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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 article, the authors developed an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson.
Abstract: We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on 2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.
87 citations
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TL;DR: In this paper, it was shown that every Stein manifold X of dimension n admits holomorphic functions with pointwise independent differentials, and that this number is maximal for every n. In particular, X admits a holomorphic function without critical points.
Abstract: We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thom's jet transversality theorem, Gromov's convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).
87 citations
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86 citations
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TL;DR: In this article, the authors studied the behavior of the Frobenius map F*: H 1(X, E) → H 1 (X, F*E) for a vector bundle E.
Abstract: Let k be an algebraically closed field of characteristic p > 0, and let X be a curve defined over k. The aim of this paper is to study the behavior of the Frobenius map F*: H1(X, E) → H1(X, F*E) for a vector bundle E.
82 citations
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TL;DR: In this article, it was shown that a 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle.
Abstract: Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,ℝ) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.
82 citations