A Non-Abelian Approach to Riemann Surfaces Part I: Wronskian Geometry

TL;DR

This paper explores the geometry of projectively flat holomorphic vector bundles over Riemann surfaces by introducing Wronskian line bundles and interpreting Abel's identity through the first Chern class.

Contribution

It introduces a novel approach linking Wronskian line bundles to projectively flat bundles and interprets classical identities in terms of Chern classes.

Findings

Defined Wronskian line bundles for holomorphic vector bundles

Connected Abel's identity to the first Chern class of the Wronskian bundle

Provided a geometric framework for studying flat vector bundles

Abstract

We study projectively flat holomorphic vector bundles over Riemann surfaces. To each such bundle, we naturally assign a Wronskian line bundle. The main idea is a notion of the division of two meromorphic sections. Abel's identity is interpreted as the first Chern class of the Wronskian line bundle.