scholarly journals Upper Bounds for Bergman Kernels Associated to Positive Line Bundles with Smooth Hermitian Metrics

2018 ◽  
pp. 437-457 ◽  
Author(s):  
Michael Christ
Keyword(s):  
Upper Bounds ◽  
Line Bundles ◽  
Hermitian Metrics ◽  
Bergman Kernels ◽  
Positive Line

scholarly journals On the asymptotic behavior of Bergman kernels for positive line bundles

2017 ◽  
Vol 289 (1) ◽  
pp. 71-89 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Xiaonan Ma ◽  
Viêt-Anh Nguyên
Keyword(s):  
Asymptotic Behavior ◽  
Line Bundles ◽  
Bergman Kernels ◽  
Positive Line

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

2018 ◽  
Vol 2020 (8) ◽  
pp. 2241-2286 ◽  
Author(s):  
Hamid Hezari ◽  
Zhiqin Lu ◽  
Hang Xu
Keyword(s):  
Bergman Kernel ◽  
Asymptotic Properties ◽  
Upper Bounds ◽  
Complete Asymptotic Expansion ◽  
Bergman Kernels ◽  
Sectional Curvatures ◽  
Positive Line Bundle ◽  
Compact Kähler Manifold ◽  
Tensor Powers ◽  
Positive Line

Abstract 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 $k^{-\frac 14}$. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a $k^{-\frac 12}$ neighborhood of the diagonal. We obtain our results by finding upper bounds of the form $C^m m!^{2}$ for the Bergman coefficients $b_m(x, \bar y)$, which is an interesting problem on its own. We find such upper bounds using the method of [3]. 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 $k \to \infty$) neighborhood of the diagonal, which recovers a result of Berman [2] (see Remark 3.5 of [2] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod $O(e^{-k \delta } )$.

scholarly journals A direct approach to Bergman kernel asymptotics for positive line bundles

2008 ◽  
Vol 46 (2) ◽  
pp. 197-217 ◽  
Author(s):  
Robert Berman ◽  
Bo Berndtsson ◽  
Johannes Sjöstrand
Keyword(s):  
Bergman Kernel ◽  
Direct Approach ◽  
Line Bundles ◽  
Bergman Kernel Asymptotics ◽  
Positive Line

The strength for line bundles

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Edoardo Ballico ◽  
Emanuele Ventura
Keyword(s):  
Line Bundle ◽  
Algebraic Variety ◽  
Commutative Algebra ◽  
Upper Bounds ◽  
Line Bundles ◽  
Homogeneous Polynomials

We introduce the strength for sections of a line bundle on an algebraic variety. This generalizes the strength of homogeneous polynomials that has been recently introduced to resolve Stillman's conjecture, an important problem in commutative algebra. We establish the first properties of this notion and give some tool to obtain upper bounds on the strength in this framework. Moreover, we show some results on the usual strength such as the reducibility of the set of strength two homogeneous polynomials.

Topological Invariants and Differential Geometry

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Chern Class ◽  
Euler Number ◽  
Line Bundles ◽  
Topological Invariants ◽  
Fundamental Groups ◽  
Hermitian Metrics ◽  
Finite Set ◽  
First Chern Class

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.

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