The Szeg\"o Kernel on a Sewn Riemann Surface

TL;DR

This paper provides explicit formulas for the Szeg"o kernel on higher genus Riemann surfaces constructed by sewing lower genus surfaces, analyzing their properties and modular behavior.

Contribution

It introduces explicit sewing procedures to compute the Szeg"o kernel on higher genus surfaces from lower genus data, with detailed examples and modular analysis.

Findings

Explicit formulas for Szeg"o kernel on genus two surfaces

Analysis of modular properties of the Szeg"o kernel

Examples of sewing two tori and a torus with a handle

Abstract

We describe the Szeg\"o kernel on a higher genus Riemann surface in terms of Szeg\"o kernel data coming from lower genus surfaces via two explicit sewing procedures where either two Riemann surfaces are sewn together or a handle is sewn to a Riemann surface. We consider in detail the examples of the Szeg\"o kernel on a genus two Riemann surface formed by either sewing together two punctured tori or by sewing a twice-punctured torus to itself. We also consider the modular properties of the Szeg\"o kernel in these cases.