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 G2−manifolds obtained from asymptotically cylindrical Calabi-Yau 3−folds studied by Kovalev, Haskins et al. and Corti et al..The most important tool is a generalisation of Hoppe's stability criterion to holomorphic bundles over smooth projective varieties X with Pic X ≃ 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.
Abstract. We consider representations of quivers in arbitrary categories and twisted representations of quivers in arbitrary tensor categories. We show that if A is an abelian category, then the category of representations of a quiver in A is also abelian, and that the category of twisted linear representations of a quiver is equivalent to the category of linear (untwisted) representations of a different quiver. We conclude by discussing how representations of quivers arise naturally in certain important problems concerning monads ans sheaves on projective varieties.
In this work, we will prove results that ensure the simplicity and the exceptionality of vector bundles, which are defined by the splitting of pure resolutions. We will call such objects syzygy bundles.
We prove three results on pure resolutions of vector bundles on projective spaces. First, we show that there are simple vector bundles of rank n on P n with arbitrary homological dimension. We then analyze the pure resolutions given by the sheafification of the Koszul complex of a certain algebra and by the sheafification of the minimal free resolution of a compressed Gorenstein Artinian graded algebra, proving that their syzygies are simple vector bundles. Our main tool is a result originally established by Brambilla, for which we give an alternative proof using representations of quivers.
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