Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 22H (Riemann Surfaces)

A meromorphic differential on a Riemann surface has local expression $\omega =a(z)dz$ with a meromorphic coefficient. On overlapping coordinates it transforms by $a(z)dz=\tilde{a}(w)dw$. For a meromorphic function $f$, the coefficients $f^{\prime }(z)$ obey this transformation by the chain rule, so $df$ is such a differential. Two nonzero differentials have coefficient ratio $a/b$; the derivative factors cancel on overlaps, yielding a global meromorphic function $h$ with $\eta =h\omega$.

An invertible holomorphic coordinate change has a nonzero holomorphic derivative, so multiplying by its derivative changes no local zero or pole order. Thus the differential's divisor is well defined. For a compact connected surface, the canonical-degree theorem states ${\sum }_P{\mathrm{ord}}_P\omega =2g-2$, including negative orders for poles. Therefore $\boxed{g=1+\frac{1}{2}\sum _P{\mathrm{ord}}_P\omega }$.

The projective closure has equation $U^2-V^2-W^2=0$. Its gradient $(2U,-2V,-2W)$ cannot vanish at a projective point, so it is nonsingular. At infinity $W=0$ gives the two points $[1:1:0]$ and $[1:-1:0]$. In the supplied sphere parameter,This has double poles at $t=\pm 1$, the two infinity points, and simple zeros at $t=\pm i$. At the parameter infinity use $s=1/t$, giving $v=2s/(1-s^2)$ and $dv=2(1+s^2)(1-s^2{)}^{-2}ds$, which is regular and nonzero at $s=0$. Hence $dv$ extends meromorphically over the whole projective curve. Its order sum $2-4=-2$ gives genus zero, consistently with the stated biholomorphism from the sphere.