Compactness for $$\Omega $$-Yang–Mills connections

Abstract

On a Riemannian manifold of dimension n we extend the known analytic results on Yang–Mills connections to the class of connections called \(\Omega \)-Yang–Mills connections, where \(\Omega \) is a smooth, not necessarily closed, \((n-4)\)-form on M. Special cases include \(\Omega \)-anti-self-dual connections and Hermitian–Yang–Mills connections over general complex manifolds. By a key observation, a weak compactness result is obtained for moduli space of smooth \(\Omega \)-Yang–Mills connections with uniformly \(L^2\) bounded curvature, and it can be improved in the case of Hermitian–Yang–Mills connections over general complex manifolds. A removable singularity theorem for singular \(\Omega \)-Yang–Mills connections on a trivial bundle with small energy concentration is also proven. As an application, it is shown how to compactify the moduli space of smooth Hermitian–Yang–Mills connections on unitary bundles over a class of balanced manifolds of Hodge-Riemann type. This class includes the metrics coming from multipolarizations, and in particular, the Kähler metrics. In the case of multipolarizations on a projective algebraic manifold, the compactification of smooth irreducible Hermitian–Yang–Mills connections with fixed determinant modulo gauge transformations inherits a complex structure from algebro-geometric considerations.