Holomorphic bundles for higher dimensional gauge theory

Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact $3-$folds, called building blocks, satisfying a stability condition `at infinity’. Such bundles are known to parametrise solutions of the Yang-Mills equation over the $\rm G_2-$manifoldsobtainedfromasymptoticallycylindricalCalabi-Yau$3-$folds studied by Kovalev and by Corti-Haskins-Nordstr"om-Pacini et al. The most important tool is a generalisation of Hoppe’s stability criterion to holomorphic bundles over smooth projective varieties $X$ with $\mathrm{Pic}X\simeq {\mathbb{Z}}^l$, a result which may be of independent interest. Finally, we apply monads to produce a prototypical model of the curvature blow-up phenomenon along a sequence of asymptotically stable bundles degenerating into a torsion-free sheaf.