Hermitian Yang-Mils Connections

I will start introducing the Kähler–Yang–Mills equations on a holomorphic vector bundle over a compact complex manifold. These equations, inspired by the Hitchin–Kobayashi correspondence for bundles and the Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metrics, intertwine the curvature of a Hermitian–Yang–Mills connection on the bundle  and the scalar curvature of a Kähler metric on the manifold. After this, I will consider special symmetric solutions on a compact Riemann surface known as gravitating vortices. 

We will discuss a few results concerning the existence of HE metrics for stable sheaves on Kähler spaces with log-terminal singularities, and their relevance for the Bogomolov-Gieseker inequality.

We will review some recent results on the behaviour of Hermitian Yang-Mills connections with respect to variations of the polarisation. In particular, we will focus on their convergence when the polarisation reaches the boundary of the stable locus. The methods presented are versatile and can be used for similar perturbation problems for geometric PDEs with moment map pictures.

In the first lecture, I will survey the history and development of solutions to the Hermitian-Einstein equations. On the algebro-geometric side, this goes back to a theorem of Weil in the 1930s on the existence of holomorphic connections on curves. This motivated the work of Narasimhan-Seshadri, and then in turn higher dimensional versions of stability of sheaves. A second motivation comes from gauge theory and solutions to the anti-self-duality equations in Yang-Mills theory. Finally, I will briefly discuss some of the key applications and generalizations of the Hermitian-Einstein equations, such as Miyaoka’s version of the Bogomolov-Gieseker inequality.

In the second lecture, I will discuss some of the important techniques that go into the proof of the Donaldson-Uhlenbeck-Yau theorem, such as Uhlenbeck’s weak compactness theorem, weakly holomorphic subbundles, and the Donaldson functional. I will then mention more recent results, such as the solution to a conjecture of Bando-Siu, and the extension of Hermitian-Einstein equations to singular varieties.

I will discuss the relationship between projective compactifications of moduli spaces of sheaves on higher dimensional projective manifolds, and the Uhlenbeck-Tian compactifications of solutions to the Hermitian-Einstein equations. This generalizes older work of Jun Li in the case of surfaces. This is joint work with Greb, Sibley, and Toma, and separately with Xuemiao Chen.