ementas

Riemann Surfaces

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics.

Bibliography

We will follow

• Simon Donaldson: Riemann Surfaces. 256 pages. Oxford graduate texts in mathematics, 22. (2011)

See https://www.maa.org/press/maa-reviews/riemann-surfaces

Preliminaries: complex variable. See e.g. the following textbook for a postgraduate level introduction in modern language:

• Carlos A. Berenstein, Roger Gay: Complex variables: an introduction. 650 pages. Graduate Texts in Mathematics, 125. (1991) Springer. ISBN 3-540-97349-4

Also see the bibliography of MAT2815 and MAT2816 below.

About the book

This graduate text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind.

Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved. Following this section, the remainder of the text illustrates various facets of the more advanced theory.

Pre-requisites and possible development of the course.

The mathematical level varies quite a lot as the book progresses… The treatment in Parts I, II, III is aimed at beginning graduate students with a reasonable background in general topology and real and complex analysis. The proofs are intended to be, for the most part, written in full detail.

There is probably more material in the book that can be covered in a single lecture course. With a well-prepared audience, in about 30 lectures, it seems possible to cover most of Parts I, II and III in detail and then outline a selection from Part IV. On the other hand, if one has to develop foundational material such as Chapter 5 in detail, then it is probably realistic to get to Chapter 9.

For DMAT PUC-Rio the book corresponds to two semester-long post-graduate courses, 3 credits each: Riemann Surfaces I (MAT2815) and Riemann Surfaces II (MAT2816), with the point of demarcation somewhere around Chapter 9-10-11. but with extra spirit and material of geometry/topology in its treatment.

Contents

• I Preliminaries
  1. Holomorphic Functions: Simple examples: algebraic functions. Analytic continuation: differential equations.
  2. Surface Topology: Classification of surfaces. The mapping class group.
• II Basic Theory
  1. Basic Definitions: Riemann surfaces and holomorphic maps. Examples: first examples, algebraic curves, quotients.
  2. Maps between Riemann Surfaces: General properties. Monodromy and Riemann Existence Theorem. Digression into algebraic topology. Monodromy of covering maps. Compactifying algebraic curves. The Riemann surface of a holomorphic function. Quotients.
  3. Calculus on Surfaces: i. Smooth surfaces: cotangent spaces and 1-forms; 2-forms and integration. ii. de Rham cohomology: definition and examples; cohomology with compact support, and Poincaré duality. iii. Calculus on Riemann surfaces: decomposition of the 1-forms; the Laplace operator and harmonic functions; the Dirichlet norm.
  4. Elliptic functions and integrals: Elliptic integrals. The Weierstrass $\mathcal{\rho}$ function. Further topics: theta functions, classification.
  5. Applications of the Euler characteristic: The Euler characteristic and meromorphic forms. Applications: the Riemann–Hurwitz formula, the degree–genus formula, real structures and Harnack’s bound, modular curves.
• III Deeper Theory
  1. Meromorphic Functions and the Main Theorem for Compact Riemann Surfaces Consequences of the Main Theorem. The Riemann–Roch formula.
  2. Proof of the Main Theorem: Discussion and motivation. The Riesz Representation Theorem. The heart of the proof. Weyl’s Lemma.
  3. The Uniformisation Theorem
• IV Further Developments
  1. Contrasts in Riemann Surface Theory:
  • Algebraic aspects: fields of meromorphic functions, valuations, connections with algebraic number theory.
  • Hyperbolic surfaces: definitions, models of the hyperbolic plane, self-isometries, hyperbolic surfaces, geodesics, the Gauss–Bonnet Theorem, right-angled hexagons, closed geodesics.
    1. Divisors, Line Bundles and Jacobians:
  • Cohomology and line bundles: sheaves and cohomology, line bundles, line bundles and projective embeddings, divisors and unique factorization.
  • Jacobians of Riemann surfaces: the Abel–Jacobi Theorem, abstract theory, geometry of symmetric products, remarks in the direction of algebraic topology and digression into projective geometry.
    1. Moduli and Deformations: Almost complex structures, Beltrami differentials and the integrability theorem. Deformation and cohomology.
    2. Mappings and Moduli:
  • Diffeomorphisms of the plane.
  • Braids, Dehn twists and quadratic singularities: classification of branched covers, monodromy and Dehn twists, plane curves.
  • Hyperbolic geometry.
  • Compactification of the moduli space: collars and cusps.
    1. Ordinary Differential Equations:
  • Conformal mapping.
  • Periods of holomorphic forms and ordinary differential equations: the hypergeometric equation, the Gauss–Manin connection, singular points.

Related courses from http://www.mat.puc-rio.br/en/syllabi#graduate-program

MAT2815 - Riemann Surfaces I

Syllabus

Definition of Riemann surface. Holomorphic maps and their properties. Isothermal parameters. Construction of Riemann surfaces. Riemann surface of an algebraic equation. Conformal structures. Branched coverings. Hurwitz formula. Riemann relation. Analytic continuation. Uniformization theorem, proof and examples: the unit disc as the universal covering of the sphere minus three points. Riemann surfaces as quotient of its universal covering surface, Koebe–Poincaré theorem. Conformal structures on the tori. Weierstrass P function and other elliptic functions. Conformal structures on the annuli. Great Picard theorem.

Basic Bibliography:

1. Ahlfors, L. Conformal Invariants. McGraw-Hill 1973.
2. Sá Earp, R.; Toubiana, E. Introduction à la géométrie hyperbolique et aux surfaces de Riemann. Cassini, 2009.
3. da Costa, C.J. Funções elípticas, algébricas e superfícies mínimas. 18.o Colóquio Brasileiro de Matemática, IMPA, 1991.

MAT2816 - Riemann Surfaces II

Syllabus

Sheaves. Algebraic Functions. Fundamental group and singular (co)homology of compact Riemann surfaces. Monodromy. Algebraic Curves. Divisors, line bundles, canonical line bundle. Linear systems, maps to the projective space. Sheaves cohomology, finiteness theorems. Dolbeault’s theorem. Serre duality. Riemann–Roch theorem. Harmonic forms. Vanishing of the cohomology, ample line bundles, immersion into the projective space. Hyperellitpic curves. Picard group. Jacobian. Abel’s theorem. Jacobi’s theorem. Applications to algebraic curves and their jacobians.

Basic Bibliography:

1. Foster, O. Lectures on Riemann Surfaces, Springer–Verlag, 1981
2. Narasimhan, R. Compact Riemann Surfaces, Birkhäuser, 1996.
3. Miranda, R. Algebraic Curves and Riemann Surfaces, American Mathematical Society, 1995.

Additional Bibliography:

1. Gunning, R.C. Lectures on Riemann Surfaces, Princeton University Press, 1966.