Homological Algebra of Twisted Quiver Bundles

By Peter B. Gothen, A. King. 2002

81 citations recorded by the source corpus.

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. It is shown that the category of such representations is an abelian category with enough injectives by the construction of an explicit injective resolution. Using this explicit resolution, a long exact sequence is found that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. It is also shown that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/71897509f719d201acb6b99c29ef7aa4fe89bf9b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the motives of moduli of chains and Higgs bundles (2011, background): “hat one can embed the categoryof chains into an abelian category, by allowing the Ei to be coherent sheaves instead of vector bundles. In this category one can then do homological algebra (see, e.g., [14]). In this section it will further be useful to consider more generally chains E• = (Ei,φi) with”
Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers (2017, result): “ankDE = rankE, degDE = degE +rankE 2g− 2+ Xk i=1 di ! − Xk i=1 diri,N, ri,j(DE) = ri,j−1(E). 6.5. Parabolic Higgs bundles. Strictly speaking, we cannot apply the well-known results of Gothen and King [GK05] in our situation because parabolic sheaves are not sheaves. However, it is quite easy to gene”
Moduli for decorated tuples of sheaves and representation spaces for quivers (2004, background): “ecial cases of this construction have been studied in the literature, such as the Higgs bundles. Recent research has Quiver Problems 3 focussed on more general aspects of this theory: Gothen and King [8] have developed the homological algebra of these representations and Alvarez-Co´nsul and Garc´ıa-”
A functorial construction of moduli of sheaves (2006, background): “[9]) or, after choosing a basis for H , indicates that there are dimH arrows.”