On the spectrum of asymptotic slopes

TL;DR

This paper investigates the set of asymptotic slopes of maximal subbundles in vector bundles, characterizing its supremum through the Harder-Narasimhan filtration, thus advancing understanding of bundle stability properties.

Contribution

It provides a new description of the supremum of the s-spectrum using the Harder-Narasimhan filtration, linking geometric properties to algebraic invariants.

Findings

Determined the supremum of the s-spectrum in terms of the Harder-Narasimhan filtration.

Established bounds for the s-spectrum based on bundle stability.

Connected asymptotic slope behavior with filtration data.

Abstract

The slopes of maximal subbundles of rank $s$ divided by the degree of the map under various pull backs form a bounded collection of numbers called the $s$-spectrum of the bundle. We study the supremum of the $s$-spectrum and determine it in terms of the Harder Narasimhan filtration of the bundle.