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Universal Deformations of a Curve and a Differential

Emma Carberry, Martin Ulrich Schmidt

TL;DR

This paper develops a universal local deformation theory for pairs $(X,dx)$, where $X$ is a compact 1-dimensional complex space and $dx$ is a meromorphic 1-form with prescribed poles and no residues. Deformations are reduced to finitely many germ data at points where $X$ is singular or $dx$ vanishes, and under the local-planarity condition these deformations preserve the periods of $dx$, enabling a period-preserving moduli theory. A universal local deformation (a Kuranishi-type family) is constructed, with fibrewise infinitesimal deformations forming a vector bundle over the base, which facilitates Whitham deformations of spectral curve data. The hyperelliptic case is treated to yield universal deformations for finite-gap spectral data of integrable systems such as the KdV, sinh-Gordon, sine-Gordon and NLS equations, linking deformation theory to spectral curves and their integrable-system applications.

Abstract

We construct universal local deformations (Kuranishi families) for pairs consisting of a compact complex curve and a meromorphic 1-form. Each pair is assumed to be locally planar, a condition which in particular forces the periods of the meromorphic differential to be preserved by local deformations. The hyperelliptic case yields universal local deformations for the spectral data of integrable systems such as simply-periodic solutions of the KdV equation or of the sinh-Gordon equation (cylinders of constant mean curvature). This is the first of two papers in which we shall develop a deformation theory of the spectral curve data of an integrable system.

Universal Deformations of a Curve and a Differential

TL;DR

This paper develops a universal local deformation theory for pairs , where is a compact 1-dimensional complex space and is a meromorphic 1-form with prescribed poles and no residues. Deformations are reduced to finitely many germ data at points where is singular or vanishes, and under the local-planarity condition these deformations preserve the periods of , enabling a period-preserving moduli theory. A universal local deformation (a Kuranishi-type family) is constructed, with fibrewise infinitesimal deformations forming a vector bundle over the base, which facilitates Whitham deformations of spectral curve data. The hyperelliptic case is treated to yield universal deformations for finite-gap spectral data of integrable systems such as the KdV, sinh-Gordon, sine-Gordon and NLS equations, linking deformation theory to spectral curves and their integrable-system applications.

Abstract

We construct universal local deformations (Kuranishi families) for pairs consisting of a compact complex curve and a meromorphic 1-form. Each pair is assumed to be locally planar, a condition which in particular forces the periods of the meromorphic differential to be preserved by local deformations. The hyperelliptic case yields universal local deformations for the spectral data of integrable systems such as simply-periodic solutions of the KdV equation or of the sinh-Gordon equation (cylinders of constant mean curvature). This is the first of two papers in which we shall develop a deformation theory of the spectral curve data of an integrable system.
Paper Structure (6 sections, 12 theorems, 71 equations)

This paper contains 6 sections, 12 theorems, 71 equations.

Table of Contents

  1. Deformations of pairs $(X,dx)$
  2. Patching deformations
  3. Infinitesimal deformations
  4. Decomposition of holomorphic functions
  5. Existence of a universal deformation
  6. Deformations of hyperelliptic curves with one differential

Key Result

Lemma 2.2

For any pair $(X,dx)$ let $q_1,\ldots,q_L\in X$ denote the points at which either $X$ is not smooth or $dx$ has a smooth root. Let be $x$-deformations such that for each $q\in\{q_1,\ldots,q_L\}$ the induced deformations $(X_q,x_q)\hookrightarrow(Y_q,x_{Y,q})\twoheadrightarrow\mathcal{S}_0$ and $(X_q,x_q)\hookrightarrow(\Tilde{Y}_q,x_{\Tilde{Y},q})\twoheadrightarrow\Tilde{\mathcal{S}}_0$ are isomo

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 28 more