Riemann surfaces and integrable systems

This is a first semester course for the M2 of the Master Program in Mathematical Physics at the IMB. The lectures will be in English.

Description

The course will be an introduction to Riemann surfaces and to some of their relations with integrable systems. We will start by reviewing some known fact complex analysis and considering some motivation for introducing Riemann surfaces. Then we will develop some of the classical theory of Riemann surfaces. If time permits we will consider some application to the theory of integrable systems, for example the construction of solutions to the KP hierarchy via the Baker-Akhiezer functions.

Topics will include: definition of Riemann surface and basic examples, plane algebraic curves, hyperelliptic curves, holomorphic coverings, fundamental group, Riemann-Hurwitz theorem, homology groups, abelian differentials, meromorphic functions, divisors, Abel theorem, Riemann-Roch theorem.

Course material

The course material will be posted on the Teams group of the course.

Main references:

Lectures

• Multivalued functions on the complex plane and surfaces.
• Definition of Riemann surface: complex chart, atlas, complex structure.
• The Riemann sphere, $S^2$, and the projective line.
• Holomorphic functions on a RS.
• Holomorphic maps between Riemann surfaces.

• Complex tori.
• Identity theorem for holomorphic maps between Riemann surfaces.
• Isolated singularities of holomorphic functions.
• Riemann's removable singularity theorem.
• The field of meromorphic functions.
• Order of a meromorphic function at a point.
• Meromorphic functions as maps to $\mathbb{C}_\infty$.
• Local form of holomorphic maps between Riemann surfaces (P).

• Corollaries: open mapping theorem, inverse function theorem, good behaviour almost everywhere, maps over compact surfaces, holomorphic functions on compact surfaces are constant, Liouville theorem, fundamental theorem of algebra.
• Degree theorem (P).
• Sum of orders of a meromorphic function over a compact RS.
• Meromorphic functions on $\mathbb{C}_\infty$.
• Automorphisms of $\mathbb{C}_\infty$.
• Elliptic functions and Jacobi theta function.

• Meromorphic functions on the torus are ratios of theta functions.
• Smooth plane algebraic curves are non-compact Riemann surfaces.
• Holomorphic and meromorphic functions on smooth affine plane curves.
• Characterisation of ramification points on a smooth affine plane curve.
• Orientability of Riemann surfaces and classification of 2-manifolds.

• Triangulations of surfaces and Euler number.
• Riemann-Hurwitz formula. (P)
• Some consequences of Hurwitz formula.
• Sheaves. Basic definitons and examples.

• Meromorphic and smooth differentials.
• Residue of a meromorphic one form at a point.
• Differential of a function and of a 1-form, wedge product.

• Pull-back of functions and forms. Order of the pull-back.
• Stalk of a sheaf.
• Divisors. Principal divisors. Degree.

• Canonical divisors. Examples. Degree of a canonical divisor. Proof.
• Pull back of divisors. Riemann-Hurwitz formula for canonical divisors.
• Cech cohomology: cochains, coboundary operator, cocycles and coboundaries.

• Cech cohomology: interpretation of zeroth and first cohomology groups.
• Computation of the first cohomology group of the pointed plane.
• Mittag-Leffler problem.
• Glueing RS. Hyperelliptic curves.

• Exercises.

• Hyperelliptic involution.
• Functions on hyperelliptic curves.
• Riemann-Roch theorem.
• Sheaf of meromorphic functions with poles bounded by a divisor.

• Index of specialty.
• Any genus zero Riemann surface is isomorphic to the Riemann sphere.
• Sheaf of meromorphic 1-forms with poles bounded by a divisor. Sheaf isomorphism.
• Serre duality.
• Equivalence of topological, arithmetic, analytic genera.
• Partial order on coverings. Refining maps and induced maps between differential complexes.
• Homotopy map.
• Sheaf cohomology as direct limit over the directed set of open coverings.

• Injectivity of $H^1(\mathcal{U},\mathcal{F}) \to H^1(X,\mathcal{F})$.
• Integration of 1-forms over paths.
• Integration of 2-forms on 2-chains.
• Stoke's theorem. Residue theorem.
• Weierstrass $\mathcal{P}$ function.

• Homomorphisms of sheaves and examples.
• Short exact sequences of sheaves.
• Partitions of unity and computation of $H^1(X, \mathcal{E})$.
• Cohomology of locally constant sheaves over a simply connected topological space.
• Leray theorem.

• The skyscraper sheaf and the short exact sequence of sheaves.
• The long exact sequence in cohomology.
• Cohomology of the skyscraper sheaf.
• Homotopy. Fundamental group.

• Again on fundamental group and 4g-gon.
• Existence of primitives of a closed one form on a simply connected RS.
• First homology group and period mapping.
• Intersection number.
• Canonical basis of cycles.

• Proof of vanishing cohomologies for locally constant sheaves over a simply connected topological space.
• Proof of RR theorem.
• Exercises.

• Jacobian.
• Abel-Jacobi map.
• Abel's theorem.