Bubble tree compactification of moduli spaces of vector bundles on surfaces

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.