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2024 | Book

The Bergman Kernel and Related Topics

Hayama Symposium on SCV XXIII, Kanagawa, Japan, July 2022

Editors: Kengo Hirachi, Takeo Ohsawa, Shigeharu Takayama, Joe Kamimoto

Publisher: Springer Nature Singapore

Book Series : Springer Proceedings in Mathematics & Statistics

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About this book

This volume consists of 15 papers contributing to the Hayama Symposium on Complex Analysis in Several Variables XXIII, which was dedicated to the 100th anniversary of the creation of the Bergman kernel. The symposium took place in Hayama and Tokyo in July 2022. Each article is closely related to the Bergman kernel, covering topics in complex analysis, differential geometry, representation theory, PDE, operator theory, and complex algebraic geometry.

Specifically, some papers address the L2 extension operators from a newly opened viewpoint after solving Suita's conjecture for the logarithmic capacity. They are also continuations of quantitative solutions to the openness conjecture for the multiplier ideal sheaves. The study involves estimates for the solutions of the d-bar equations, focusing on the existence of compact Levi-flat hypersurfaces in complex manifolds.

The collection also reports progress on various topics, including the existence of extremal Kähler metrics on compact manifolds, Lp variants of the Bergman kernel, Wehrl-type inequalities, homogeneous Kähler metrics on bounded homogeneous domains, asymptotics of the Bergman kernels, and harmonic Szegő kernels and operators on the Bergman spaces and Segal-Bargmann spaces.

Some of the papers are written in an easily accessible way for beginners. Overall, this collection updates how a basic notion provides strong insights into the internal relationships between independently found phenomena.

Table of Contents

Frontmatter
Concavity Property of Minimal Integrals with Lebesgue Measurable Gain VII–Negligible Weights
In Memory of Jean-Pierre Demailly (1957–2022)

In this article, we present characterizations of the concavity property of minimal \(L^2\) integrals with negligible weights degenerating to linearity on the fibrations over open Riemann surfaces and the fibrations over products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets \(L^2\) extension problem with negligible weights on the fibrations over open Riemann surfaces and the fibrations over products of open Riemann surfaces.

