Preface
Acknowledgements
Contents
1 Introduction
1.1 Background and Motivation
1.2 Monopoles of GCK-Type
1.3 Previous Works on Monopoles and Algebraic Objects
1.3.1 SU(2)-Monopoles with Finite Energy on R3
1.3.2 The Correspondence due to Charbonneau and Hurtubise
1.3.3 Remark
1.4 Review of the Kobayashi-Hitchin Correspondences for λ-Flat Bundles
1.4.1 Harmonic Bundles and Their Underlying λ-Flat Bundles
1.4.2 Kobayashi-Hitchin Correspondences in the Smooth Case
1.4.3 Tame Harmonic Bundles and Regular Filtered λ-Flat Bundles
1.4.4 Wild Harmonic Bundles and Good Filtered λ-Flat Bundles
1.5 Equivariant Instantons and the Underlying Holomorphic Objects
1.5.1 Instantons and the Underlying Holomorphic Bundles
1.5.2 Instantons and Harmonic Bundles
1.5.3 Instantons and Monopoles
1.5.4 Instantons and Monopoles as Harmonic Bundles of Infinite Rank
1.5.4.1 Instantons as Harmonic Bundles of Infinite Rank
1.5.4.2 The Underlying λ-Flat Bundles of Infinite Rank
1.5.4.3 Monopoles as Harmonic Bundles of Infinite Rank
1.6 Difference Modules with Parabolic Structure
1.6.1 Difference Modules
1.6.2 Parabolic Structure of Difference Modules at Finite Place
1.6.3 Good Parabolic Structure at ∞
1.6.4 Parabolic Difference Modules
1.6.5 Degree and Stability Condition
1.6.6 Easy Examples of Stable Parabolic Difference Modules (1)
1.6.6.1 The Case Where (∞) Is Even
1.6.6.2 The Case Where (∞) Is Odd
1.6.7 Easy Examples of Stable Parabolic Difference Modules (2)
1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles
1.7.1 The Correspondence in the Case λ=0
1.7.1.1 Mini-complex Structure
1.7.1.2 Mini-holomorphic Bundles Associated with Monopoles
1.7.1.3 Dirac Type Singularity
1.7.1.4 Meromorphic Extension and Filtered Extension at Infinity
1.7.1.5 Kobayashi-Hitchin Correspondence in the Case λ=0
1.7.1.6 OM0Z(H0∞)-Modules and C(w)-Modules with an Automorphism
1.7.2 The Correspondences in the General Case
1.7.2.1 Preliminary Consideration
1.7.2.2 Mini-complex Structure Corresponding to the Twistor Parameter λ
1.7.2.3 Another Coordinate System and the Compactification of Mλ
1.7.2.4 Mini-holomorphic Bundles Associated with Monopoles
1.7.2.5 Meromorphic Extension and Filtered Extension at Infinity
1.7.2.6 Kobayashi-Hitchin Correspondence of Periodic Monopoles of GCK Type
1.7.2.7 Difference Modules and OMλZ (Hλ∞)-Modules
1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type
1.8.1 Setting
1.8.2 Decomposition of Mini-holomorphic Bundles
1.8.3 The Induced Higgs Bundles
1.8.3.1 Preliminary (1)
1.8.3.2 Preliminary (2)
1.8.3.3 The Induced Higgs Bundles
1.8.4 Asymptotic Orthogonality
1.8.5 Curvature Decay
1.8.6 The Filtered Extension in the Case λ=0
1.8.7 The Filtered Extension for General λ
1.8.7.1 Ramified Covering Space
1.8.7.2 Approximation
1.8.7.3 Formal Completion of Asymptotic Harmonic Bundles at Infinity
1.8.7.4 The Formal Structure of PhEλ at Infinity
2 Preliminaries
2.1 Outline of This Chapter
