Complex Analysis and Geometry

Name: Complex Analysis and Geometry | KSCV10, Gyeongju, Korea, August 2014
Brand: Springer
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KSCV10, Gyeongju, Korea, August 2014

Filippo Bracci Jisoo Byun Hervé Gaussier Kengo Hirachi Kang-Tae Kim Nikolay Shcherbina(Editor)

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Published on 5. August 2015

IX, 377 pages

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978-4-431-55744-9 (ISBN)

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Content

Intro
Preface
Contents
Fatou Flowers and Parabolic Curves
1 The Original Leau-Fatou Flower Theorem
2 Écalle-Hakim Theory
3 Blow-Ups, Indices and Fatou Flowers
4 Parabolic Domains
5 The Formal Infinitesimal Generator
6 Homogeneous Vector Fields and Geodesics
7 Other Systems with Parabolic Behavior
References
A CR Proof for a Global Estimate of the Diederich--Fornaess Index of Levi-Flat Real Hypersurfaces
1 Introduction
2 Preliminaries
2.1 Levi-Flat CR Manifold
2.2 Description on Foliated Charts
2.3 The Diederich--Fornaess Index
3 The Proof of Lemma 1 and a Remark
3.1 Proof of Lemma 1
3.2 The Approach of Bejancu and Deshmukh
3.3 Open Question
References
Unbounded Pseudoconvex Domains in mathbbCn and Their Invariant Metrics
1 Some Problems
2 A Remark on Hahn-Lu Comparision Theorem
3 A Technique for Positivity and Completeness of Bergman Metric
4 Examples
4.1 The Kohn-Nirenberg and Fornaess Domains
4.2 Other Domains
5 Remarks and More Questions
References
Abstract Basins of Attraction
1 The Construction of Abstract Basins
2 Bedford's Conjecture
3 Loewner's Theory
4 Models
References
Invertible Dynamics on Blow-ups of mathbbPk
1 Introduction
2 Rational Surfaces
3 Degree Complexity, or Dynamical Degree
4 Some Rational Surface Automorphisms with d&1
5 Connection Between Dynamical Degree and Length Growth: A Graphic Example
6 Heuristic Picture: Dynamical Complexity Versus Degree Complexity
7 What Are the Compact Surfaces X Which Carry an finAut(X) with d1(f)&1?
8 Pseudo-automorphisms
9 Intermediate Dynamical Degrees
10 Existence of Pseudo-automorphisms of Blowups of mathbbPk
11 Cohomological Hyperbolicity
References
On Nazarov's Complex Analytic Approach to the Mahler Conjecture and the Bourgain-Milman Inequality
1 Introduction
2 Equivalent SCV Formulation
3 The Upper Bound
4 The Lower Bound
References
A Survey on Bergman Completeness
1 Introduction
2 Bergman Completeness for Domains in mathbbCn
3 Bergman Completeness for Open Complex Manifolds
References
Structure Theorems for Compact Kähler Manifolds with Nef Anticanonical Bundles
1 Introduction and Preliminaries
2 Criterion for Rational Connectedness
3 A Generalized Holonomy Principle
4 Structure Theorem for Compact Kähler Manifolds with KX-1 Semipositive
5 Compact Kähler Manifolds with Nef Anticanonical Bundles
References
An Estimate for the Squeezing Function and Estimates of Invariant Metrics
1 Introduction
2 Proof of Theorem 1.2
3 Proof of Theorem 1.1
3.1 Estimate (a)
3.2 Estimate (b)
4 An Example
References
Classification of Proper Holomorphic Mappings Between Generalized Pseudoellipsoids of Different Dimensions
1 The Research of Proper Holomorphic Mappings
1.1 Extension Problem
1.2 Proper Holomorphic Mappings and Biholomorphic Mappings
1.3 Determination of Proper Holomorphic Mappings
2 Main Theorem and History of Gap Theorems
2.1 Main Theorem
2.2 History of Gap Theorems
3 Expansion of a Mapping
4 Main Result (General Case)
5 Main Result (Low Dimensional Case)
6 Some Conjectures
6.1 The Number of Blocks
6.2 Dimension in the Blocks
References
Bergman Kernel Asymptotics and a Pure Analytic Proof of the Kodaira Embedding Theorem
1 Introduction and Set up
1.1 Set up
2 Terminology in Semi-classical Analysis
3 Asymptotic Expansion of Bergman Kernel
3.1 Big Line Bundles and Shiffman Conjecture
4 A Bergman Kernel Proof of the Kodaira Embedding Theorem
References
On the Density and the Volume Density Property
1 Introduction
2 Main Feature of Density and Volume Density Property and Some Recent Applications
3 The Criterion and Volume Density of Products
4 Open Problems
References
On Meromorphic Continuation of Local Zeta Functions
1 Introduction
2 Preliminaries
2.1 Polyhedra
2.2 Newton Polyhedra
2.3 The ?-Part
2.4 Adapted Coordinates and Superadapted Coordinates
3 Main Results
References
Themes on Non-analytic Singularities of Plurisubharmonic Functions
1 Introduction
2 Multiplier Ideal Sheaves and the Openness Conjecture
2.1 Basic Notions and Some Examples
2.2 Multiplier Ideal Sheaves and the Openness Conjecture
3 Demailly Approximation of Psh Singularity
4 Resolution of Psh Singularity
