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Lectures on Compact Riemann Surfaces IPhT-T18/007 Lectures on compact Riemann surfaces. B. Eynard12 Paris{Saclay's IPHT doctoral school Lecture given in winter 2018. This is an introduction to the geometry of compact Riemann surfaces. We largely follow the books [8,9, 10]. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann{Roch theorem that computes the dimension of spaces of functions and forms with given orders of poles and zeros. 5) The moduli space of Riemann surfaces, with its combinatorial representation as Strebel graphs, and also with the uniformization theorem that maps Riemann surfaces to hyperbolic surfaces. 6) An application of Riemann surfaces to integrable systems, more precisely finding sections of an eigenvector bundle over a Riemann surface, which is known as the "algebraic reconstruction" method in integrable systems, and we mention how it is related to Baker-Akhiezer functions and Tau functions. 1Institut de Physique Th´eoriquede Saclay, F-91191 Gif-sur-Yvette Cedex, France. 2CRM, Centre de recherches math´ematiquesde Montr´eal,Universit´ede Montr´eal,QC, Canada. 1 2 Notations • D(x; r) is the open disc of center x and radius r in C, or the ball of center x and radius r in Rn. •C (x; r) = @D(x; r) is the circle (resp. the sphere) of center x and radius r in C (resp. in Rn). •C x is a "small" circle around x in C, or a small circle in a chart around x on a surface, small meaning that it is a circle of radius sufficiently small to avoid all other special points. • Tτ = C=(Z+τZ) is the 2-torus of modulus τ, obtained by identifying z ≡ z +1 ≡ z + τ. • CP 1 = C = C [ f1g is the Riemann sphere. • C+ is the upper complex half{plane = fz j =z > 0g, it is identified with the 1 Hyperbolic plane, with the metric =z jdzj, of constant curvature −1, and whose geodesics are the circles or lines orthogonal to the real axis. • M1(Σ) the C vector space of meromorphic forms on Σ, •O 1(Σ) the C vector space of holomorphic forms on Σ. • H1(Σ; Z) (resp. H1(Σ; C)) the Z{module (resp. C{vector space) generated by homology cycles (equivalence classes of closed Jordan arcs, γ1 ≡ γ2 if there exists a 2-cell A whose boundary is @A = γ1 − γ2, with addition of Jordan arcs by concatenation) on Σ. • π1(Σ) is the fundamental group of a surface (the set of homotopy classes of closed curves on a Riemann surface with addition by concatenation). 3 Index L2, 31 conserved quantities, 105 , 36 convex envelope, 39 CP 1, 15 coordinate, 11 1st kind cycle, 65 deck transformation, 20 1st kind form, 27, 43 Deligne Mumford, 79 2nd kind cycle, 66 desingularization, 20 2nd kind form, 27, 43 dimension, 12 3rd kind cycle, 65 Dirichlet principle, 33 3rd kind form, 27, 43 divisor, 26 Abel map, 53 Euclidian, 11 abelianization, 109 exact form, 27, 28 action variable, 111 amoeba, 40 Fay, 113 angle variable, 111 Fenchel-Nielsen coordinates, 100, 102 atlas, 11 Fuchsian group, 98 fundamental domain, 49 Baker-Akhiezer functions, 106 fundamental form, 36, 62 Bergman kernel, 62 fundamental second kind differential, 62 Betti space, 99 branch point, 19 generalized cycle, 67 Green function, 35 cameral curve, 107 Casimir, 111 half{integer characteristic, 60 chart, 11 harmonic, 31 closed, 31 Haussdorf space, 11 co-closed, 31 Hirota, 114 compact, 12 Hitchin base, 107 complex projective plane, 15 Hodge decomposition theorem, 32 complex structure, 12 Hodge star, 31 4 holomorphic, 23 Poincar´eresidue, 39 holomorphic 1-form, 23 Poisson equation, 35 holomorphic function, 23 polarization, 112 horizontal trajectory, 89 positive divisor, 69 prime form, 61 interior of Newton's polygon, 41 principal bundle, 107 intersection numbers, 47 projective connexion, 96 Jacobian, 53 projective curve, 22 puncture, 22 Klein kernel, 64 quadratic differential, 24, 87 Laplacian, 32 Lax equation, 105 ramification point, 19 Liouville form, 107 reconstruction method, 109 Liouville's equation, 95 residue, 25, 26, 28 local coordinate, 11 Riemann bilinear identity, 49 Riemann bilinear inequality, 51 M¨obiustransformation, 75 Riemann matrix of periods, 52 manifold, 11, 12 Riemann sphere, 15 mapping class group, 100 Riemann surface, 12 marking, 47, 100 Riemann's constant, 54 meromorphic, 23 Riemann-Hurwitz, 30 meromorphic 1-form, 23 Riemann-Roch, 70 meromorphic function, 23 meromorphic kth order form, 24 Sato, 112 modular transformation, 64 Schiffer kernel, 36, 64 moduli space, 75 Schwartzian derivative, 96 second countable, 11 Newton's polytope, 38 self intersection, 21 nodal point, 21 separated space, 11 nodal surface, 77 Siegel, 52, 54 normalized basis of forms, 51 spinor, 24 stable, 79 oper, 96 stack, 79, 81 order, 25 stereographic projection, 15 orientable, 12 Strebel differential, 92 Pl¨ucker, 113 Strebel graph, 92 Poincar´emetric, 95 stress energy tensor, 95 Poincar´epairing, 26 symplectic basis, 47 5 symplectic group, 64 Szeg¨okernel, 109 tautological form, 107 Teichm¨ullerspace, 100 Theta divisor, 55 Theta function, 54 Torelli marking, 47 torus, 16 transition function, 11 Weierstrass function, 25 Weil-Petersson form, 102 Weil-Petersson volume, 102 Weyl group, 108 6 Contents Index 4 Contents7 1 Riemann surfaces 11 1 Manifolds, atlases, charts, surfaces..................... 