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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, it was shown that the cohomology groups of a holomorphic vector bundle of rankr on a compact complex manifoldX of dimensionn vanish if and only if the manifold is ample and p+q≧n+1,l≧ n−p+r−1.
Abstract: LetE be a holomorphic vector bundle of rankr on a compact complex manifoldX of dimensionn. It is shown that the cohomology groupsH
p,q
(X, E⊗k
⊗(detE)
l
) vanish ifE is ample andp+q≧n+1,l≧n−p+r−1. The proof rests on the well-known fact that every tensor powerE
⊗k
, splits into irreducible representations of Gl(E). By Bott's theory, each component is canonically isomorphic to the direct image onX of a homogeneous line bundle over a flag manifold ofE. The proof is then reduced to the Kodaira-Akizuki-Nakano vanishing theorem for line bundles by means of the Leray spectral sequence, using backward induction onp. We also obtain a generalization of Le Potier's isomorphism theorem and a counterexample to a vanishing conjecture of Sommese.
42 citations
42 citations
TL;DR: In this article, it was shown that a co-Higgs bundle arising from a Schwarzenberger bundle with nonzero Higgs field is rigid, in the sense that a nearby deformation is again Schwartzberger.
Abstract: On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's rank-2 vector bundle on the projective plane, constructed from a line bundle on the double cover CP^1 \times CP^1 \to CP^2, is naturally a co-Higgs bundle, with the twisted endomorphism, or "Higgs field", also descending from the double cover. Allowing the branch conic to vary, we find that Schwarzenberger bundles give rise to an 8-dimensional moduli space of co-Higgs bundles. After studying the deformation theory for co-Higgs bundles on complex manifolds, we conclude that a co-Higgs bundle arising from a Schwarzenberger bundle with nonzero Higgs field is rigid, in the sense that a nearby deformation is again Schwarzenberger.
42 citations
TL;DR: In this paper, the structural structure underlying the Lie 3-algebra of superconformal models in six space-time dimensions has been revealed, and the generalized Bianchi identities can be retrieved concisely from Q 2 = 0, which encode all the essence of the structural identities.
41 citations
TL;DR: In this article, a survey on the Stokes structure of a good meromorphic flat bundle is given, and it is shown that a good flat bundle has good formal structure if and only if it has a good lattice.
Abstract: We give a survey on the Stokes structure of a good meromorphic flat bundle. We also show that a meromorphic flat bundle has the good formal structure if and only if it has a good lattice.
41 citations