The Bergman Kernel and Related Topics: Hayama Symposium on SCV XXIII, Kanagawa, Japan, July 2022 (Springer Proceedings in Mathematics & Statistics, 447)

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 L 2 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, L p 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. Preface Contents Concavity Property of Minimal L2 Integrals with Lebesgue Measurable Gain VII–Negligible Weights 1 Introduction 1.1 Main Results 1.2 Applications 2 Preparations I: Minimal L2 Integrals 2.1 Minimal L2 Integrals on Weakly Pseudoconvex Kähler Manifolds 2.2 The Sufficient and Necessary Conditions of the Concavity of G(h Superscript -1(r)) Degenerating to Linearity 2.3 Basic Properties of the Green Functions 2.4 Other Lemmas 3 Preparations II: Multiplier Ideal Sheaves and Optimal L2 Extensions 3.1 Multiplier Ideal Sheaves 3.2 Optimal Jet L2 Extensions 3.3 Other Calculations 4 Proofs of Theorem 1, Propositions 1 and 2 4.1 Proof of Theorem 1 4.2 Proof of Proposition 1 4.3 Proof of Proposition 2 5 Proofs of Theorems 2, 3, 4 and Proposition 3 5.1 Proof of Theorem 2 5.2 Proof of Theorem 3 5.3 Proof of Theorem 4 5.4 Proof of Proposition 3 6 Proofs of Theorems 5 and 6 6.1 Proof of Theorem 5 6.2 Proof of Theorem 6 7 Proofs of Theorems 7 and 8 and 9 7.1 Proofs of Theorem 7 and Remark 10 7.2 Proofs of Theorem 8 and Remark 11 7.3 Proofs of Theorem 9 and Remark 12 References M-harmonic Szegö Kernel on the Ball 1 Introduction 2 Formula for the Szegö Kernel 3 Boundary Singularity 4 Concluding Remarks References Some Aspects of the p-Bergman Theory 1 Introduction and Preliminaries 2 Regularity of the p-Bergman Kernel 3 Geometric Properties of the p-Bergman Metric 4 Analysis of the p-Bergman Space References On Semiclassical Ohsawa-Takegoshi Extension Theorem 1 Introduction 2 Holomorphic Jet Extensions in the Semiclassical Setting 3 Towers of Embeddings and Transitivity of Extensions 4 On the Metric Structure of the Section Ring 5 Submultiplicative Norms on Section Rings References Balanced Metrics for Extremal Kähler Metrics and Fano Manifolds 1 Introduction 2 Balanced Metrics for Extremal Metrics 3 Anticanonically Balanced Metrics 4 Geodesically Convex Functions on Complete Riemannian Manifolds References Unbounded Operators on the Segal–Bargmann Space 1 Introduction 2 The -Complex 3 Properties of the Graph Norm 4 Compact Subsets 5 The partial-Complex on Weighted Bergman Spaces on Hermitian Manifolds 5.1 Holomorphic Vector Fields 5.2 Example (a) 5.3 Example (b) 5.4 Real Holomorphic Vector Fields and Holomorphic Torsion 5.5 Conformally Kähler Metrics 6 The Generalized partial-Complex 6.1 Basic Estimates 6.2 Commutator Terms as a Sum of Squared Norms 6.3 Hermitian Forms 6.4 Further Examples 6.5 Compactness References Asymptotic Construction of the Optimal Degeneration for a Fano Manifold 1 Geometric Flow and Optimal Degeneration 1.1 Kähler-Ricci Flow 1.2 Optimal Degeneration 2 Construction via the Multiplier Ideal Sheaves 2.1 Space of Kähler Metrics 2.2 Multiplier Ideal on the Product Space 3 Quantization 3.1 Geometric Quantization 3.2 Quantization of the Flow 3.3 Quantized Optimal Degeneration References Semi-classical Spectral Asymptotics of Toeplitz Operators on Strictly Pseudodonvex Domains 1 Introduction 2 Preliminaries 2.1 Notions from Microlocal and Semi-classical Analysis 2.2 Set Up of Complex Manifolds with Smooth Boundary 3 The Toeplitz Operator TR 4 Asymptotic Expansion of χk(TR) References On a Concrete Realization of Simply Connected Complex Domains Admitting Homogeneous Kähler Metrics 1 Introduction 2 The Siegel Upper Half Plane 3 Realization of Bounded