5 Analytic Adjoint Ideal Sheaves
References
Proper Holomorphic Maps Between Bounded Symmetric Domains
1 Introduction
2 Geometry of Boundary Components
2.1 Adapted Frames
2.2 The Connection Matrix Form
2.3 The Tangent Space of Sp,q,r
2.4 Structure Identities
3 Differential Equations for CR Maps
References
Characterizations of Strongly Pseudoconvex Models in Almost Complex and CR Geometries
1 Introduction
1.1 The Characterization of the Unit Ball
1.2 The Characterization of the Unit Sphere
1.3 Generalizations
2 The Wong-Rosay Theorem in the Almost Complex Manifold
2.1 Automorphisms of the Pseudo-Siegel Domain
2.2 The Scaling Method in Almost Complex Manifold
2.3 Bounded Realization of the Pseudo-Siegel Domain
3 Schoen's Theorem in Almost CR Manifolds
3.1 Pseudo-hermitian Structure Equations
3.2 Pseudo-hermitian Equivalence Problem
3.3 Sub-Riemannian Yamabe Problem
3.4 A Generalization of Schoen's Theorem
References
Compact Smooth but Non-complex Complements of Complete Kähler Manifolds
1 Introduction
2 Compact Counterexamples
3 Remarks
References
Injectivity Theorems with Multiplier Ideal Sheaves and Their Applications
1 Introduction
1.1 Analytic Versions of the Injectivity Theorem
1.2 Applications to the Vanishing Theorem
1.3 Applications to the Extension Theorem
2 Proof of the Main Results
2.1 Proof of Theorem 2.1
2.2 Proof of Theorem 1.3
2.3 Proof of Theorem 1.4
3 Open Problems
References
Amoebas of Cuspidal Strata for Classical Discriminant
1 The Contour of Amoeba and the Logarithmic Gauss Map
2 Cuspidal Strata for Classical Discriminant
3 Amoebas of Reduced Cuspidal Strata for Classical Discriminant
References
A Remark on Hörmander's Isomorphism
1 Introduction
2 An L2 Isomorphism Theorem
3 Extension and Isomorphism on Compact Manifolds
4 Additional Remarks
References
The Julia-Wolff-Carathéodory Theorem and Its Generalizations
1 The Classical Julia-Wolff-Carathéodory Theorem
2 Generalizations to Several Variables
3 Julia-Wolff-Carathéodory Theorem for Infinitesimal Generators
References
A Brief Survey on Local Holomorphic Dynamics in Higher Dimensions
1 Introduction
2 The First-Level Invariants
2.1 The Multipliers
2.2 Multi-resonance
2.3 Diagonalization
3 The Second-Level Invariants
3.1 The Order and Characteristic Directions
3.2 The Director and Residual Index
3.3 The Non-dicritical Order
4 The Third-Level Invariants
4.1 Essentially Non-degenerate
4.2 Non-dynamically-Separating
References
L2-Serre Duality on Singular Complex Spaces and Applications
1 Introduction
2 L2-Theory for the overline-Operator on Singular Spaces
3 Two overline-Complexes on Singular Complex Spaces
4 L2-Serre Duality
5 Hartogs' Extension Theorem
6 Rational Singularities
7 mathcalA-sheaf duality
References
Proper Holomorphic Maps Between Bounded Symmetric Domains
1 Proper Holomorphic Maps Between the Unit Balls
2 Rigidity of Proper Holomorphic Maps Between Bounded Symmetric Domains
3 Other Proper Holomorphic Maps Between Bounded Symmetric Domains of Type I
References
Some Dynamical Systems of Extremal Measures
1 Introduction
1.1 Bergman Volume Forms
1.2 Kähler-Einstein Volume Forms
1.3 Dynamical System of Bergman Kernels
1.4 Supercanonical Volume Form
1.5 Extremal Measures
1.6 Relation Between Extremal Measures and Supercanonical Measures
2 Dynamical Systems of Extremal Measures
2.1 Dynamical System of Extremal Measure
2.2 Proof of Theorem 2.1
3 Dynamical System of Extremal Measures on Compact Kähler manifolds
3.1 Abundance of Canonical Line Bundle
3.2 Twisted Kähler-Einstein Currents and Canonical Measures
3.3 Dynamical Systems of Extremal Measures on a Compact Kähler Manifold
References
On Representative Domains and Cartan's Theorem
1 Introduction
2 Preliminaries
3 Cartan Theorem Revisited
4 Open Problems
References
On Curvature Estimates of Bounded Domains
1 Introduction
2 Intrinsic Derivatives Induced by the Bergman Kernel
3 Lu's Estimates of the Bergman Curvatures
3.1 Lower and Upper Bounds of the Bergman Curvatures
4 Uniform Estimates of the Bergman Curvatures
5 Concluding Remarks and Open Questions
References
Some Problems
1 Worm
2 Nirenberg Problem
3 The bar-problem
4 Bergman Space
5 Polynomial Convexity
6 Complex Dynamics/Real Dynamics
7 Fatou-Bieberbach Domains
7.1 Definition of Fatou-Bieberbach (FB) Domains
7.2 Standard Construction of Fatou-Bieberbach Domains
8 Convexity
9 Unbounded Domains
10 Automorphism Groups
11 CR Manifolds and CR Vector Fields
12 Semicontinuity of Automorphism Group
13 The Scaling Methods
14 Curvature
References

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Complex Analysis and Geometry

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