Bubble tree compactification of moduli spaces of vector bundles on surfaces

TL;DR

This paper introduces a new algebraic method for compactifying moduli spaces of rank-2 vector bundles on surfaces using trees of surfaces, paralleling the bubbling phenomenon in differential geometry.

Contribution

It proposes a novel algebraic compactification technique for moduli spaces of vector bundles via spaces on trees of surfaces, extending previous geometric approaches.

Findings

New algebraic compactification spaces constructed as quotients by group actions.

Application to the case of rank-2 vector bundles on the projective plane.

Framework aligns with bubbling phenomena in differential geometry.

Abstract

In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Koll\'ar. As an example the compactification of the space of stable rank-2 vector bundles with Chern classes $c_1=0, c_1=2$ on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.