Shijie Bao, Qi’an Guan, Zhitong Mi, Zheng Yuan
M-harmonic Szegö Kernel on the Ball
Abstract
We give a description of the boundary singularity of the Szegö kernel of M-harmonic functions, i.e. functions annihilated by the invariant Laplacian, on the unit ball of the complex n-space, in terms of the Gauss hypergeometric functions.
Petr Blaschke, Miroslav Engliš
Some Aspects of the Bergman Theory
Abstract
This survey presents various results on the \(p-\)Bergman theory: basic properties, regularity of the \(p-\)Bergman kernel, geometric properties of the \(p-\)Bergman metric, analysis of the \(p-\)Bergman space.
Bo-Yong Chen, Yuanpu Xiong, Liyou Zhang
On Semiclassical Ohsawa-Takegoshi Extension Theorem
Abstract
For a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal \(L^2\)-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic formula for this extension operator and derive several consequences of this study.
Siarhei Finski
Balanced Metrics for Extremal Kähler Metrics and Fano Manifolds
Abstract
The first three sections of this paper are a survey of the author’s work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive, where we present a proof which relies only on the Hopf–Rinow theorem and extends to locally compact complete length metric spaces.
Yoshinori Hashimoto
Unbounded Operators on the Segal–Bargmann Space
Abstract
In this paper we analyse the domains of differential and multiplication operators on the Segal–Bargmann space. We consider the basic estimate for the \(\partial \)- complex, remark that this estimate is closely related to the uncertainty principle in quantum mechanics and compute the Bergman kernel of the graph norm. It is shown that the set of all functions u in the Segal–Bargmann \(A^2(\mathbb C, e^{-|z|^2})\) such that the multiplication with a polynomial p is norm bounded gives a relatively compact subset of the Segal–Bargmann space. In the following section we give a survey of recent results on the \(\partial \)-complex on weighted Bergman spaces on Hermitian manifolds, analysing metrics which produce a similar duality between differentiation and multiplication as in the Segal–Bargmann space. Finally we study the basic estimates and the corresponding questions of compactness for the generalized \(\partial \)-complex.
Friedrich Haslinger
Asymptotic Construction of the Optimal Degeneration for a Fano Manifold
Abstract
Optimal degeneration is the algebraic counterpart of the prescribed geometric flow. We review some construction of the degeneration via the multiplier ideal sheaves and then consider their geometric quantization.
Tomoyuki Hisamoto
Semi-classical Spectral Asymptotics of Toeplitz Operators on Strictly Pseudodonvex Domains
Abstract
On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension n we consider a Toeplitz operator \(T_R\) with symbol a Reeb-like vector field R near the boundary. We show that the kernel of a weighted spectral projection \(\chi (k^{-1}T_R)\), where \(\chi \) is a cut-off function with compact support in the positive real line, is a semi-classical Fourier integral operator with complex phase, hence admits a full asymptotic expansion as \(k\rightarrow +\infty \). More precisely, the restriction to the diagonal \(\chi (k^{-1}T_R)(x,x)\) decays at the rate \(O(k^{-\infty })\) in the interior and has an asymptotic expansion on the boundary with leading term of order \(k^{n+1}\) expressed in terms of the Levi form and the pairing of the contact form with the vector field R.
Chin-Yu Hsiao, George Marinescu
On a Concrete Realization of Simply Connected Complex Domains Admitting Homogeneous Kähler Metrics
Abstract
We show that any simply connected complex domain admitting homogeneous Kähler metrics is realized as a complex orbit in the Siegel-Jacobi domain under the action of a solvable Lie subgroup of the Jacobi group.
Hideyuki Ishi
The Asymptotic Behavior of the Bergman Kernel on Pseudoconvex Model Domains
Abstract
In this paper, we investigate the asymptotic behavior of the Bergman kernel at the boundary for some pseudoconvex model domains. This behavior can be described by the geometrical information of the Newton polyhedron of the defining function of the respective domains. We deal with not only the finite type cases but also some infinite type cases.
Joe Kamimoto
Bundle-Convexity and Kernel Asymptotics on a Class of Locally Pseudoconvex Domains
Abstract
A generalization of Grauert’s solution of the Levi problem for strongly pseudoconvex domains will be given. On a class of bounded locally pseudoconvex domains in complex manifolds, spaces of holomorphic sections of vector bundles are analyzed by the \(L^2\) method. Under appropriate positivity conditions on the curvature, a bundle-convexity theorem holds as a generalization of Grauert’s theorem. Based on it, function-theoretic properties of such domains will be discussed.
Takeo Ohsawa
The -Equation on the Hartogs Triangles in and
Abstract
This is a survey paper of the recent progress of the Cauchy-Riemann equation on the Hartogs triangles in \(\mathbb {C}^2\) and \({\mathbb{C}\mathbb{P}}^2\). In particular, \(L^2\) and \(W^1\) Dolbeault cohomology groups on the Hartogs triangles in \({\mathbb{C}\mathbb{P}}^2\) are investigated.
Mei-Chi Shaw
Dynamical Systems of p-Bergman Kernels
Abstract
In this paper, we consider the dynamical system of p-Bergman kernels. This is useful when we consider a projective manifold with intermediate Kodaira dimension.
Hajime Tsuji
Wehrl-Type Inequalities for Bergman Spaces on Domains in  and Completely Positive Maps
Abstract
We prove certain Wehrl type \(L^p\)-inequalities for the Bergman spaces on bounded domains D in \(\mathbb C^d\) and apply the result to bounded symmetric domains \(D=G/K\). We introduce also G-invariant completely positive trace preserving map \(A\rightarrow \mathcal T(A)\) from operators A on a weighted Bergman space \(H_\mu \) on \(D=G/K\) to operators \(\mathcal T(A)\) on another weighted Bergman space \(H_{\mu +\nu }\) and prove that the \(L^p\)-norm of Bergman functions \(f\in H_\mu \) can be obtained as limit of trace of \(\mathcal T({f\otimes f^*})\) as the weight \(\nu \rightarrow \infty \).
Genkai Zhang
Converse of Existence and Extension of Cohomology Classes
Abstract
The present paper grew out of my talk at Hayama conference in 2022. We give a survey of recent progress on a converse of \(L^2\) existence theorem and a criterion of Nakano positivity which is used to answer a problem of Lempert affirmatively. We also mention a recent result on the extension of cohomology classes which answers a problem of Cao–Demailly–Matsumura affirmatively, and an injectivity theorem as an application.
Xiangyu Zhou
Metadata
Title
The Bergman Kernel and Related Topics
Editors
Kengo Hirachi
Takeo Ohsawa
Shigeharu Takayama
Joe Kamimoto
Copyright Year
2024
Publisher
Springer Nature Singapore
Electronic ISBN
978-981-9995-06-6
Print ISBN
978-981-9995-05-9
DOI
https://doi.org/10.1007/978-981-99-9506-6

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