Compactness for $\Omega$-Yang-Mills connections

TL;DR

This paper extends analytic results on Yang-Mills connections to $ abla$-Yang-Mills connections involving an $(n-4)$-form, establishing compactness, removable singularity theorems, and moduli space compactification on complex and balanced manifolds.

Contribution

It introduces the concept of $ abla$-Yang-Mills connections, proves compactness and removable singularity results, and constructs moduli space compactifications on balanced and algebraic manifolds.

Findings

Weak compactness for moduli space of $ abla$-Yang-Mills connections with bounded curvature.

Removable singularity theorem for small energy concentration.

Compactification of Hermitian-Yang-Mills moduli space on balanced manifolds.

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. 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 Kaehler 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.