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Frame bundle
About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.
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TL;DR: In this paper, an affine group scheme X defined over the field k as well as a principal X-bundle S1755069607000151inline1 over the curve X is given by a xs0211A-graded neutral Tannakian category built out of all strongly semistable vector bundles over X.
Abstract: Let X be a geometrically irreducible smooth projective curve defined over a field k. Assume that X has a k-rational point; fix a k-rational point x e X. From these data we construct an affine group scheme X defined over the field k as well as a principal X-bundle S1755069607000151inline1 over the curve X. The group scheme X is given by a xs0211A-graded neutral Tannakian category built out of all strongly semistable vector bundles over X. The principal bundle S1755069607000151inline1 is tautological. Let G be a linear algebraic group, defined over k, that does not admit any nontrivial character which is trivial on the connected component, containing the identity element, of the reduced center of G. Let E G be a strongly semistable principal G-bundle over X. We associate to E G a group scheme M defined over k, which we call the monodromy group scheme of E G , and a principal M-bundle E M over X, which we call the monodromy bundle of E G . The group scheme M is canonically a quotient of X, and E M is the extension of structure group of S1755069607000151inline1. The group scheme M is also canonically embedded in the fiber Ad(E G ) x over x of the adjoint bundle.
41 citations
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TL;DR: In this paper, a new derivation of the curvature formula (−26/12m3+1/6m)δn, −m for the canonical holomorphic line bundle over DiffS1/S1 is given which clarifies the relation of that bundle with the complex line bundles over infinite-dimensional Grassmannians.
Abstract: The recent results by Bowick and Rajeev on the relation of the geometry of DiffS1/S1 and string quantization in ℝd are extended to a string moving on a group manifold. A new derivation of the curvature formula (−26/12m3+1/6m)δn, −m for the canonical holomorphic line bundle over DiffS1/S1 is given which clarifies the relation of that bundle with the complex line bundles over infinite-dimensional Grassmannians, studied by Pressley and Segal.
41 citations
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TL;DR: In this paper, the authors prove generic semipositivity of the tangent bundle of a non-uniruled Calabi-Yau variety in positive characteristic, and construct an example of a nef line bundle in characteristic zero, whose each reduction to positive characteristic is not nef.
41 citations
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TL;DR: In this article, the Pfaffian line bundle of a certain family of real Dirac operators is shown to be an object in the category of line bundles, and it is shown how string structures give rise to trivialisations of that line bundle.
Abstract: The present paper is a contribution to categorial index theory. Its main result is the calculation of the Pfaffian line bundle of a certain family of real Dirac operators as an object in the category of line bundles. Furthermore, it is shown how string structures give rise to trivialisations of that Pfaffian.
41 citations
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TL;DR: In this paper, a generalised notion of connection on a fiber bundle E over a manifold M is presented, characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is parametrised in some specific way by a vector bundle map from a prescribed vector bundle over M into TM.
Abstract: A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is ‘parametrised’ in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework.
41 citations