2.2 Mini-Complex Structures on 3-Manifolds
2.2.1 Mini-Holomorphic Functions on RC
2.2.2 Mini-Complex Structure on Three-Dimensional Manifolds
2.2.3 Tangent Bundles
2.2.4 Cotangent Bundles
2.2.5 Meromorphic Functions
2.3 Mini-Holomorphic Bundles
2.3.1 Mini-Holomorphic Bundles
2.3.2 Metrics and the Induced Operators
2.3.3 Splittings
2.3.4 Scattering Maps
2.3.5 Dirac Type Singularity of Mini-Holomorphic Bundles
2.3.6 Kronheimer Resolution of Dirac Type Singularity
2.3.7 Precise Description of Dirac Type Singularities
2.3.8 Subbundles and Quotient Bundles
2.3.9 Basic Functoriality
2.4 Monopoles
2.4.1 Monopoles and Mini-Holomorphic Bundles
2.4.2 Euclidean Monopoles
2.4.3 Dirac Type Singularity
2.4.3.1 Dirac Monopoles (Examples)
2.4.4 Basic Functoriality
2.5 Dimensional Reduction from 4D to 3D
2.5.1 Instantons Induced by Monopoles
2.5.2 Holomorphic Bundles and Mini-Holomorphic Bundles
2.6 Dimensional Reduction from 3D to 2D
2.6.1 Monopoles Induced by Harmonic Bundles
2.6.2 Mini-Holomorphic Bundles Induced by Holomorphic Bundles with a Higgs Field
2.6.3 Mini-Holomorphic Sections and Monodromy
2.6.4 Appendix: Monopoles as Harmonic Bundles of Infinite Rank
2.7 Twistor Families of Mini-Complex Structures on RC and (R/TZ)C
2.7.1 Preliminary
2.7.2 Spaces
2.7.3 Twistor Family of Complex Structures
2.7.4 Family of Mini-Complex Structures
2.7.5 The Mini-Complex Coordinate System (t0,β0)
2.7.6 The Mini-Complex Coordinate System (t1,β1)
2.7.7 Coordinate Change
2.7.8 Compactification
2.7.9 Mini-Holomorphic Bundles Associated with Monopoles
2.7.9.1 Compatibility with the Dimensional Reduction from 4D to 3D
2.8 OMλ-Modules and λ-Connections
2.8.1 Dimensional Reduction from OMλ-Modules to λ-Flat Bundles
2.8.1.1 Setting
2.8.1.2 Some Vector Fields and Forms
2.8.1.3 A General Equivalence
2.8.1.4 Mini-Holomorphic Bundles and Flat λ-Connections
2.8.1.5 λ-Flat Bundles of Infinite Rank
2.8.1.6 Remark
2.8.2 Comparison of Some Induced Operators
2.8.2.1 Comparison of Mini-Holomorphic Bundles Induced by Harmonic Bundles
2.8.3 OMλ-Modules and λ-Connections
2.8.3.1 Setting
2.8.3.2 A General Equivalence
2.8.3.3 Mini-Holomorphic Bundles and Meromorphic Flat λ-Connections
2.8.3.4 Another Description of the Construction
2.9 Curvatures of Mini-Holomorphic Bundles with Metric on Mλ
2.9.1 Contraction of Curvature and Analytic Degree
2.9.2 Chern-Weil Formula
2.9.3 Another Description of G(h)
2.9.4 Change of Metrics
2.9.5 Relation with λ-Connections
2.9.5.1 λ-Flat Bundles of Infinite Rank with a Harmonic Metric
2.9.5.2 Remark
2.9.6 Dimensional Reduction of Kronheimer
2.9.7 Appendix: Ambiguity of the Choice of a Splitting
2.10 Difference Modules and OMλZ(Hλ∞)-Modules
2.10.1 Difference Modules with Parabolic Structure at Finite Place
2.10.2 Construction of Difference Modules from OMλZ(Hλ∞)-Modules
2.10.3 Construction of OMλZ(Hλ)-Modules from Difference Modules
2.10.4 Appendix: Mellin Transform and Parabolic Structures at Finite Place
