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Compact Riemann Surfaces

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Compact Riemann Surfaces Differentialgeometrie III Compact Riemann Surfaces Prof. Dr. Alexander Bobenko CONTENTS 1 Contents 1 Definition of a Riemann Surface and Basic Examples3 1.1 Non-singular Algebraic Curves........................4 1.2 Quotients under Group Actions........................7 1.3 Euclidean Polyhedral Surfaces as Riemann Surfaces.............9 1.4 Complex Structure Generated by Metric................... 10 2 Holomorphic Mappings 15 2.1 Algebraic curves as coverings......................... 18 2.2 Quotients of Riemann Surfaces as Coverings................. 20 3 Topology of Riemann Surfaces 22 3.1 Spheres with Handles............................. 22 3.2 Fundamental group............................... 25 3.3 First Homology Group of Riemann surfaces................. 27 4 Abelian differentials 32 4.1 Differential forms and integration formulas.................. 32 4.2 Abelian differentials of the first, second and third kind........... 36 4.3 Periods of Abelian differentials. Jacobi variety............... 42 4.4 Harmonic differentials and proof of existence theorems........... 44 5 Meromorphic functions on compact Riemann surfaces 51 5.1 Divisors and the Abel theorem........................ 51 5.2 The Riemann-Roch theorem.......................... 54 5.3 Special divisors and Weierstrass points.................... 59 5.4 Jacobi inversion problem............................ 62 6 Hyperelliptic Riemann surfaces 64 6.1 Classification of hyperelliptic Riemann surfaces............... 64 6.2 Riemann surfaces of genus one and two................... 67 7 Theta functions 71 7.1 Definition and simplest properties...................... 71 7.2 Theta functions of Riemann surfaces..................... 72 7.3 Theta divisor.................................. 75 CONTENTS 2 8 Holomorphic line bundles 80 8.1 Holomorphic line bundles and divisors.................... 80 8.2 Picard group. Holomorphic spin bundle.................... 83 1 DEFINITION OF A RIEMANN SURFACE AND BASIC EXAMPLES 3 1 Definition of a Riemann Surface and Basic Examples Let R be a two-real dimensional manifold and fU g 2A an open cover of R, i. e. [ 2AU = R.A local parameter (local coordinate, coordinate chart) is a pair (U ; z ) of U with a homeomorphism z : U ! V to an open subset V ⊂ C. Two coordinate charts (U ; z ) and (U ; z ) are called compatible if the mapping −1 f , = z ∘ z : z (U \ U ) ! z (U \ U ); which is called a transition function is holomorphic. The local parameter (U ; z ) will be often identified with the mapping za if its domain is clear or irrelevant. If all the local parameters fU ; z g 2A are compatible, they form a complex atlas A of R. Two complex atlases A = fU ; z g and A~ = fU~ ; z~ g are compatible if A[ A~ is a complex atlas. An equivalence class Σ of complex atlases is called a complex structure. It can be identified with a maximal atlas A∗, which consists of all coordinate charts, compatible with an atlas A ⊂ Σ. Definition 1.1 A Riemann surface is a connected one-complex-dimensional analytic manifold, that is, a two-real dimensional connected manifold R with a complex structure Σ on it. When it is clear, which complex structure is considered we use the notation R for the Riemann surface. Remark If fU; zg is a coordinate on R then for every open set V ⊂ U and every function f : C ! C, which is holomorphic and injective on z(V ), fV; f ∘ zg is also a local parameter on R. Remark The coordinate charts establish homeomorphisms of domains in R with do- mains in C. This means, that locally the Riemann surface is just a domain in C. But for any point P 2 R there are many possible choices of these homeomorphisms. There- fore one can associate to R only the notions from the theory of analytic functions in C, which are invariant with respect to biholomorphic maps, i. e. for definition of which one should not specify a local parameter. For example one can talk about an angle between two smooth curves and ~ on R, intersecting at some point P 2 R. This angle equals to the one between the curves z( ) and z(~ ), which lie in C and intersect at the point z(P ), where z is some local parameter at P . This definition is invariant with respect to the choice of z. Remark If (R; Σ) is a Riemann surface, then the manifold R is orientable. The transition function f , written in terms of real coordinates (z = x + iy) (x ; y ) ! (x ; y ) preserves orientation 2 2 i i dz dz dx ^ dy = dz ^ dz¯ = dz ^ dz¯ = dx ^ dy : 2 2 dz dz 1 DEFINITION OF A RIEMANN SURFACE AND BASIC EXAMPLES 4 The simplest examples of Riemann surfaces