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A Riemann Surfaces and Theta Functions A Riemann Surfaces and Theta Functions In this appendix we collect some basic facts on theta functions on Riemann surfaces. The idea is to give a comprehensive presentation of the mathemat- ics on theta functions and the notation used in this book without providing lengthy proofs. For more detailed accounts of the subject, the reader is re- ferred to [128], [138], [139], [187], and [256] to [270], for topics related to the Ernst equation see also [271]. A.1 Riemann Surfaces and Algebraic Curves A Riemann surface Σ is a connected complex one-dimensional manifold. Riemann surfaces can be associated to multi-valued functions. Let f be an analytic function on C2 with arguments w and z. The equation f(w, z) = 0 (A.1) defines a one-dimensional complex submanifold of C2. Let the complex gra- dient gradCf be given by ∂f ∂f grad f := , (A.2) C ∂w ∂z and call (w0,z0) with f(w0,z0) = 0 regular iff gradCf| =0 . (A.3) (w0,z0) It can be shown that the corresponding Riemann surface admits a com- pactification if equation (A.1) describes a plane algebraic curve, i.e. if f is of the form k i f(w, z)= w ai(z) , (A.4) i=0 with ai (i =0,...,k) being polynomials in z. Then w = w(z) is a multiple- valued algebraic function. It can be shown that any compact Riemann surface can be represented as an algebraic curve. A complex structure is introduced on the algebraic curve in the following way: in the neighborhoods of the C. Klein, O. Richter: Ernst Equation and Riemann Surfaces, Lect. Notes Phys. 685, 191–208 (2005) www.springerlink.com c Springer-Verlag Berlin Heidelberg 2005 192 A Riemann Surfaces and Theta Functions points where ∂f/∂w = 0, the variable z is taken as a local parameter, in the neighborhoods of the points ∂f/∂z = 0, the parameter w is taken as a local parameter. In this volume we shall be mainly concerned with functions of the form 2 f(w, z)=w − Pn(z) , (A.5) with Pn being a polynomial of degree n in z. The corresponding multiple- valued function is denoted by w = Pn(z) and the Riemann surface L is called hyperelliptic (for n =3, 4 the surface is called elliptic). The hyperellip- tic surface is everywhere regular if and only if Pn(z) has no multiple roots. ∼ For w = 0 one chooses√ z as a local parameter, for z zi, where Pn(zi)=0, − the variable τzi = z zi can be used as a local parameter. For Σ being hyperelliptic the hyperelliptic involution σ is defined by σ : Σ P =(z,w) → σ(P ) ≡ P σ =(z,−w) ∈ Σ, (A.6) i. e., σ interchanges the two sheets of the Riemann surface. Any compact Riemann surface Σg of genus g is topologically equivalent to a sphere with g handles. Any compact Riemann surface has finite genus. The first homology group of Σg is denoted by H1(Σg, Z). A standard basis of generators of H1(Σg, Z) consists of g pairs of cycles (a1,b1),...,(ag,bg) where a pair (ai,bi) encircles the i-th handle (or surrounds the ith hole) so that ai intersects bi, see Fig. A.1 a1 a2 b1 b2 Fig. A.1. A homology basis for a Riemann surface of genus two The way cycles intersect is described by intersection numbers. In Fig. A.2 we show for two cycles γ1 and γ2 when the intersection number γ1 · γ2 is +1 or −1. We choose oriented closed curves a1,...,ag,b1,...,bg such that their in- tersection numbers are ai · aj = bi · bj =0, ai · bj = −bi · aj = δij . (A.7) A basis with the above intersection numbers is called canonical basis.The ˜ choice of such a basis is not unique: Any other basis (˜a, b)ofH1(Σg, Z) A.1 Riemann Surfaces and Algebraic Curves 193 γ2 γ1 γ1 γ2 γ1 · γ2 =+1 γ1 · γ2 = −1 Fig. A.2. The orientation dependence of the intersection numbers T (where a and b denote the g-dimensional vector a =(a1,...,ag) and b = T (b1,...,bg) ) is given by the transformation a˜ a = A ,A∈ SL(2g,Z) . (A.8) ˜b b From the requirement that the new basis is also canonical we find that the matrix A is symplectic, A ∈ Sp(g,Z), 0 −I J = AJAT ,J= . (A.9) I 0 A canonical basis is also referred to as a cut-system. If one cuts the Rie- mann surface starting from one point along the canonical cycles, the resulting surface is simply connected, a 4g-gon called the fundamental polygon. For the surface of Fig. A.1 one