2.10.4.1 Mellin Transform
2.10.4.2 Algebraic Nahm Transform for Filtered λ-Flat Bundles (Special Case)
2.11 Filtered Prolongation of Acceptable Bundles
2.11.1 Filtered Bundles on a Neighbourhood of 0 in C
2.11.1.1 G-Equivariance
2.11.1.2 Subbundles, Quotient and Splitting
2.11.1.3 Basic Functoriality
2.11.1.4 Pull Back
2.11.1.5 Push-Forward
2.11.1.6 Descent
2.11.1.7 Some Examples
2.11.2 Acceptable Bundles on a Punctured Disc
2.11.2.1 Basic Functoriality
2.11.2.2 Pull Back and Descent
2.11.3 Global Case
2.11.3.1 Filtered Bundles
2.11.3.2 Acceptable Bundles
3 Formal Difference Modules and Good Parabolic Structure
3.1 Outline of This Chapter
3.2 Formal Difference Modules
3.2.1 Formal Difference Modules of Level ≤1
3.2.2 Formal Difference Modules of Pure Slope
3.2.3 Slope Decomposition of Formal Difference Modules
3.3 Good Filtered Bundles of Formal Difference Modules
3.3.1 Filtered Bundles over C((yq-1))-Modules
3.3.1.1 G-Equivariance
3.3.1.2 Submodules, Quotient Modules and Splittings
3.3.1.3 Basic Functoriality
3.3.1.4 Pull Back
3.3.1.5 Push-Forward
3.3.1.6 Descent
3.3.2 Good Filtered Bundles over Formal Difference Modules
3.3.3 The Induced Endomorphisms on the Graded Pieces
3.4 Geometrization of Formal Difference Modules
3.4.1 Ringed Spaces
3.4.2 Some Formal Spaces
3.4.3 Difference Modules and OH∞,q(H∞,q)-Modules
3.4.4 Lattices and the Induced Local Systems
3.5 Filtered Bundles in the Formal Case
3.5.1 Pull Back and Descent of OH∞,p(H∞,p)-Modules
3.5.2 Filtered Bundles
3.5.2.1 Subbundles and Quotient Bundles
3.5.2.2 Basic Functoriality
3.5.2.3 Pull Back
3.5.2.4 Push-Forward
3.5.2.5 Descent
3.5.3 Basic Filtered Objects with Pure Slope
3.5.4 Good Filtered Bundles over OH∞,q(H∞,q)-Modules with Level ≤1
3.5.5 Good Filtered Bundles over OH∞,q(H∞,q)-Modules
3.5.5.1 An Equivalence
3.5.5.2 Some Properties
3.5.6 Global Lattices on the Covering Space
3.5.7 Local Lattices
3.5.8 Complement for Good Filtered Bundles with Level ≤1
3.6 Formal Difference Modules of Level ≤1 and Formal λ-Connections
3.6.1 Formal λ-Connections
3.6.2 Some Sheaves of Algebras on H∞,q
3.6.3 From Formal λ-Connections to Formal Difference Modules
3.6.4 Equivalence
3.6.4.1 Simpler Cases of Proposition 3.6.8
3.6.5 Example 1
3.6.5.1
3.6.5.2
3.6.6 Example 2
3.6.6.1
3.6.6.2
3.6.7 Comparison of Good Filtered Bundles
3.6.8 Comparison of the Associated Graded Pieces
3.6.9 Some Functoriality
3.7 Appendix: Pull Back and Descent in the R-Direction
3.7.1 Examples
4 Filtered Bundles
4.1 Outline of This Chapter
4.2 Filtered Bundles in the Global Case
4.2.1 Subbundles and Quotient Bundles
4.2.2 Degree and Slope
4.2.3 Stability Condition
4.2.4 Good Filtered Bundles of Dirac Type and Parabolic Difference Modules
4.2.4.1 Polystable Parabolic Difference Modules
4.2.4.2 Equivalence
4.3 Filtered Bundles on Ramified Coverings
4.3.1 The Case λ=0
4.3.2 Ramified Coverings for General λ
