Some results in the theory of vector bundles

Umemura, Hiroshi

doi:10.1017/s0027763000015919

Cited by 45 publications

(28 citation statements)

References 12 publications

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“…In [29], Umemura proves that on curves ampleness and Griffiths-positivity coincide. As is was pointed out to me by N.M. Kumar, the part of the argument that needs a result analogue to Proposition 3.2.4.8 is omitted in [29].…”

Section: Nefness and T-nefness On Curvesmentioning

confidence: 99%

Singular hermitian metrics on vector bundles

Cataldo¹

1998

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic vector bundles and define positivity in view of L 2 -estimates. Associated with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a certain d ′′ -complex. We prove a vanishing theorem for the cohomology of this sheaf. All this generalizes to the case of higher rank known results of Nadel for the case of line bundles. We introduce a new semi-positivity notion, t-nefness, for vector bundles, establish some of its basic properties and prove that on curves it coincides with ordinary nefness. We particularize the results on s.h.m. to the case of vector bundles of the form E = F ⊗ L, where F is a t-nef vector bundle and L is a positive (in the sense of currents) line bundle. As applications we generalize to the higher rank case 1) Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning the global generation of jets for the adjoint to powers of ample line bundles, and 3) Matsusaka Big Theorem made effective.

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Section: Nefness and T-nefness On Curvesmentioning

confidence: 99%

Singular hermitian metrics on vector bundles

Cataldo¹

1998

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The converse is the Griffiths conjecture. In the case of curves this was proven in [20,4]. Somewhat strong evidence for this conjecture in the general case is provided by the fact that if E is Hartshorne-ample, then E ⊗ det(E) is Nakano-positive (stronger than Griffiths positive) [1,16].…”

Section: Introductionmentioning

confidence: 93%

Representability of Chern–Weil forms

Pingali

2017

Math. Z.

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Abstract. In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.

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“…Umemura proved ( [44]) that a vector bundle V over a curve B is positive (i.e., Griffiths positive, or equivalently Nakano positive) if and only if V is ample.…”

Section: Fujita's Second Theoremmentioning

confidence: 99%

Vector bundles on curves coming from variation of Hodge structures

Catanese

Dettweiler

2016

Int. J. Math.

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Abstract. Fujita's second theorem for Kähler fibre spaces over a curve asserts that the direct image V of the relative dualizing sheaf splits as the direct sum V = A ⊕ Q, where A is ample and Q is unitary flat. We focus on our negative answer ([9]) to a question by Fujita: is V semiample?We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group (Z/n) 2 of a Del Pezzo surface of degree 5 (branched on a union of lines forming a bianticanonical divisor), and endowed with a semistable fibration with only 3 singular fibres.The simplest such surfaces are the three ball quotients considered in [3], fibred over a curve of genus 2, and with fibres of genus 4.These examples are a larger class than the ones corresponding to Shimura curves in the moduli space of Abelian varieties.

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Some results in the theory of vector bundles

Cited by 45 publications

References 12 publications

Singular hermitian metrics on vector bundles

Singular hermitian metrics on vector bundles

Representability of Chern–Weil forms

Vector bundles on curves coming from variation of Hodge structures

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