Holomorphic bundles for higher dimensional gauge theory

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 ${\mathrm{G}}_2-$manifolds obtained from asymptotically cylindrical Calabi-Yau $3-$folds studied by Kovalev and by Corti-Haskins-Nordström-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.

19 pages. Final version to appear in Bulletin of the London Mathematical Society

References in corpus (5)

• G_2-manifolds and associative submanifolds via semi-Fano 3-folds
• K3 surfaces with non-symplectic involution and compact irreducible G_2-manifolds
• Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
• $G_2$-instantons on twisted connected sums
• $G_2$-instantons over asymptotically cylindrical manifolds

Cited by in corpus (9)

• Instanton bundles on two Fano threefolds of index $1$
• Current progress on $G_2$--instantons over twisted connected sums
• Instanton Sheaves on Ruled Fano 3-folds of Picard Rank 2 and Index 1
• K3 surfaces with two involutions and low Picard number
• Torsion free instanton sheaves on the blow-up of ${\mathbb{P}}^n$ at a point
• Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds
• Instanton bundles on ${\mathbb{P}}^1\times {\mathbb{F}}_1$
• Monads on multiprojective spaces and associated vector bundles
• $\ell$-away ACM Bundles on Fano Surfaces