11 1.1 Classification of surfaces...................... 13 2 Examples of Riemann surfaces....................... 14 2.1 The Riemann sphere........................ 14 2.2 The torus.............................. 16 3 Compact Riemann surface from an algebraic equation.......... 17 3.1 Example............................... 17 3.2 General case............................. 19 3.3 Non{generic case: desingularization................ 21 3.4 Projective algebraic curves..................... 22 2 Functions and forms on Riemann surfaces 25 1 Definitions.................................. 25 1.1 Examples.............................. 26 1.2 Classification of 1-forms...................... 29 2 Some theorems about forms and functions................ 29 3 Existence of meromorphic forms...................... 33 3.1 Uniqueness of Riemann surfaces of genus 0............ 40 4 Newton's polygon.............................. 40 4.1 Meromorphic functions and forms................. 41 4.2 Poles and slopes of the convex envelope.............. 42 4.3 Holomorphic forms......................... 44 4.4 Fundamental second kind form.................. 46 7 3 Abel map, Jacobian and Theta function 51 1 Holomorphic forms............................. 51 1.1 Symplectic basis of cycles..................... 51 1.2 Small genus............................. 52 1.3 Higher genus ≥ 1.......................... 53 1.4 Riemann bilinear identity..................... 53 2 Normalized basis.............................. 55 3 Abel map and Theta functions....................... 57 3.1 Divisors, classes, Picard group................... 64 4 Prime form................................. 64 5 Fundamental form............................. 66 5.1 Modular transformations...................... 68 5.2 Meromorphic forms and generalized cycles............ 69 4 Riemann-Roch 73 1 Spaces and dimensions........................... 73 2 Special cases................................. 74 3 Genus 0................................... 75 4 Genus 1................................... 75 5 Higher genus ≥ 2.............................. 76 5.1 General remarks.......................... 76 5.2 Proof of the Riemann{Roch theorem............... 77 5 Moduli spaces 79 1 Genus 0................................... 79 1.1 M0;3 ................................. 80 1.2 M0;4 ................................. 80 1.3 n > 5................................. 85 2 Genus 1................................... 86 2.1 Boundary of M1;1 .......................... 88 3 Higher genus................................ 89 4 Coordinates in the moduli space...................... 90 4.1 Strebel graphs............................ 90 4.2 Topology of the moduli space................... 98 5 Uniformization theorem.......................... 98 6 Teichm¨ullerspace.............................. 104 6.1 Fenchel{Nielsen coordinates.................... 104 8 6 Eigenvector bundles and solutions of Lax equations 109 1 Eigenvalues and eigenvectors........................ 110 1.1 The spectral curve......................... 110 1.2 Eigenvectors and principal bundle................. 111 1.3 Monodromies............................ 111 1.4 Algebraic eigenvectors....................... 112 1.5 Geometric reconstruction method................. 113 1.6 Genus 0 case............................ 115 1.7 Tau function, Sato and Hirota relation.............. 116 Bibliography 121 9 10 Chapter 1 Riemann surfaces 1 Manifolds, atlases, charts, surfaces Definition 1.1 (Topological Manifold) A manifold M is a second countable (the topology can be generated by a countable basis of open sets) topological separated space (distinct points have disjoint neighborhoods, also called Haussdorf space), locally Euclidian (each point has a neighborhood homeomorphic to an open subset of Rn for some integer n). Definition 1.2 (Charts and atlas) A chart on M is a pair (V; φV ), of a neighbor- n hood V , together with an homeomorphism φV : V ! U ⊂ R , called the coordinate or the local coordinate. For each intersecting pair V \ V 0 6= ;, the transition 0 0 −1 function is the map: U!U 0 : φV (V \ V ) ! φV 0 (V \ V ), x 7! φV 0 ◦ φV (x), it is a homeomorphism of Euclidian subspaces, with inverse −1 U!U 0 = U 0!U : (1-1) A countable set of charts
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