Enumerating Non-Stable Vector Bundles

TL;DR

This paper develops a motivic analog of a topological enumeration result for non-stable vector bundles, leading to new insights on projective modules, cancellation properties, and symplectic bundles over algebraic varieties.

Contribution

It introduces a motivic framework for enumerating non-stable vector bundles and proves cancellation results under certain conditions, extending previous algebraic and topological findings.

Findings

Established a motivic analog of James-Thomas enumeration result

Proved cancellation of rank d-1 vector bundles over algebraically closed fields

Explored cancellation properties of symplectic vector bundles

Abstract

In this article, we establish a motivic analog of an enumeration result of James-Thomas on non-stable vector bundles in topological setting. Using this, we obtain results on enumeration of projective modules of rank $d$ over a smooth affine $k$-algebra $A$ of dimension $d$, recovering in particular results of Suslin and Bhatwadekar on cancellation of such vector bundles. Admitting a conjecture of Asok and Fasel, we prove cancellation of such vector bundles of rank $d-1$ if the base field $k$ is algebraically closed. We also explore the cancellation properties of symplectic vector bundles.