Complex Analysis II

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Complex Analysis II Technische Universitat¨ Berlin Institut fur¨ Mathematik Complex Analysis II Riemann surfaces Prof. Dr. Ulrich Pinkall Lecture notes Dr.Felix Knoppel¨ and Oliver Gross February 12, 2019 Table of Contents 1 Introduction1 1.1 Basic Concepts of Complex Analysis...............1 1.2 A Historical Interlude.......................3 2 Complex Manifolds8 2.1 Crash Course in Topology.....................8 2.2 Smooth Structures......................... 10 2.3 Introduction to Several Complex Variables........... 17 2.4 Complex Structures and Manifolds................ 22 2.5 Complex Linear Subspaces.................... 24 3 The Tangent Bundle 27 3.1 Tangent Spaces........................... 27 3.2 The tangent bundle as a smooth vector bundle......... 32 3.3 Vector Bundles............................ 33 3.4 The Lie-bracket of Vector Fields.................. 38 4 Almost Complex Structures 40 4.1 Tensors................................ 40 4.2 The Newlander-Nirenberg Theorem............... 42 4.3 Connections on Vector Bundles.................. 43 4.4 Partition of unity.......................... 51 4.5 Conformal Equivalence...................... 53 5 Integration on manifolds 56 5.1 Volume Forms............................ 56 5.2 Integration of Forms........................ 59 5.3 Stokes Theorem........................... 61 5.4 Fundamental theorem for flat vector bundles.......... 63 I TABLE OF CONTENTS 6 Riemann surfaces 68 6.1 Holomorphic line bundles over a Riemann surface...... 71 6.2 Poincare-Hopf´ index theorem................... 73 7 Classification of Line Bundles 85 7.1 Tensor products of vector spaces and bundles......... 85 7.2 Line Bundles on Surfaces..................... 87 7.3 Combinatorial Topology...................... 91 7.4 Discrete Forms........................... 96 7.5 Poincare´ Duality.......................... 98 7.6 ”Baby Riemann Roch Theorem”................. 101 7.7 A Natural Complex Structure on Dual Spaces......... 105 7.8 Holomorphic Structures on Vector Bundles........... 106 7.9 Elliptic Differential Operators................... 109 8 Appendix 122 II 1. Introduction The first formal Definition of a manifold and a Riemann surface is found in the paper ”Die Idee der Riemann’schen Fl¨ache” by H. Weyl in 1910. 1.1 Basic Concepts of Complex Analysis The central objects of discussion in this course will be Riemann surfaces, but before we start we will revise some basic concepts and vivid ideas of what we will be dealing with. Later we will then (hopefully) understand that these, sometimes vaguely stated, claims are in fact true. Within this course we will see that the easiest examples of Riemann surfaces are open sets U ⊂ C = R2, e.g. D2 = f(x, y) 2 R2 j x2 + y2 < 1g, an annulus or the whole plane. Definition 1.1 (homeomorphic). Let U, U˜ ⊂ Rn are called homeomorphic, if there is a bijection j : U ! U˜ such that j and j−1 both are continuous. Example 1.2. (i) The unit disc D2 and R2 are homeomorphic. A homeomorphism is given by pr j : (r cos f, r sin f) 7! tan 2 (cos f, sin f) 1 1.1 Basic Concepts of Complex Analysis with inverse −1 2 j : (r cos f, r sin f) 7! p arctan(r)(cos f, sin f). It is easy to see that bot, j and j−1 are continuous. (ii) U1 and U2 are homeomorphic. (iii) U3 and U4 are homeomorphic. Definition 1.3 (diffeomorphic). Let U, U˜ ⊂ Rn open are called diffeomor- phic if there is a bijective map j : U ! U˜ such that j and j−1 are both smooth. 2 The same statements as in example 1.2 about U1,..., U5, R are true replac- ing homeomorphic by diffeomorphic. Definition 1.4 (biholomorphic). Let U, U˜ ⊂ Rn open are called biholomor- phic if there is a bijective map j : U ! U˜ such that j and j−1 are both holomorphic. Theorem 1.5. The open unit disc D2 is not biholomorphic to C. Proof. Let j : C ! D2 be holomorphic. Then j is bounded and hence by Liouville’s theorem constant. Thus j is no bijection. Theorem 1.6 (Riemann mapping theorem). Let U, U˜ ⊂ C be open, U 6= C, U˜ 6= C and both homeomorphic to D2. Then U is biholomorphic to U.