Universal Deformations of a Curve and a Differential

TL;DR

This paper develops a universal deformation theory for pairs of complex curves and meromorphic differentials, preserving periods and applying to integrable systems like KdV and sinh-Gordon equations.

Contribution

It constructs universal local deformations for pairs of complex curves and meromorphic differentials, especially in hyperelliptic cases relevant to integrable systems.

Findings

Constructed universal local deformations (Kuranishi families) for pairs of curves and differentials.

Preserved periods of meromorphic differentials under local deformations.

Applied deformation theory to spectral data of integrable systems like KdV and sinh-Gordon.

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.