Holomorphic bundles for higher dimensional gauge theory

TL;DR

This paper develops techniques to construct holomorphic bundles on noncompact threefolds with applications to higher-dimensional gauge theory and G2-manifolds, including stability criteria and curvature blow-up models.

Contribution

It generalizes Hoppe's stability criterion to certain projective varieties and applies monads to model curvature blow-up phenomena in higher-dimensional gauge theory.

Findings

Constructed holomorphic bundles satisfying stability at infinity

Generalized stability criterion for varieties with Picard group isomorphic to Z^l

Modeled curvature blow-up along degenerating stable bundles

Abstract

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 $\rm G_2-$manifolds obtained from asymptotically cylindrical Calabi-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 $\operatorname{Pic}{X}\simeq\mathbb{Z}^l$, 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.