˜ Remark 1.7. Considering theorem 1.6 one may get the idea that also the annuli U3 and U4 are biholomorphic. Surprisingly this does not hold in general, but one can proof that indeed two annuli are biholomorphic if and only if the quotient of their respective radii is equal. Example 1.8 (Riemann sphere). An informal idea about Riemann surfaces is that they are ”surfaces” (2-dimensional manifolds) that ”locally” (meaning near every point) look like an open set U ⊂ C. Another example of such a surface is given by the Riemann sphere M = S2 = C¯ . This can be seen by considering the stereographic projection from the north pole 2 sn : S n (0, 0, 1) ! C as indicated in figure 1.1. Using the projection from the south pole is changing the chart by z 7! 1/z. Thus we have completely covered S2 by charts. 2 Introduction Figure 1.1: An image of how the stereographic projection works. 1.2 A Historical Interlude Riemann surfaces naturally came up in mathematics when trying to solve so called elliptic integrals. To get an idea about what an elliptic integral is and what makes them special, we will consider two examples that math- ematicians in early times tried to conquer. This will then lead to an idea how elliptic integrals and Riemann surfaces fit together. Ellipse: An ellipse is given by 2 = f( ) 2 R2 j x2 + y = g E x, y a2 b2 1 . Z a È The length of E is then given by L = 4 1 + f 0(x)2dx, where 0 … [ ] ! R 7! ( ) = − x2 f : 0, a , x f x b 1 a2 . 3 1.2 A Historical Interlude Then 0 −xb f (x) = … 2 − x2 a 1 a2 and thus, with z = x/a and e2 = 1 − a2/b2, Z 1 2 2 L = 4a p 1−e z dz 0 (1−z2)(1−e2z2) which is an elliptic integral and has no solution by elementary func- tions. Pendulum: Considering the (mathematical) pendulum two related prob- lems arise naturally: (i) What is the period? (ii) What is q as a function of time? We assume that m = g = L = 1. The pendulum equation is q00 = − sin q. By adding − sin q to both sides and multiplying the equation with q0 we get q0q00 + q0 sin q = 0. We see that Å ã0 Ä 1 0ä2 0 00 0 2 q − cos q = q q + q sin q = 0. Ä 1 0ä2 Ä 1 0ä2 Thus 2 q − cos q, the sum of the kinetic energy 2 q and the po- tential energy 1 − cos q, must be constant. At maximum elongation a we have q0 = 0, as the potential energy is 4 Introduction maximal, what leads to const = − cos a. So we know the total energy of the system leading to Ä 1 0ä2 d 0 È 2 q = cos q − cos a , dt q = q = 2 (cos q − cos a) Thus, when q0(t) ≥ 0, by seperation of variables dq dt = È . 2(cos q − cos a) So that a quarter of a period, meaning from angle 0 to q, is Z q t = p dj 0 2(cos j−cos a) With 2 j 2 j 2 j 2 j cos j = cos 2 − sin 2 = 1 − sin 2 − sin 2 we get 2 j 1 − cos j = 2 sin 2 , thus as the same holds for cos a Z q Z q Z q t = p dj = 1 dj = 1 1 1 dj 2 q a j 2 a à j 0 2(cos j−cos a) 0 sin2 −sin2 0 sin 2 2 2 2 sin 2 1− 2 a sin 2 j sin 2 a Substitution with z = a and defining e := sin 2 leads to sin 2 j sin = 2 = 2e = 2e = p 2e z e dj j dz q j dz 2 2 dz cos 2 − 2 1−e z ; 1 sin 2 what finally yields Z r t = p dz . 0 (1−z2)(1−e2z2) 2 q sin 2 where r := 2 a . Thus, with a = q it is r = 1, therefore the full sin 2 period is given by Z 1 T = 4 p dz . 0 (1−z2)(1−e2z2) 5 1.2 A Historical Interlude But how does this now lead to Riemann surfaces? In two examples we have found integrals of the form Z 1 Z 1 ( − 2 2) p dz , p 1 e z dz 0 (1−z2)(1−e2z2) 0 (1−z2)(1−e2z2) 2 2 2 1 1 1 The map z 7! (1 − z )(1 − e z ) has zeros at − e , −1, 1, e . Define a := e . p p What shall we do with some expression like z − a, say z? We have two choices here. p For each z 2 C n fnegative real axisg consider two points representing ± z, visualize a z, but in two ’sheets’ (copies of the complex plane). The same procedure, this time we glue two copies of C together along the È real intervalls [−a, −1] and [1, a], turns (1 − z2)(1 − e2z2) into an honest function on a Riemann surface – the two-sheeted cover of C branched over −a, −1, 1, a. Remark 1.9. Note that the blue curve g on the Riemann surface is closed, whereas the green h curve is not. 6 Introduction Actually this can be regarded as a torus with two points removed – the missing points correspond to z = ¥. 7 2. Complex Manifolds 2.1 Crash Course in Topology As a Riemann surface will be also a two-dimensional real manifold, which itself is nothing more than a special kind of topological space it makes sense to shortly revise the most important definitions and ideas of Topology.
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