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Riemann Surfaces RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. They are crucial objects of interest in algebraic geometry, num- ber theory, symplectic geometry, dynamics, and complex analysis, just to name a few. One good, albeit advanced reference for this ma- terial is [McM], where I learned much of the material in these notes. To start, we begin by giving some examples of Riemann surfaces. Intuitively, a Riemann surface is just an object which looks like C when you zoom in. Example 1.1. Here are some examples of Riemann surfaces. (1) C (2) C× = C − f0g (3) The Riemann sphere, Cˆ (4) C/L for some lattice L (5) The g-holed torus (6) H := fx 2 C : im x > 0g , the upper half plane (7) H/G where H is the upper half plane, and G is some torsion free discrete group in SL2(R) Remark 1.2. It is a nontrivial theorem (the uniformization theorem) that the above Riemann surfaces are an exhaustive list (though the above list does contain some repetitions). We’re ready to define Riemann surfaces. If you know about man- ifolds, a Riemann surface is just a 1-dimensional complex manifold with complex holomorphic transition functions. Remark 1.3. One way to define a (compact) Riemann surface is as a connected subset X ⊂ Cn such that for every point x 2 X there is a biholomorphism (a holomorphic map with holomorphic inverse) between X and an open subset of C. However, that definition is not so great, because it requires that we specify an embedding X ! Cn for some n. It would be better to have an intrinsic definition, not depending on an embedding. Recall that a topological space X is Hausdorff if for any two points x, y 2 X there are open sets U, V with x 2 U, y 2 V so that U \ V = Æ. A topological space is second countable if there is a countable collection of open sets Ui so that every open is a union of such Ui. A topological space is connected if it cannot be written as a disjoint RIEMANN SURFACES 3 union of two nonempty open sets. A map is continuous if the preim- age of an open set is open. A homeomorphism is a continuous map between two topological spaces that has a continuous inverse. Definition 1.4. A Riemann surface is a Hausdorff, second countable, connected topological space X with an open cover by sets Ui ⊂ X and maps fi : Ui ! C so that (1) fi is a homeomorphism onto an open subset of C ◦ −1j ( \ ) ! (2) The transition functions fj fi fi(Ui\Uj) : fi Ui Uj fj(Ui \ Uj) are complex analytic (i.e., holomorphic). A collection of (Ui, fi)i2I as above is called an atlas for X. A particu- lar (Ui, fi) is called a chart. Remark 1.5. A Riemann surface is the datum of the topological space X together with the atlas (Ui, fi)i2I. In particular, it is possible to have two riemann surfaces with the same underlying topological spaces but different atlases. Exercise 1.6. Show that Cˆ , the Riemann sphere (the one point com- pactification of C, whose underlying set is C [ f¥g and whose open sets are either opens in C or sets whose complements are compact sets in C) is a Riemann surface. Hint: Cover it by two charts Cˆ − f0g ˆ 1 and C − ¥, with transition function given by z 7! z . Exercise 1.7. Let E = C/L for L ⊂ C a lattice of rank 2, meaning L = Z ⊕ aZ for a 2/ R. Show that E is a Riemann surface. Hint: Pick charts for E as charts coming from open subsets U ⊂ C mapping homeomorphically to E under the projection C ! E. (Why is C itself not a chart for E?) 4 AARON LANDESMAN 2. MAPS OF RIEMANN SURFACES 2.1. Defining the maps. As is ubiquitous in mathematics, after defin- ing the objects of our category (the Riemann surfaces) we should say what maps between them are. Definition 2.1. A biholomorphism between two open sets X, Y ⊂ C is a holomorphic map f : X ! Y with holomorphic inverse. Definition 2.2. Let U1, U2 be two Riemann surfaces where U1 has an atlas consisting of the single chart f1 : U1 ! V1 and U2 has an atlas consisting of the single chart f2 : U2 ! V2, with Vi ⊂ C open. Then −1 we say a map g : U1 ! U2 is holomorphic if the map f2 ◦ g ◦ f1 : V1 ! V2 g U1 U2 (2.1) −1 f f1 2 −1 f2◦g◦ f1 V1 V2 is holomorphic. Definition 2.3. Let X and Y be two Riemann surfaces. A map f : X ! Y is a continuous map such that if (Ui, gi) are an atlas for X and (Vj, hj) are an atlas for Y, then the resulting restriction of f −1 f f (Vj) \ Ui −! Vj is holomorphic for all i, j. Loosely speaking, a map of riemann surfaces is just a map which is holomorphic when restricted to each of the open sets. 2.2. The multiplicity of a map. We next come to the notion of the multiplicity of a map of riemann surfaces. Roughly speaking, this is just the number of preimages of “most points.” The following lemma makes this precise, and is essential in what follows. Lemma 2.4 (Key Lemma). For any map of Riemann surfaces g : X ! Y, 0 with f (p) = q there exists open sets U 3 p, V 3 q and charts f1 : U ! 0 −1 0 0 U, f2 : V ! V with h := f2 ◦ g ◦ f1 : U ! V given by h(z) = 0 or h(z) = zn for some n. 0 00 0 Proof. To start, choose charts f1 : U ! U and f2 : V ! V with 00 0 0 U ⊂ C and V ⊂ C open sets containing 0 2 C with f1(0) = p and f2(0) = q. (Primes have nothing to do with derivatives here.) 0 −1 0 0 Define h := f2 ◦ g ◦ f1. By assumption, h is holomorphic, so after RIEMANN SURFACES 5 g X Y gj U U V 0 (2.2) f1 f2 00 h0 0 f1 U V a h U0 FIGURE 1. Diagram illustrating the proof of Lemma 2.4. shrinking U00, U, V0, V we can assume that h0 agrees with its Taylor expansion about 0. If h0 is constant, we can take h = h0 and we are done. So, we assume h0 is not constant. Then, we can write h0(z) = znt(z) with t(0) 6= 0. After possibly shrinking our open sets further, we may assume the power series t(z) has an nth root, say r(z)n = t(z), so h0(z) = (z · r(z))n. Define a map a : U00 ! C sending z 7! z · r(z) and let U0 ⊂ C denote the image of a. Because t(0) 6= 0, we also have r(0) 6= 0, and so a has nonvanish- ing derivative at 0. We can then formally invert the power series for a, and hence, after further shrinking the open sets U, U0, U00, V, V0, we may assume that a : U00 ! U0 is a biholomorphism. −1 0 −1 Then, define f1 := a ◦ f1 and h := h ◦ a . Since (h ◦ a)(z) = 0 n n h (z) = (z · r(z)) , and a(z) = z · r(z), we have h(z) = z . Definition 2.5. For f : X ! Y define the multiplicity of f at x to be ( d if f locally at x looks like z 7! zd via Lemma 2.4 multx( f ) = ¥ if f is constant in some neighborhood of x Lemma 2.6. If f : X ! Y is a map of Riemann surfaces, either f is constant or multx( f ) < ¥ for all x 2 X. Proof. Note that the set of points where multx( f ) < ¥ is open and the set where multx( f ) = ¥ is open, using Lemma 2.4. Since X is connected, only one of these two conditions may hold.
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