4.3.3 Filtered Bundles
4.3.4 Local Lattices and the Weight Filtration on the Graded Pieces
4.3.5 Convenient Frame
4.4 Hermitian Metrics and Filtrations
4.4.1 Prolongation by Growth Conditions
4.4.2 Norm Estimate for Good Filtered Bundles
4.4.3 Strong Adaptedness
4.5 Comparison with λ-Connections
4.5.1 Some Sheaves on Yqλcov and Yqλ
4.5.2 The Induced OBqλ-Modules from λ-Connections
4.5.3 Norm Estimates for λ-Connections
4.5.4 Comparison of the Norm Estimates
4.5.5 Non-integrable Case
4.5.5.1 A General Construction
4.5.5.2 Comparison with the Construction in Sect.4.5.2
4.5.5.3 Comparison with the Construction in the Formal Case
5 Basic Examples of Monopoles Around Infinity
5.1 Examples of Monopoles with Pure Slope /p
5.1.1 Equivariance with Respect to the Z-Action
5.1.2 The Underlying Mini-holomorphic Bundle atλ=0
5.1.3 The Underlying Mini-holomorphic Bundle at General λ
5.1.4 Complex Structures
5.1.5 The Induced Instanton and the Underlying Holomorphic Bundle
5.1.6 C∞-Frame
5.1.7 Proof of Proposition 5.1.4
5.2 Examples of Monopoles Induced by wild Harmonic Bundles
5.2.1 The Underlying Mini-holomorphic Bundle atλ=0
5.2.2 The Associated λ-Connection and the Induced Mini-holomorphic Bundle at λ
5.2.3 Special Case
5.2.4 Notation
5.3 Examples of Monopoles Induced by Tame Harmonic Bundles (1)
5.3.1 The Mini-holomorphic Bundle at λ=0
5.3.2 The Mini-holomorphic Bundle at λ
5.4 Examples of Monopoles Induced by Tame Harmonic Bundles (2)
5.4.1 Case of λ=0
5.4.2 Case of λ≠0
6 Asymptotic Behaviour of Periodic Monopoles Around Infinity
6.1 Outline of This Chapter
6.2 Preliminary
6.2.1 Notation
6.2.2 Decomposition of Holomorphic Bundles with an Automorphism
6.2.3 Some Basic Mini-Holomorphic Bundles on B0q(R)
6.2.3.1 Basic Rank One Bundles
6.2.3.2 Mini-Holomorphic Bundles Induced by Holomorphic Bundles with Automorphism
6.2.4 Decomposition of Mini-Holomorphic Bundles
6.3 Estimate of Periodic Monopoles Around Infinity
6.3.1 Setting
6.3.2 Asymptotic Orthogonality of the Decomposition (6.5)
6.3.3 Eigen Decomposition in the Level 1
6.3.4 Asymptotic Harmonic Bundles
6.3.5 Curvature
6.3.6 Another Equivalent Decay Condition
6.4 Connections and Orthogonal Decompositions
6.4.1 Statement
6.4.2 Preliminary
6.4.3 Step 1
6.4.4 Step 2
6.5 Some Lemmas from Linear Algebra
6.5.1 Eigenvalues
6.5.2 Almost Commuting Hermitian Matrix and Anti-Hermitian Matrix
6.5.3 Decomposition of Finite Tuples in Metric Spaces (Appendix)
6.6 Vector Bundles with a Connection on a Circle (I)
6.6.1 Statement
6.6.2 Preliminary
6.6.3 A Decomposition of Function Spaces
6.6.4 Gauge Transformation
6.6.5 Proof of Proposition 6.6.1
6.7 Vector Bundles with a Connection on a Circle (II)
6.7.1 Additional Assumption on the Eigenvalues of the Monodromy
6.7.2 Gauge Transformations
6.7.3 Comparison of the Decompositions
6.8 Proof of Theorem 6.3.4
7 The Filtered Bundles Associated with Periodic Monopoles
7.1 Notation
7.1.1 Some Spaces and Morphisms
