Line bundles defined by the Schwarz function

TL;DR

This paper characterizes and constructs Cauchy and exponential transforms as canonical sections of line bundles on the Riemann sphere, linking Schwarz functions, reflection, and complex geometry.

Contribution

It introduces a novel geometric framework for understanding classical transforms via line bundles associated with the Schwarz function.

Findings

Cauchy and exponential transforms are characterized as sections of specific line bundles.

A connection between Schwarz reflection and line bundles on the Schottky double is established.

The approach provides a geometric perspective on classical complex analysis transforms.

Abstract

Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.