are any domain (connected open subset) U ⊂ C in a complex plane, the complex plane C itself and the extended complex plane 1 (or Riemann sphere) C^ = CP = C [ f1g. The complex structures on U and C are defined by single coordinate charts (U; id) and (C; id). The extended complex plane is the simplest compact Riemann surface. To define the complex structure on it we use two charts (U1; z2); (U2; z2) with U1 = C; z1 = z; U2 = (Cnf0g) [ f1g; z2 = 1=z: The transition functions −1 −1 f1;2 = z1 ∘ z2 ; f2;1 = z2 ∘ z1 : Cnf0g ! Cnf0g are holomorphic f1;2(z) = f2;1(z) = 1=z: In large extend the beauty of the theory of Riemann surfaces is due to the fact that Riemann surfaces can be described in many completely different ways. Interrelations between these descriptions comprise an essential part of the theory. The basic examples of Riemann surfaces we are going to discuss now are exactly these foundation stones the whole theory is based on. 1.1 Non-singular Algebraic Curves 2 Definition 1.2 An algebraic curve C is a subset in C 2 C = f(, ) 2 C j P(, ) = 0g; (1) where P is an irreducible polynominal in and N M X X i j P(, ) = pij : i=0 j=0 The curve C is called non-singular if @P @P grad P = ; 6= 0: (2) C jP=0 @ @ jP(,)=0 To introduce a complex structure on the non-singular curve (1,2) one uses a complex version of the implicit function theorem. Theorem 1.1 Let P(, ) be an analytic function of and in a neighbourhood of a 2 point (0; 0) 2 C with P(0; 0) = 0, and, in addition @P ( ; ) 6= 0: @ 0 0 1 DEFINITION OF A RIEMANN SURFACE AND BASIC EXAMPLES 5 Then in a neighbourhood of (0; 0) the set 2 f(, ) 2 C j P(, ) = 0g is described as f((); ) j 2 Ug; where U ⊂ C is a neighbourhood of 0 2 U and () is an analytic function. The derivative of the function () is equal d @P=@ = − : d @P=@ The complex structure on C is introduced as follows: the variable is taken to be a local parameter in the neighbourhoods of the points where @P=@ 6= 0, and the variable is a local parameter near the points where @P=@ 6= 0. The holomorphic compatibility of the introduced local parameters results from Theorem 1.1. The surface C can be made a compact Riemann surface C^ by joining point(s) 1(1);:::; 1(N) C^ = C [ f1(1)g [ ::: [ f1N g at infinity ! 1; ! 1, and introducing proper local parameters at this(ese) point(s). In oder to explain this compactification let us define Riemann surfaces with punctures. Definition 1.3 Let R be a Riemann surface such that there exists an open subset U1 (1) (N) U1 [ ::: [ U1 = U1 ⊂ R (n) such that RnU1 is compact and U1 are homeomorphic to punctured discs (n) zn : U1 ! Dnf0g = fz 2 C j 0 < jzj < 1g; where homomorphisms zn are holomorphically compatible with the complex structure of R. Then R is called a compact Riemann surface with punctures. z1 1(1) 1(2) z2 Figure 1: A compact Riemann surface with punctures. Let us extend the homeomorphisms zn to D ^ (n) (n) (n) zn : U1 = U1 [ 1 ! D = fz j jzj < 1g; (3) 1 DEFINITION OF A RIEMANN SURFACE AND BASIC EXAMPLES 6 (n) (n) defining punctures 1 by the condition zn(1 ) = 0; n = 1;:::;N. A complex atlas for a new Riemann surface R^ = R [ f1(1)g [ ::: [ f1(n)g is defined as a union of a complex atlas A of R with the coordinate charts (3) compatible with A due to Definition 1.3. We call R^ a compactification of R. Hyperelleptic Curves. Let us consider the important special case of hyperelleptic curves 1 N 2 Y = ( − j);N ≥ 3; j 2 C: (4) j=1 The curve is non-singular if all the points j are different j 6= i; i; j = 1; : : : ; N: In this case the choice of local parameters can be additionally specified. Namely, in the neighbourhood of the points (0; 0) with 0 6= j 8j, the local parameter is the homeomorphism (, ) ! . (5) In the neighbourhood of each point (0; j) it is defined by the homeomorphism p (, ) ! − j: (6) Indeed, near (0; i) 0v 1 u N p uY = − i @t (i − j) + o(1)A ; ! i; j=1 p and the local parameter − j is equivalent to . The hyperelleptic curve (4) is a compact Riemann surface with a puncture (or punctures) at ! 1. To show this one should consider the cases of even N = 2g + 2 and odd N = 2g + 1 separately. The formulas 1 m = ; l = g+1 describe a biholomorphic map (, ) 7! (m; l) of a neighbourhood of infinity U1 = f(, ) 2 C j jj > c > jij; i = 1;:::;Ng onto the punctured neighbourhood 0 −1 V0 = f(m; l) 2 C j 0 < jlj < c g 1When N = 3 or 4 the curve (4) is called elliptic 1 DEFINITION OF A RIEMANN SURFACE AND BASIC EXAMPLES 7 of the point (m; l) = (0; 0) of the curve 2g+1 2 Y m = l (1 − li) (7) i=1 for N = 2g + 1, or onto punctured neighbourhoods of the points (m; l) = (±1; 0) of the curve 2g+2 2 Y m = (1 − li) (8) i=1 for N = 2g +p 2.
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