Book contents
- Frontmatter
- Contents
- 1 Introduction
- 2 Integrable dynamical systems
- 3 Synopsis of integrable systems
- 4 Algebraic methods
- 5 Analytical methods
- 6 The closed Toda chain
- 7 The Calogero—Moser model
- 8 Isomonodromic deformations
- 9 Grassmannian and integrable hierarchies
- 10 The KP hierarchy
- 11 The KdV hierarchy
- 12 The Toda field theories
- 13 Classical inverse scattering method
- 14 Symplectic geometry
- 15 Riemann surfaces
- 16 Lie algebras
- Index
15 - Riemann surfaces
Published online by Cambridge University Press: 19 August 2009
- Frontmatter
- Contents
- 1 Introduction
- 2 Integrable dynamical systems
- 3 Synopsis of integrable systems
- 4 Algebraic methods
- 5 Analytical methods
- 6 The closed Toda chain
- 7 The Calogero—Moser model
- 8 Isomonodromic deformations
- 9 Grassmannian and integrable hierarchies
- 10 The KP hierarchy
- 11 The KdV hierarchy
- 12 The Toda field theories
- 13 Classical inverse scattering method
- 14 Symplectic geometry
- 15 Riemann surfaces
- 16 Lie algebras
- Index
Summary
Riemann surfaces play a ubiquitous role in the analytic study of integrable systems. Here it is fundamental to see Riemann surfaces both as smooth analytical one-dimensional varieties and as the desingularization of the locus of an algebraic equation P(x, y) = 0. We explain the notion of line bundle which arises naturally in the study of integrable systems. This allows us to provide a proof of the Riemann—Roch theorem. This theorem is the main enumerative tool in our applications. Riemann himself discovered the profound implications of theta functions and notably the geometry of the theta divisor in the subject. The starting point is Riemann's theorem which we use to exhibit explicit solutions of integrable systems in terms of theta-functions. We close the chapter by sketching Birkhoff's proof of the Riemann—Hilbert factorization theorem, which plays a central role throughout the book.
Smooth algebraic curves
Riemann surfaces are compact smooth analytic varieties of dimension 1. This means that around each point p there is a neighbourhood and a local parameter z(p) mapping it homeomorphically to an open disc |z| < 1 of the complex numbers. Moreover, in the intersection of two such neighbourhoods the corresponding local parameters z1(p) and z2(p) must be related by an analytic bijection. Hence, locally a smooth curve looks like the complex line. Finally, a Riemann surface is compact, hence it is a closed surface without boundary.
- Type
- Chapter
- Information
- Introduction to Classical Integrable Systems , pp. 545 - 570Publisher: Cambridge University PressPrint publication year: 2003