On the asymptotic expansion of Bergman kernel

Dai, Xianzhe; Liu, Kefeng; Ma, Xiaonan

doi:10.1016/j.crma.2004.05.011

Cited by 94 publications

(201 citation statements)

References 25 publications

(32 reference statements)

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“…2 by explaining the formal calculus on C n for the model operator L. In Sect. 3, we recall the definition of the spin c Dirac operator and the asymptotic expansion of the Bergman kernel obtained in [20]. In Sect.…”

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confidence: 99%

Toeplitz Operators on Symplectic Manifolds

Marinescu

2008

J Geom Anal

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152

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confidence: 99%

Toeplitz Operators on Symplectic Manifolds

Marinescu

2008

J Geom Anal

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152

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“…The proof of the above theorems uses techniques adapted from [1, §11], [2,5], along with a deformation of D 2 p by the Casimir operator (i.e., to consider D 2 p − pCas, which plays a key role in proving Theorems 2.1, 2.2). We refer to [6] for more details.…”

Section: Theorem 25mentioning

confidence: 99%

Bergman kernels and symplectic reduction

Zhang

2005

Comptes Rendus. Mathématique

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We present several results concerning the asymptotic expansion of the invariant Bergman kernel of the spin c Dirac operator associated with high tensor powers of a positive line bundle on a compact symplectic manifold. To cite this article: X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 341 (2005).  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Noyaux de Bergman et réduction symplectique. Nous annonçons des résultats sur le développement asymptotique du noyau de Bergman G-invariant de l'opérateur de Dirac spin c associé à une puissance tendant vers l'infini d'un fibré en droites positif sur une variété symplectique compacte. Soit (X, ω) une variété symplectique compacte, et soit (L, h L ) un fibré en droites hermitien muni d'une connexion hermitienne ∇ L telle que √ −1 2π (∇ L ) 2 = ω. Soit (E, h E ) un fibré vectoriel hermitien sur X muni d'une connexion hermitienne ∇ E . Soit g T X une métrique riemannienne sur X, et soit J une structure presque complexe compatible séparément à g T X et ω. Alors les données géométriques ci-dessus définissent canoniquement un opérateur de Dirac spin c D p agissant sur Ω 0,• (X, L p ⊗ E), l'espace de (0, •)-formes à valeurs dans L p ⊗ E. Soit G un groupe de Lie compact connexe et soit g son algèbre de Lie. On suppose que G agit sur X, et que son action se relève à L et E en préservant J , les métriques et les connexions ci-dessus.

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“…The important point is that the full Bergman kernel expansion holds in our setting, see [4,Theorem 4.18']. The involved Dirac operator is…”

Section: Moreover (3) Is Uniform In the Sense That There Is An Integmentioning

confidence: 99%

“…For the last part of the theorem, one notices that with our assumptions all the constants are uniformly bounded, see [4,Section 4.5] or [14].…”

Section: Moreover (3) Is Uniform In the Sense That There Is An Integmentioning

confidence: 99%

Quantization of Donaldson’s heat flow over projective manifolds

Keller

Seyyedali

2015

Math. Z.

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Consider E a holomorphic vector bundle over a projective manifold X polarized by an ample line bundle L. Fix k large enough, the holomorphic sections H 0 (E ⊗ L k ) provide embeddings of X in a Grassmanian space. We define the balancing flow for bundles as a flow on the space of projectively equivalent embeddings of X . This flow can be seen as a flow of algebraic type hermitian metrics on E. At the quantum limit k → ∞, we prove the convergence of the balancing flow towards the Donaldson heat flow, up to a conformal change. As a by-product, we obtain a numerical scheme to approximate the Yang-Mills flow in that context.

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On the asymptotic expansion of Bergman kernel

Abstract: We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle.

Cited by 94 publications

References 25 publications

Toeplitz Operators on Symplectic Manifolds

Toeplitz Operators on Symplectic Manifolds

Bergman kernels and symplectic reduction

Quantization of Donaldson’s heat flow over projective manifolds

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