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Riemann Surfaces RIEMANN SURFACES YANKI LEKILI 1. Motivation Riemann surfaces are domains of most general type which can be used to replace the complex plane in studying holomorphic functions of one complex variable. p To illustrate this, let us consider the problem of defining the square-root function z for z 2 C. In other words, we would like to solve the equation w2 = z: There is no single function z 7! w(z) that is defined in the whole complex plane pC. However, if we cut along the axis [0; 1), we can define two holomorphic functions ± z defined on C n [0; 1) by iθ p p i θ z = re 7! + z = re 2 iθ p p i θ z = re 7! − z = − re 2 iθ p p pNow, observep that as z = re goes to r as θ goesp to 0, we get thatp + z !p+ r and −pz ! − r, whereas if we let θ go to 2π, then + z goes to − r, and − z goes to + r. p There are two issues here. First, the functionp z is not defined on all of C, and second if we remove [0; 1), there are two values of z, i.e. it is a multi-valued function. Let us write D for the bordification of the domain C n [0; 1p ) where we regard distinct sides of the cut as distinct edges of D. We can extend ± z to D by continuity.p To distinguish the two extension, we write D± for the domain of the extension of ± z. Now, we construct a \Riemann surface" S by gluing D+ and D− in a way that the upper side of the cut in D+ gets identified with lower side of the cut in D−, and the lower side of the cut in D+ gets identified with the uppser side of the cut in D−. Let φ : S ! C be p the natural map taking points in D± to the \same" points in C. The functions ± φ(z) on D± taken together give a single-valued function w : S ! C such that w(φ(z))2 = φ(z) Thus, the basic idea of Riemann surface theory is to construct a domain S on which the branches of the multi-valued functions on C fit together to define a single-valued function. Similarly, we can associate a \domain" to any multi-valued holomorphic function w(z) satisfying P (z; w(z)) = 0 1 2 YANKI LEKILI where P (z; w) is an irreducible polynomial. The resulting Riemann surface is an \algebraic curve" 2 C = f(z; w) 2 C : P (z; w) = 0; (@wP )(z; w) 6= 0g The points where (@wP )(z; w) = 0 are called ramification points. They could be filled in, and the function w : C ! C can be extended but its derivative will vanish at these points, so these are the \true" singularities of the original multi-valued function. 2. Basic definitions Definition 1. A topological surface is a Hausdorff topological space S which is locally homeomorphic to C. Here, locally homeomorphic means that and p 2 S has an open neighborhood U in S which is homeomorphic to an open subset V of C. Definition 2. A Riemann surface is given by the following: • A Hausdorff topological space S S • A collection of open sets Uα ⊂ S such that α Uα = S, • For each α we have a homeomorphism φα : Uα ! φα(Uα) = Vα ⊂ C such that whenever Uα ◦ Uβ 6= 0, the map −1 φβ ◦ φα : φα(Uα \ Uβ) ! φβ(Uα \ Uβ) is holomorphic. We recall that a function f : U ! V with U; V ⊂ C is holomorphic if the limit 0 f(z0 + h) − f(z0) f (z ) = lim ∗ 0 h2C ;h!0 h exits for all z0 2 U. Writing this in real coordinates, z = x + iy, f(z) = u(x; y) + iv(x; y), this condition is equivalent to u and v being continuously differentiable satisfying the Cauchy-Riemann equations: @xu = @yv @xv = −@yu and it is also equivalent to the existence of a power series expansion, for each z0 2 U: 1 X n f(z) = cn(z − z0) n=0 absolutely convergent in an open set (and uniformly convergent on every compact). −1 A single (φα;Uα) are referred to as a \chart" or a \local co-ordinate", the maps φβ ◦ φα relating different local co-ordinates are called transition functions, the whole collection of charts f(φα;Uα)g is called an \atlas". Informally, one can think of a Riemann surface as open subsets of C \glued together" via holomorphic maps. RIEMANN SURFACES 3 Definition 3. A map f : S ! T between Riemann surfaces is called holomorphic if for every choice of co-ordinate φ in S and is T , the composition ◦ f ◦ φ−1 is a holomorphic function on its domain of definition. In particular, a function f : S ! C is holomorphic if f ◦ φ−1 is holomorphic wherever it is defined. We shall identify two Riemann surfaces whenever there is a holomorphic bijection f : S ! T . This allows us to remove the dependence of the definition of a Riemann surface on a particular atlas. 