scholarly journals Positivity notions for holomorphic line bundles over compact Riemann surfaces

1981 ◽  
Vol 23 (1) ◽  
pp. 5-22
Author(s):  
Joshua H. Rabinowitz
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Line Bundles ◽  
Continuous Evolution ◽  
Compact Riemann Surfaces ◽  
Dimension One ◽  
Positive Line ◽  
Holomorphic Line Bundles

Since the early 1950's, when Kodaira “discovered” positive line bundles, the notion of positivity has undergone a continuous evolution. This paper is intended as an introduction to the study of positivity notions. More specifically, I consider the simplest case - line bundles over compact Riemann surfaces - and compare five positivity notions for such bundles. The results obtained are certainly not new; they are, in fact, known in much greater generality. However, by restricting to the dimension one case, I am able to make use of Riemann surface techniques to significantly simplify the proofs. In fact, this article should be easily understood by anyone familiar with the contents of Gunning's Lectures on Riemann surfaces.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Rings of Meromorphic Functions on Non-Compact Riemann Surfaces

1969 ◽  
Vol 21 ◽  
pp. 284-300 ◽  
Author(s):  
James Kelleher
Keyword(s):  
Riemann Surface ◽  
Algebraic Structure ◽  
Riemann Surfaces ◽  
Meromorphic Functions ◽  
Doctoral Dissertation ◽  
University Of Illinois ◽  
Compact Riemann Surfaces ◽  
The University ◽  
Complex Valued

In this paper we shall be concerned with the algebraic structure of certain rings of functions meromorphic on a non-compact (connected) Riemann surface Ω. In this setting, A = A(Ω) and K= K(Ω) denote (respectively) the ring of all complex-valued functions analytic on Ω and its field of quotients, the field of functions meromorphic on Ω. The rings considered here are those subrings of K containing A,which we term A-rings of K. Most of the results given here were previously announced without proof (15) and are contained in the author's doctoral dissertation (16), completed at the University of Illinois under the direction of Professor M. Heins, whose encouragement and advice are gratefully acknowledged.

ASYMPTOTIC BEHAVIOR OF THE GINZBURG-LANDAU FUNCTIONAL ON COMPLEX LINE BUNDLES OVER COMPACT RIEMANN SURFACES

1996 ◽  
Vol 08 (03) ◽  
pp. 457-486
Author(s):  
GIANDOMENICO ORLANDI
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Chern Class ◽  
Riemann Surfaces ◽  
Line Bundles ◽  
Complex Line ◽  
Ginzburg Landau ◽  
Renormalized Energy ◽  
Compact Riemann Surfaces ◽  
Complex Line Bundles

Motivated by the works of F. Bethuel, H. Brezis, F. Hélein [5] and of F. Bethuel, T. Rivière [6], an asymptotic analysis is carried out for minimizers of the Ginzburg-Landau functional depending on a parameter ε, in the more general case of complex line bundles with prescribed Chern class over compact Riemann surfaces. Such a functional describes a 2-dimensional abelian Higgs model and is also related to phenomena in superconductivity. A suitable renormalized energy is defined which characterizes the singularities (degree one vortices) of the limiting configuration.

On Rings with a Certain Type of Factorization and Compact Riemann Surfaces

1990 ◽  
Vol 42 (6) ◽  
pp. 1041-1052
Author(s):  
Pascual Cutillas Ripoll
Keyword(s):  
Riemann Surface ◽  
Finite Subset ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Meromorphic Functions ◽  
Irreducible Elements ◽  
Finite Sums ◽  
Compact Riemann Surfaces

AbstractLet be a compact Riemann surface, be the complement of a nonvoid finite subset of and A() be the ring of finite sums of meromorphic functions in with finite divisor. In this paper it is proved that every nonzero f ∈ A() can be decomposed as a product αβ, where α is either a unit or a product of powers of irreducible elements of A(), uniquely determined by f up to multiplication by units, and β is a product of functions of the type eφ – 1, with φ holomorphic and nonconstant in . Furthermore, a similar result is obtained for a certain class of subrings of A().

scholarly journals Conformal immersions of compact Riemann surfaces into the 2n-sphere (n ≥ 2)

1996 ◽  
Vol 141 ◽  
pp. 79-105 ◽  
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Euclidean Space ◽  
Unit Sphere ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Minimal Immersion ◽  
Dimensional Euclidean Space ◽  
Compact Riemann Surfaces

The purpose of this article is to prove the following theorem:Let n be a positive integer larger than or equal to 2, and let S be the unit sphere in the 2n + 1 dimensional Euclidean space. Given a compact Riemann surface, we can always find a conformal and minimal immersion of the surface into S whose image is not lying in any 2n — 1 dimensional hyperplane.This is a partial generalization of the result by R. L. Bryant. In this papers, he demonstrates the existence of a conformal and minimal immersion of a compact Riemann surface into S2n, which is generically 1:1, when n = 2 ([2]) and n = 3 ([1]).

Harmonic Morphisms as Sphere Bundles Over Compact Riemann Surfaces

1997 ◽  
Vol 08 (07) ◽  
pp. 935-942
Author(s):  
Sigmundur Gudmundsson
Keyword(s):  
Riemann Surface ◽  
Homotopy Type ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Harmonic Morphisms ◽  
Sphere Bundle ◽  
Harmonic Morphism ◽  
Totally Geodesic ◽  
Compact Riemann Surfaces

We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.

ON THE KOSTERLITZ–THOULESS TRANSITION ON COMPACT RIEMANN SURFACES

1993 ◽  
Vol 07 (03) ◽  
pp. 171-182 ◽  
Author(s):  
ACHILLES D. SPELIOTOPOULOS ◽  
HARRY L. MORRISON
Keyword(s):  
Field Theory ◽  
Riemann Surface ◽  
Partition Function ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Constant Density ◽  
Two Dimensional ◽  
Coulombic Interaction ◽  
Density Limit ◽  
Compact Riemann Surfaces

A Lagrangian for the two-dimensional vortex gas is derived from a general microscopic Lagrangian for 4 He atoms on an arbitrary compact Riemann Surface without boundary. In the constant density limit the vortex Hamiltonian obtained from this Lagrangian is found to be the same as the Kosterlitz and Thouless Coulombic interaction Hamiltonian. The partition function for the Kosterlitz–Thouless ensemble on the general compact is formulated and mapped into the sine–Gordon field theory.

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