Homogeneous Domains 4 The Siegel-Jacobi Domain 5 Realization of Simply Connected Complex Domains with Homogeneous Kähler Metrics References The Asymptotic Behavior of the Bergman Kernel on Pseudoconvex Model Domains 1 Introduction 2 Main Results 2.1 Newton Data 2.2 Main Results 3 Integral Formula of the Bergman Kernel 4 Localization Lemma 5 The hat mathcal E-Condition 5.1 Newton Polyhedra in the Real Case 5.2 The -Condition in the Real Case 5.3 The hat mathcal E-Condition in the Complex Case 6 Behavior of tildemathcalBF(ρ) in the hat mathcal E-Case 7 Proof of Theorem 2 8 Asymptotic Analysis of Some Laplace Integrals 8.1 Newton Data in the Real Case 8.2 Meromorphic Extension of Some Local Zeta Functions 8.3 Asymptotic Behavior of Some Laplace Integrals 8.4 Proof of Lemma 3 9 Appendix 9.1 Asymptotic Expansion in (2) 9.2 Convex Geometry 9.3 Newton Nondegeneracy Condition References Bundle-Convexity and Kernel Asymptotics on a Class of Locally Pseudoconvex Domains 1 Introduction 2 Preliminaries and the Proof of Theorem3 3 Complete Kähler Metrics on Punctured Domains with a Control of Potentials 4 Proof of Theorem2 5 Application to the Kernel Asymptotics References The dbar-Equation on the Hartogs Triangles in double struck upper C squaredmathbbC2 and double struck upper C upper P backslash slash squaredmathbbCP2 1 Introduction 2 Boundary Regularity for dbar on the Hartogs Triangle in mathbbC2 3 L2 Theory for dbar on Lipschitz Pseudoconvex Domains in mathbbCPn 3.1 L2 Theory of dbar for (0,q)-Forms 3.2 Bounded Plurisubharmonic Exhaustion Functions 4 Hartogs Triangles in CP2 5 Levi-Flat Hypersurfaces in mathbbCPSuperscript n References Dynamical Systems of p-Bergman Kernels 1 Introduction 2 Some Invariant Volume Forms 2.1 Bergman Volume Forms 2.2 Kähler-Einstein Volume Forms 2.3 Supercanonical Volume Form 2.4 Extremal Measures 2.5 Equivalence of the Extremal Maasure and the Supercanonical Measure 3 Dynamical System of p-Bergman Kernels (The Case of Canonically Polarized Manifolds) 3.1 p-Bergman Kernels 3.2 Dynamical System of p-Bergman Kernels 4 Canonical Measures and p-Bergman Kernels 4.1 Hodge Bundles 4.2 Canonical Measure 4.3 Dynamical System of p-Bergman Kernels (The Case of Compact Kähler Manifolds with Kodaira Dimension 4.4 Dynamical System of Bergman Kernels for Twisted Kähler-Einstein Volume Forms 4.5 Proof of Theorem 9 4.6 Dynamical Systems of Extremal Measures References Wehrl-Type Inequalities for Bergman Spaces on Domains in mathbbCd and Completely Positive Maps 1 Introduction 2 Wehrl-Type Inequality for Bounded Domains 2.1 Bergman Spaces on Bounded Domains D 2.2 Bounded Symmetric Domains 3 Wehrl-Type Inequalities Related to Berezin Transform 4 Tensor Product of Bergman Spaces on the Unit Ball and Construction of Completely Positive Maps 4.1 Bergman Spaces of Symmetric Tensor-Valued Holomorphic Functions 4.2 Irreducible Decomposition of Tensor Product H Subscript mu Baseline circled times upper H Subscript nu and SU(d, 1)-Invariant Completely Positive Maps 4.3 Limit of the Trace trace script upper T 0 Superscript p Baseline left parenthesis f circled times f Superscript asterisk Baseline right parenthesis References Converse of L2 Existence and Extension of Cohomology Classes 1 Background and Motivation 1.1 Plurisubharmonic Functions 1.2 Hörmander's L2 Existence Theorem 1.3 Multiplier Ideal Sheaves 1.4 Holomorphic Vector Bundles 2 Some Known Results 2.1 An Optimal L2 Extension Theorem 2.2 Demailly's Strong Openness Conjecture 3 Some Recent Progress 3.1 Converse of L2 Existence Theorem 3.2 Multiplier Submodule Sheaves 3.3 Injectivity Theorem and Extension of Cohomology Classes References