7.1.2 Neighbourhoods of Infinity
7.1.3 Norm of Differential Forms
7.2 Meromorphic Prolongation
7.2.1 Statements
7.2.2 Proof
7.3 Filtered Prolongation
7.3.1 Statement
7.3.2 Refined Statement
7.3.2.1 The Filtered Bundles Associated with the Basic Monopoles of Rank One
7.3.2.2 The Filtered Bundles Associated with the Asymptotic Harmonic Bundles
7.3.2.3 Refined Statement
7.3.3 Norm Estimate
7.3.4 Step 0
7.3.5 Step 1
7.3.6 Step 2
7.3.7 Step 3
7.3.8 Step 4
7.3.9 Step 5
7.3.10 Step 6
7.3.11 Proof of Proposition 7.3.5
7.4 Strong Adaptedness and the GCK-Condition
7.4.1 Statements
7.4.2 Some Estimates for Tame Harmonic Bundles (1)
7.4.2.1 Appendix
7.4.3 Some Estimates for Tame Harmonic Bundles (2)
7.4.4 λ-Connections
7.4.5 Proof of Proposition 7.4.3
7.4.6 The Strong Adaptedness and the Norm Estimate
7.4.7 Proof of Proposition 7.4.1
7.5 Some Functoriality
8 Global Periodic Monopoles of Rank One
8.1 Preliminary
8.1.1 Ahlfors Type Lemma
8.1.2 Poisson Equation (1)
8.1.2.1 Appendix
8.1.3 Poisson Equation (2)
8.1.4 Subharmonic Functions
8.2 Global Periodic Monopoles of Rank One (1)
8.2.1 Reformulation
8.3 Global Periodic Monopoles of Rank One (2)
8.3.1 Construction of Mini-Holomorphic Bundles
8.3.2 Good Filtered Bundles
8.3.3 Monopoles
8.4 Global Periodic Monopoles of Rank One (3)
9 Global Periodic Monopoles and Filtered Difference Modules
9.1 Statements
9.2 Preliminary
9.2.1 Ambient Good Filtered Bundles with Appropriate Metric
9.2.2 Degree of Filtered Subbundles
9.2.3 Analytic Degree of Subbundles
9.3 Good Filtered Bundles Associated with Monopoles of GCK-Type
9.4 Construction of Monopoles
9.5 Smooth Parabolic 0-Difference Modules of Rank 2
9.5.1 Smooth Parabolic 0-Difference Modules
9.5.2 Rank One Case
9.5.3 Filtered Torsion-Free Sheaves of Rank One on an Integral Scheme
9.5.3.1 Degree of Torsion-Free Sheaves on Σ(Q,γ)
9.5.3.2 Filtered Torsion-Free Sheaf of Rank One on (Σ(Q,γ),Σ∞(Q,γ))
9.5.3.3 The Induced Smooth Parabolic Difference Modules of Rank 2
9.5.4 Description of Parabolic Difference Modules of Rank 2
Parabolic Torsion Free Sheaves and Stable Parabolic Difference Modules
Exceptional Case
10 Asymptotic Harmonic Bundles and Asymptotic Doubly Periodic Instantons (Appendix)
10.1 Formal λ-Connections Associated with Asymptotic Harmonic Bundles
10.1.1 Asymptotic Harmonic Bundles
10.1.2 Simpson's Main Estimate
10.1.3 The Associated Filtered Bundles and Formal λ-Connections
10.1.4 Formal Good Filtered λ-Flat Bundles
10.1.5 Residues and KMS-Structure
10.1.6 Norm Estimate
10.1.7 Comparison of KMS-Structures
10.1.8 Appendix
10.2 Family of Vector Bundles on Torus with Small Curvature
10.2.1 Preliminary
10.2.2 Partially Almost Holomorphic Frames
10.2.3 Spectra
10.2.4 Additional Assumption on Spectra
10.2.5 Spaces of Functions
10.2.6 Some Estimates
10.3 Estimates for Asymptotic Doubly Periodic Instantons
10.3.1 Setting
10.3.2 Decomposition
10.3.3 Estimates
References
Index