gets the fundamental polygon shown in Fig. A.3. −1 a1 b1 −1 b1 a1 − a2 1 b2 − b2 1 a2 Fig. A.3. The fundamental polygon Σ˜g of a Riemann surface Σg of g =2 194 A Riemann Surfaces and Theta Functions A.2 Differentiation and Integration on Riemann Surfaces A differential (or one form) dΩ = adx +bdy = αdz + βd¯z, where z is a local complex coordinate in the neighborhood of some point P on the Riemann surface Σg of genus g is called an Abelian differential if we have dΩ = f(z)dz, (A.10) with f being a meromorphic function in the vicinity of P . Since ∂f ∂f d(dΩ)= dz ∧ dz + d¯z ∧ dz =0, (A.11) ∂z ∂z¯ Abelian differentials are always closed. We may classify Abelian differentials as follows: – Abelian differentials of the first kind (or holomorphic differentials): f is a holomorphic function in any local chart. – Abelian differentials of the second kind: have a single pole of order higher than 1 with vanishing residue. – differentials of the third kind: have two first-order poles with residues ±1. Furthermore, it turns out that each Abelian differential can be decom- posed into Abelian differentials of the above mentioned kinds. We denote a (n) differential of the second kind with pole p of order n +1 bydΩP and a differential of the third kind with poles P (residue +1) and Q (residue −1) by dΩPQ. We have for the differential dΩPQ 1 dΩPQ = + O(1) dτP , near P, τP 1 dΩPQ = − + O(1) dτQ , near Q, (A.12) τQ where τP and τQ are local parameters at P and Q with τP (P )=0and τQ(Q) = 0. For the differential of the second kind we have (n) 1 dΩP = n+1 + O(1) dτP , near P, (A.13) τP where τP is a local parameter at P with τP (P ) = 0. A differential of the second kind can be obtained from a differential of the third kind by differen- tiating 1 dΩ(n) = ∂n dΩ . (A.14) P n! P PQ An Abelian integral on a Riemann surface is an integral of an Abelian differ- ential. A.2 Differentiation and Integration on Riemann Surfaces 195 1 It can be shown that the vector space H (Σg) of holomorphic differentials on Σg is g-dimensional. For example if Σg is a hyperelliptic Riemann surface 2g+2 2 1 of the form µ = (λ − λi), we may define a basis in H (Σg)by i=1 λk−1dλ dν = , (A.15) k µ for i =1,...,g. The hyperelliptic differentials of the third kind have the form µ + µ µ + µ dλ dω (R)= P − Q , if µ =0 ,µ =0 , PQ λ − τ λ − τ 2µ P Q P Q µ + µ 1 dλ dω (R)= P − , if µ =0 ,µ =0, PQ λ − τ λ − τ 2µ P Q P Q 1 1 dλ dωPQ(R)= − , if µP =0,µQ =0, (A.16) λ − τP λ − τQ 2µ where the argument of dωPQ is the point R =(λ, µ) ∈ Σg. Definition A.1. The periods along the cycles a1,...,bg of a closed differen- tial dΩ are defined by # . Ai = dΩ, a#i . Bi = dΩ, (A.17) bi for i =1,...,g. The periods are independent of the representatives of the cycles with the given homology classes since the differentials are closed. Let dΩ respectively dΩ be closed differentials and denote the corresponding periods by Ai and ∈ Bi respectively Ai and Bi (i =1,...,g). Let P0 Σg be fixed and define a function f on Σ˜g (the fundamental polygon) by P f(P )= dΩ, (A.18) P0 ∀P ∈ Σg. With these settings one obtains Riemann’s bilinear identities, Theorem A.2. The following relation holds: # g ∧ − dΩ dΩ = fdΩ = (AiBi AiBi) , (A.19) i=1 Σg ∂Σ˜g where ∂Σ˜g is the boundary of the 4g-gon Σ˜g, oriented in positive direction. 196 A Riemann Surfaces and Theta Functions The first identity in (A.19) follows from Stokes’ theorem, the last identity fol- lows from an evaluation of the integrand at the boundary of the fundamental domain (for details see [186]). The Riemann bilinear relations of Theorem A.2 imply useful relations for the A-andB-periods of Abelian differentials. Applying them to two holomorphic differentials dΩ and dΩ one finds g − AjBj BjAj =0. (A.20) j=1 and g AjBj ≤ 0 . (A.21) j=1 Thus one has Corollary A.3. (i) An Abelian differential of the first kind where all A- (ii) or all B-periods vanish is identically zero. (iii) An Abelian differential of the first kind with only real periods is identically zero. Remark A.4. With the above relations one may show that for an integral of the third kind the poles and integration limits can be interchanged Q R dΩRS = dΩQP , (A.22) P S P, Q, R, S ∈ Σg.
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