Ben Sibley, now at the Simons Center in Stony Brook, New York but formerly of this parish, will speak in the geometry seminar on 20th June, at 1.30pm. Ben’s title is *“A complex analytic structure on the compactification of the moduli space of Hermitian-Yang-Mills connections on a projective manifold”*. The talk will take place in the Salle de Profs, on the 9th floor of building NO. His abstract is below.

The enormous success of gauge-theoretic techniques applied to the differential topology of four-dimensional manifolds culminated in the definition of the Donaldson polynomial invariants. These were diffeomorphism invariants which could be used to distinguish non-diffeomorphic four-manifolds. The invariants were notoriously difficult to calculate, since their definition involved a certain compactification due to Donaldson and Uhlenbeck of the moduli space of anti-self-dual connections on a given four-manifold.

Complex projective surfaces were a particularly fertile testing ground for the new invariants, and as such it was natural to ask if there was an algebraic-geometric way of computing them in this setting. The Kobayashi-Hitchin correspondence gives an identification of the (projectively) anti-self-dual connections with the moduli space of stable holomorphic vector bundles, and the natural algebraic-geometric compactification is the moduli space of Gieseker semi-stable sheaves. To construct Donaldson invariants algebraic geometrically, Jun Li and independently John Morgan constructed a continuous comparison map from the Gieseker compactification to the analytic Uhlenbeck compactification. Jun Li’s proof in fact shows that the Uhlenbeck compactification admits the structure of a projective scheme.

More recently, the trend is to study gauge theory on higher dimensional manifolds. If one has an appropriate differential geometric structure, there is a notion of anti-self-duality. A special case of this notion in the case of a Kähler manifold is that of an Hermitian-Einstein connection. Due to work of Tian, the moduli space of such connections admits an analytic compactification involving the notion of bubbling along holomorphic subvarieties. If the manifold is projective, then one also has a Gieseker compactification, so one might hope, following Jun Li to prove a comparison theorem in the higher dimensional setting, and in particular that Tian’s compactification is a complex analytic space. I will explain work in progress with Daniel Greb, Matei Toma, and Richard Wentworth towards this problem.