2.1. First Examples. • Any open subset of C is a Riemann surface by using a single chart. Some important examples of this are the whole complex plane C, the unit disc D = fz 2 C : jzj < 1g, and the upper halfplane H = fz 2 C : Imz > 0g. Exercise: i) Show that H and D are isomorphic Riemann surfaces via the map: H ! D z − i z 7! z + i ii) Show that C and D are not isomorphic Riemann surfaces, by using Liouville's theorem (a bounded entire function is necessarily constant). • The Riemann sphere S2 = C [ f1g is topologically given as a one-point compact- ification of C - a collection of basic open sets of 1 is given by the family of sets B(1; r) = fz 2 C : jzj > rg [ f1g. Next, the following two charts makes S2 into a Riemann surface: U0 = C; φ0(z) = z ( 1=z if z 6= 1 U1 = (C n f0g) [ f1g; φ1(z) = 0 if z = 1 Exercise: Show that the projective line 1 2 × P = (C n f0g)=C ; the space of lines through origin in C2 is a Riemann surface which is isomorphic to the Riemann sphere. • The cylinder, C=Z, is the quotient space of C where two points are identified if they differ by an integer. Consider the two open sets V0 = (0; 1) × R ⊂ C and V1=2 = (1=2; 3=2) × R ⊂ C. When restricted to V0 and V1=2, the projection map π : C ! C=Z is a homeomorphism, hence, C=Z is a topological surface and we can −1 −1 define charts by letting U0 = π(V0), φ0 = π and U1=2 = π(V1=2), φ1=2 = π . jV0 jV1=2 Furthermore, the transition function −1 φ1=2 ◦ φ0 : ((0; 1=2) [ (1=2; 1)) × R ! ((1=2; 1) [ (1; 3=2)) × R 4 YANKI LEKILI given by z 7! z + 1; if Rez 2 (0; 1=2) z 7! z; if Rez 2 (1=2; 3=2) is a holomorphic function. Similarly, one can check that the transition function −1 φ0 ◦ φ1=2 is a holomorphic function. Exercise: Show that C=Z and C× := C n f0g are isomorphic Riemann surfaces. • The complex torus associated to a lattice Λ is the quotient space C=Λ, where Λ is a discrete subgroups of C generated by two non-zero complex numbers !1 and !2 that are linearly independent over R, i.e. !1=!2 2= R. One can show that these are Riemann surfaces via an argument similar to the one given for the cylinder. Exercise: (i) Show that C=Λ is a Riemann surface for any lattice Λ. (ii) Show that any such Riemann surface C=Λ is isomorphic to C=(Z ⊕ τZ) for some τ 2 H. • Hyperbolic surfaces. Recall the group SL2(R) is given by a b ; a; b; c; d 2 ; ad − bc = 1 c d R SL2(R) acts on the upper half-plane H by holomorphic isomorphisms via az + b z 7! cz + d This action actually descends to an action of P SL2(R) = SL2(R)={±Ig. Now, let Γ ⊂ SL2(R) be a torsion-free discrete subgroup of SL2(R), then H=Γ is a Riemann surface. Exercise: Consider the subgroup of SL2(R) isomorphic to Z given by 1 n ; n 2 0 1 Z Show that H=Z is isomorphic to D× = D n f0g as a Riemann surface. More generally, for λ > 1, let λZ be the subgroup of H given by 1 nλ ; n 2 0 1 Z Then H/λZ is isomorphic to the annulus A(r) = fz : r < jzj < 1g with r = exp(−2π2= log λ): In fact, above we have listed all the Riemann surfaces. Theorem 4. Every Riemann surface S is isomorphic to one of 2 C;S ; C=Z; C=Λ; H=Γ where Λ ⊂ C is some lattice, and Γ ⊂ SL2(R) is a torsion-free discrete subgroup. This is a deep theorem called the uniformization theorem for Riemann surfaces with a long history. The first proofs are due to Poincar´eand Koebe around 1907. There are also several modern proofs. We will not cover any of the proofs. RIEMANN SURFACES 5 3. Algebraic Curves 3.1. Affine curves. A large class of examples of Riemann surfaces are obtained via poly- nomials in two variables. Definition 5.
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