Families of Holomorphic Bundles

Teleman, Andrei

doi:10.1142/s0219199708002892

Cited by 14 publications

(16 citation statements)

References 22 publications

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“…The following theorem is a generalization of a result by Teleman ([24, Theorem 1.4]) and answers a conjecture of his (Conjecture 2 in p. 544 of [24]) in an affirmative way for projective varieties.…”

Section: The Weil-petersson Currentsupporting

confidence: 70%

“…In particular, it possesses a compactification as an analytic space. Our main result states that for moduli of stable holomorphic vector bundles the Weil-Petersson form extends as a positive, closed current to a compactification, thus answering a conjecture of Andrei Teleman ( [24]) in the affirmative way. The extension property holds, whenever a given family of stable vector bundles on a Kähler manifold is the restriction of a coherent sheaf.…”

Section: Introductionmentioning

confidence: 56%

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The Weil-Petersson current for moduli of vector bundles and applications to orbifolds

Biswas¹,

Schumacher²

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Nous étudions les fibrés vectoriels holomorphes stables sur une varieté kählérienne compacte ou plus généralement sur une orbifold possédant une structure kählérienne. Dans ce contexte, nous utilisons l'existence d'une connection Hermite-Einstein et construisons une forme de Weil-Petersson généralisée sur l'espace des modules des fibrés holomorphes stables à fibré déterminant fixé. Nous montrons que la forme de Weil-Petersson s'étend en un courant (semi-)positif fermé pour des dégénerescences de familles qui sont des restrictions de faisceaux cohérents. Ce courant sera appelé un courant de Weil-Petersson. Dans le cas d'une orbifold de type Hodge, un fibré en droite déterminant existe sur l'espace des modules. Ce fibré en droites est muni d'une métrique de Quillen dont la courbure coincide avec la forme de Weil-Petersson généralisée. En application, nous montrons que le fibré en droites déterminant s'étend à une compactification de l'espace des modules.

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Section: The Weil-petersson Currentsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 56%

The Weil-Petersson current for moduli of vector bundles and applications to orbifolds

Biswas¹,

Schumacher²

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…where C and C ′ are constants not depending on F with C ′ > 0, cf. [Tel08] 2.1 for a similar computation in the smooth case. This proves the second assertion.…”

Section: A Boundedness Criterionmentioning

confidence: 87%

Bounded sets of sheaves on Kähler manifolds

Toma

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…7.3.38]. A direct analytic proof for Kähler or Hermitian manifolds seems more dicult (for a partial result,[36, Thm. 3]).…”

mentioning

confidence: 99%

Continuity of the Yang–Mills flow on the set of semistable bundles

Sibley

Wentworth

2021

Pure and Applied Mathematics Quarterly

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Dedicated to Duong H. Phong, with admiration, on the occasion of his 65th birthday. IntroductionLet (X, ω) be a compact Kähler manifold of dimension n and (E, h) → X a C ∞ hermitian vector bundle on X. The celebrated theorem of Donaldson-Uhlenbeck-Yau states that if A is an integrable unitary connection on (E, h) that induces an ω-slope stable holomorphic structure on E, then there is a complex gauge transformation g such that g(A) satises the Hermitian-Yang-Mills (HYM) equations. The proof in [40] uses the continuity method applied to a deformation of the Hermitian-Einstein equations for the metric h. The approach in [11, 12] deforms the metric using a nonlinear parabolic equation, the Donaldson ow.Deforming the metric is equivalent to acting by a complex gauge transformation modulo unitary ones, and in this context the Donaldson ow is equivalent (up to unitary gauge transformations) to the Yang-Mills ow on the space of integrable unitary connections. The proof in [12] assumes that X is a projective algebraic manifold (more precisely, that ω is a Hodge metric) whereas the argument in [40] does not. The methods of Uhlenbeck-Yau and Donaldson were combined by Simpson [35] to prove convergence of the Yang-Mills ow for stable bundles on all compact Kähler manifolds. The Yang-Mills ow thus denes aHYM (E, h) from the space of smooth integrable connections on (E, h) inducing stable holomorphic structures to the moduli space M * HYM (E, h) of irreducible HYM connections.1 Continuity of this map follows by a comparison of Kuranishi slices (see [15,31]).When the holomorphic bundle E A = (E,∂ A ) is strictly semistable, then the Donaldson ow fails to converge unless E A splits holomorphically into a sum of stable bundles (i.e. it is polystable). If n = 1 it is still true, however, that the Yang-Mills ow converges to a smooth HYM connection on E for any semistable initial condition. This was proven by Daskalopoulos and Råde [7,32]. Moreover, the holomorphic structure of the limiting connection is isomorphic to the polystable holomorphic bundle Gr(E A ) obtained from the

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Families of Holomorphic Bundles

Cited by 14 publications

References 22 publications

The Weil-Petersson current for moduli of vector bundles and applications to orbifolds

The Weil-Petersson current for moduli of vector bundles and applications to orbifolds

Bounded sets of sheaves on Kähler manifolds

Continuity of the Yang–Mills flow on the set of semistable bundles

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