Riemann–Roch theorem

Summary

The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student . It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. Preliminary notions A Riemann surface X is a topological space that is locally homeomorphic to an open subset of \Complex, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the not

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https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem

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To any compact Riemann surface of genus g one may assign a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus and we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. The aim of this thesis is to contribute to the Schottky problem, which is to discern the Jacobians among the PPAVs. Buser and Sarnak approached this problem by means of a geometric invariant, the first successive minimum. They showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector, in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. The main goal of this work was to improve this result and to get insight into the connection between the geometry of a compact Riemann surface that is given in hyperbolic geometric terms, and the geometry of its Jacobian. We show the following general findings: For a hyperelliptic surface the first successive minimum is bounded from above by a universal constant. The square of the second successive minimum of the Jacobian of a Riemann surface of genus g is equally of order log(g). We provide refined upper bounds on the consecutive successive minima if the surface contains several disjoint small simple closed geodesics and a lower bound for the norm of certain lattice vectors of the Jacobian, if the surface contains small non-separating simple closed geodesics. If the concrete geometry of the Riemann surface is known, more precise statements can be made. In this case we obtain theoretical and practical estimates on all entries of the period Gram matrix. Here we establish upper and lower bounds based on the geometry of the cut locus of simple closed geodesics and also on the geometry of Q-pieces. In addition the following two results have been obtained: First, an improved lower bound for the maximum value of the norm of the shortest non-zero lattice vector among all PPAVs in even dimensions. This follows from an averaging method from the geometry of numbers applied to a family of symmetric PPAVs. Second, a new proof for a lower bound on the number of homotopically distinct geodesic loops, whose length is smaller than a fixed constant. This lower bound applies not only to geodesic loops on Riemann surfaces, but on arbitrary manifolds of non-positive curvature.

EPFL2011

Finite Element Method on Riemann Surfaces and Applications to the Laplacian Spectrum

Marc Maintrot

The objective of this PhD thesis is the approximate computation of the solutions of the Spectral Problem associated with the Laplace operator on a compact Riemann surface without boundaries. A Riemann surface can be seen as a gluing of portions of the Hyperbolic Plane made with suitable conditions to obtain a 2 dimensional manifold. The solutions of the Spectral Problem associated with the Laplace operator are to be understood as the eigenfunctions defined on the surface and their corresponding eigenvalues. This work is separated into two parts: the first part describes the method used to approximate the eigenvalues and eigenfunctions, the second focuses on the design of a program to compute these approximations. The approximation method is inspired by the Finite Element Method (FEM), in that it relies on the variational expression of the Spectral Problem and the definition of a finite subspace of functions in which the approximated eigenvalues and eigenfunctions are computed. However, it differs from the FEM in that it removes the euclidian basis of the FEM and is invariant under the isometries of the Hyperbolic Plane. To ful fill this objective, we begin by geodesically triangulating the surface as regularly as possible. This hyperbolic triangulation allows us to de ne the finite subspace of functions by using the concept of barycentric coordinates associated with each vertex of the triangulation (idea introduced by Whitney and taken up by Dodziuk). We then prove that the approximated solutions convergence to the exact ones when the diameter of the triangulation decreases, as well as the order of convergence. The program is a practical application of the theoretical work and allows the computation of the approximated eigenfunctions and eigenvalues.

EPFL2012

Computational Approach to the Geometry of Compact Riemann Surfaces

Manuel Racle

The goal of this document is to provide a generalmethod for the computational approach to the topology and geometry of compact Riemann surfaces. The approach is inspired by the paradigms of object oriented programming. Our methods allow us in particular to model, for numerical and computational purposes, a compact Riemann surface given by Fenchel-Nielsen parameters with respect to an arbitrary underlying graph, this in a uniformand robust manner. With this programming model established we proceed by proposing an algorithmthat produces explicit compact fundamental domains of compact Riemann surfaces as well as generators of the corresponding Fuchsian groups. In particular, we shall explain how onemay obtain convex geodesic canonical fundamental polygons. In a second part we explain in what manner simple closed geodesics are represented in our model. This will lead us to an algorithm that enumerates all these geodesics up to a given prescribed length. Finally, we shall briefly overview a number of possible applications of our method, such as finding the systoles of a Riemann surface, or drawing its Birman-Series set in a fundamental domain.

EPFL2013

EPFL2011