Riemann surfaces Paul Hacking and Giancarlo Urzua 1/28/10 A Riemann surface (or smooth complex curve) is a complex manifold of dimension one. We will restrict to compact Riemann surfaces. It is a theorem that compact Riemann surfaces are the same as smooth projective curves, this is, closed one dimensional complex manifolds in P n defined by a system of homogeneous polynomials. This statement is false in dimensions bigger than one. 0.1 genus Recall the topological classification of oriented compact surfaces (smooth real manifolds of dimension two): there is a unique topological invariant, the genus (number of holes). Given a triangulation on a compact Riemann surface X of genus g(x), the Euler characteristic of X is defined as its number of vertices minus its number of edges plus its number of triangles. This is a well-defined invariant, independent of the chosen triangulation, which we denote by e(x). It is not hard to check: e(x) = 2 2g(X). 0.2 degree If f : X Y is a (non constant) holomorphic map between Riemann surfaces, and P X is a point, then we can choose local coordinates z at P X and w at f(p) Y such that the map f is given by (0 C 1 z) (0 C 1 w), z z e P for some e P 1, the ramification index of f at P. For Q Y the number d = P f 1 Q is the number of points mapping to P counted with multiplicities. It is independent of Q Y and called the degree of f. (A general fibre consists e P 1
of d distinct points). The proof uses that the function Q P f 1 Q e P is continuous, and that X is connected and compact. The points in X with e P > 1 form a finite set called ramification points of f. Its image is the set of branch points. Figure 1: A fundamental domain of a compact Riemann surface on D 0.3 Riemann-Hurwitz [GH78, p. 216] Let X,Y be compact Riemann surfaces and f : X Y a (non constant) holomorphic map of degree d. Then 2g(X) 2 = d(2g(y ) 2) + P X(e P 1). This can be proved by pulling back a triangulation of Y with vertices containing the branch points of f. Then, we compute e(x) = 2 2g(X) using this new triangulation. It can also be proved via differentials, which avoids this topological argument. There are analogues of Riemann-Hurwitz in higher dimensions through differentials, where X and Y have the same dimension. 2
The Riemann sphere P 1 is only compact Riemann surface of genus zero (why?), and has itself as universal covering. For genus one, we have the tori given by C/1 Z + τ Z. As τ moves in the upper-half plane, one gets all possible genus one compact Riemann surfaces (why?). Notice that the universal covering is C. This description comes from the topological quotient of a square (fundamental domain of the group 1 Z + τ Z acting in C ), where opposite sides are glued. For genus g 2, we have the disk D = {z C : z < 1} as universal covering and the description of these curves, in general, is much more complicated. The construction of a genus g 2 compact Riemann surface depends on 3g 3 complex parameters (Riemann). The fundamental domain is a hyperbolic polygon in D. Figure 1 is an example. Find its genus using Riemann-Hurwitz. Also find its name, it is a famous one. Figure 2: A fundamental domain on the D for a hyperelliptic curve 0.4 Hyperelliptic curve [GH78, p. 253] Let a 1,...,a 2g+2 C 1 x P1 be 2g + 2 distinct points. Then the curve U = (y 2 = (x a i )) C 2 x,y C 1 x 3
can be compactified to a (smooth) double cover X P 1 branched over the points a 1,...,a 2g+2. By the Riemann-Hurwitz formula, we have added two points to compactify, and the genus of X is g. (This is not the compactification in P 2, where the missing points become one singular point when g > 0). One can visualize the involution on X (with 2g +2 fixed points) by rotating the surface of genus g on a skewer through 180 [Reid88, p. 45]. In Figure 2 we have a hyperelliptic curve. Find its genus and find the involution, where are the ramification points? In this way, we have concrete examples in any genus. This is also particular to dimension one, since even for complex manifolds in dimension two there are open questions about finding examples with basic topological invariants fixed. 0.5 genus formula for plane curve [GH78, p. 219] Let X = (F = 0) P 2 be a smooth plane curve defined by a homogeneous polynomial F of degree d. Then g(x) = 1 (d 1)(d 2). 2 The first few values are g = 0,0,1,3,6,... for d = 1,2,... In particular, not every Riemann surface can be embedded in the plane. One way to prove the genus formula is by projecting from a point P outside of X into a line (degree 1 curve). (Notice that a line is isomorphic to P 1, a conic (degree 2) too!). We can assume that P = [0 : 0 : 1]. Then there is a holomorphic map P 2 \P P 1 sending [x : y : z] to [x : y], the projection from P to the line z = 0. Then, we restrict this map to X, obtaining h: X P 1. By Riemann-Hurwitz formula, we have 2d + 2g(X) 2 = P X (e P 1), but the right hand side can be computed as the intersection of X with Y = (F y = 0), where F y is the derivative of F with respect to y. Here we use two things. One is Bezout s Theorem which says: given two different irreducible homogeneous polynomials F and G of degrees d and e, the intersection of the curves (F = 0) and (G = 0) is de (counting multiplicities when tangencies appear). This is because the intersection class of (F = 0) is deg F L where L is any line (so L L = 1). Hence, in our case, X Y = d(d 1). But also X Y = P X (e P 1). Why? 4
Say z = 0 and y = 0 do not intersect X Y, and y = 0 intersect X at d points. We know that the ramification of h is in (f = F(x,y,1) = 0), and we only need to check the projection in x. Now, if f y = 0 at P X, then f x 0, and so there is x = x(y) chart of X at P. But then f x x y + f y = 0 and so the order of vanishing of f y at P is the ramification index of x(y) minus one at P. But this is the intersection of Y and X at P, and so (X Y ) P = e P 1. Let P be a general point in P 2 not in X. How many tangent lines has X containing P? When d > 3, are any of these smooth curves X in P 2 hyperelliptic? Find non hyperelliptic curves. Figure 1 is a plane curve, is it hyperelliptic? References [GH78] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley 1978. [Reid88] M. Reid, Undergraduate algebraic geometry, CUP 1988. 5