Holomorphic Line Bundles over a Tower of Coverings

Yuan, Yuan; Zhu, Junyan

doi:10.1007/s12220-015-9617-3

Cited by 4 publications

(2 citation statements)

References 28 publications

(18 reference statements)

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“…Before closing the introduction we only list some related work that were not cited above: [BoSj75, En00, Ch03, Lo04, LuTi04, MaMa08, Liu10, LiuLu15, LuZe16, Ze16]. Applications of the Bergman kernel, and the closely related Szegö kernel, can be found in [Do01], [BlShZe00], [ShZe02], [YuZh16]. The book of Ma and Marinescu [MaMa07] contains an introduction to the asymptotic expansion of the Bergman kernel and its applications.…”

Section: Introductionmentioning

confidence: 99%

Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials

Hezari

2018

International Mathematics Research Notices

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We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size k − 1 4 . These improve the earlier results in the subject for smooth potentials, where an expansion exists in a k − 1 2 neighborhood of the diagonal. We obtain our results by finding upper bounds of the form C m m! 2 for the Bergman coefficients bm(x,ȳ), which is an interesting problem on its own. We find such upper bounds using the method of [BeBeSj08]. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal x = y in our results. In the special case of metrics with local constant holomorphic sectional curvatures, we obtain off-diagonal asymptotic in a fixed (as k → ∞) neighborhood of the diagonal, which recovers a result of Berman [Ber08] (see Remark 3.5 of [Ber08] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod O(e −kδ ).Near the diagonal, i.e. in a 1 √ k -neighborhood of the diagonal, one has a scaling asymptotic expansion for the Bergman kernel (see [ShZe02,MaMa07,MaMa13,LuSh15,HeKeSeXu16]). For d(x, y) ≫ log k k , where d is the Riemannian distance induced by ω, no useful asymptotics are known. However, there are off-diagonal upper bounds of Agmon type (1.2) |K k (x, y)| h k ≤ Ck n e −c √ kd(x,y) ,

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Section: Introductionmentioning

confidence: 99%

Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials

Hezari

2018

International Mathematics Research Notices

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“…Before closing the introduction we only list some related work that were not cited above: [BoSj75, En00, Ch03, Lo04, LuTi04, Lo04, MaMa08, Liu10, LiuLu15, Se15, LuZe16, Ze16, LuSe17]. Applications of the Bergman kernel, and the closely related Szegö kernel, can be found in [Do01], [BlShZe00], [ShZe02], [YuZh16]. The book of Ma and Marinescu [MaMa07] contains an introduction to the asymptotic expansion of the Bergman kernel and its applications.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic properties of Bergman kernels for potentials with Gevrey regularity

2018

Preprint

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We study the asymptotic properties of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is in Gevrey class G a for some a > 1, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size k − 1 2 + 1 4a+4ε for every ε > 0. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a ( log k k )1 2 neighborhood of the diagonal. We obtain our results by finding upper bounds of the form C m m! 2a+2ε for the Bergman coefficients bm(x, ȳ) in a fixed neighborhood by the method of [BeBeSj08]. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal x = y in our results.

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On a property of Bergman kernels when the Kähler potential is analytic

Hezari

2021

Pacific J. Math.

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Holomorphic Line Bundles over a Tower of Coverings

Cited by 4 publications

References 28 publications

Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials

Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials

Asymptotic properties of Bergman kernels for potentials with Gevrey regularity

On a property of Bergman kernels when the Kähler potential is analytic

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