Line Bundles Over Families of (SUPER) Riemann Surfaces. I: The Non-Graded Case

TL;DR

This paper develops a foundational theory of line bundles over families of Riemann surfaces, extending classical theorems like Gauss-Bonnet and flatness to this context.

Contribution

It introduces a systematic approach to relative line bundles over Riemann surfaces, serving as a basis for future work on super Riemann surfaces.

Findings

Extended Gauss-Bonnet theorem for line bundles

Generalized flatness theorem for line bundles

Framework for relative line bundles over Riemann surfaces

Abstract

A first step towards a systematic theory of relative line bundles over SUSY-curves is presented. In this paper we deal with the case of relative line bundles over families of ordinary Riemann surfaces. Generalizations of the Gauss-Bonnet theorem and of the flatness theorem for line bundles are discussed.