactness makes the continuous function $|f|$ attain a maximum. Apply the preceding argument in a chart at a maximizing point to make $f$ constant on a neighborhood. The permitted identity theorem and connectedness then make it constant on all of $S$. Hence every holomorphic function from a compact connected Riemann surface to $\mathbb{C}$ is constant.

A divisor on an algebraic curve is a finite integer sum $D={\sum }_p{n}_{p}p$; its degree is ${\sum }_p{n}_p$. For a meromorphic function its principal divisor records its zero and pole orders, andLet $K$ be the divisor of a nonzero meromorphic differential, called a canonical divisor. The Riemann-Roch theorem statesMultiplication by that differential identifies $L(K)$ with the space of holomorphic differentials. By the preceding result $L(0)$ consists exactly of constants, so $\ell (0)=1$. Applying the formula at $D=0$ gives $1-\ell (K)=1-g$, hence $\ell (K)=g$.

Apply Riemann-Roch at $D=K$ and use $\ell (K)=g$ and $\ell (0)=1$:Rearranging gives $\mathrm{deg}K=2g-2$. This holds for every canonical divisor, since any two differ by a principal divisor of degree zero.