Vector bundles and Arakelov geometry on the projective line over the integers

TL;DR

This paper investigates rank-two vector bundles on the projective line over integers, applying Arakelov geometry to compute arithmetic invariants and explore their properties.

Contribution

It provides new insights into indecomposable vector bundles and applies Arakelov geometric techniques to compute their arithmetic Chern classes.

Findings

Computed arithmetic Chern classes for vector bundles

Applied arithmetic Riemann-Roch theorem in this context

Analyzed properties of indecomposable sheaves

Abstract

We study locally free sheaves of rank two on the projective line over the integers, especially indecomposable ones. Subsequently we apply various concepts of Arakelov geometry to these sheaves. We compute for example the arithmetic Chern classes and use the arithmetic Riemann-Roch theorem.