Line Bundles Over Families of (SUPER) Riemann Surfaces. II: The Graded Case

TL;DR

This paper develops a relative Picard theory for graded manifolds, introduces Berezinian calculus and connections over SUSY-curves, and proves a Gauss-Bonnet theorem in this context, advancing the understanding of line bundles in supergeometry.

Contribution

It introduces a systematic theory of connections and Berezinian calculus for SUSY-curves, extending classical results to the supergeometric setting.

Findings

Proved a Gauss-Bonnet theorem for line bundles over SUSY-curves.

Developed a Berezinian calculus in the context of graded manifolds.

Discussed the conditions for flatness of connections in supergeometry.

Abstract

A relative Picard theory in the context of graded manifolds is introduced. A Berezinian calculus and a theory of connections over SUSY-curves are systematically developed, and used to prove a Gauss-Bonnet theorem for line bundles in that setting and to discuss the validity of a flatness theorem