Introduction to Stokes Structures
Published on 3. October 2012
XIV, 249 pages
978-3-642-31695-1 (ISBN)
Description
This research monograph provides a geometric description of holonomic differential systems in one or more variables. Stokes matrices form the extended monodromy data for a linear differential equation of one complex variable near an irregular singular point. The present volume presents the approach in terms of Stokes filtrations. For linear differential equations on a Riemann surface, it also develops the related notion of a Stokes-perverse sheaf.
This point of view is generalized to holonomic systems of linear differential equations in the complex domain, and a general Riemann-Hilbert correspondence is proved for vector bundles with meromorphic connections on a complex manifold. Applications to the distributions solutions to such systems are also discussed, and various operations on Stokes-filtered local systems are analyzed.
More details
Series
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Product notice
Reflowable
Illustrations
XIV, 249 p. 14 illus., 1 illus. in color.
File size
3,10 MB
ISBN-13
978-3-642-31695-1 (9783642316951)
DOI
10.1007/978-3-642-31695-1
Schweitzer Classification
Other editions
Additional editions
Content
- Intro
- Introduction to Stokes Structures
- Contents
- Introduction
- 1 I-Filtrations
- 1.1 Introduction
- 1.2 Étalé Spaces of Sheaves
- 1.3 Étalé Spaces of Sheaves of Ordered Abelian Groups
- 1.4 The Category of Pre-I-Filtrations
- 1.5 Traces for Étale Maps
- 1.6 Pre-I-Filtrations of Sheaves
- 1.7 I-Filtrations of Sheaves (the Hausdorff Case)
- 1.8 I-Filtered Local Systems (the Hausdorff Case)
- 1.9 I-Filtered Local Systems (the Stratified Hausdorff Case)
- 1.10 Comments
- Part I Dimension One
- 2 Stokes-Filtered Local Systems in Dimension One
- 2.1 Introduction
- 2.2 Non-ramified Stokes-Filtered Local Systems
- 2.3 Pull-Back and Push-Forward
- 2.4 Stokes Filtrations on Local Systems
- 2.5 Extension of Scalars
- 2.6 Stokes-Filtered Local Systems and Stokes Data
- 3 Abelianity and Strictness
- 3.1 Introduction
- 3.2 Strictness and Abelianity
- 3.3 Level Structure of a Stokes-Filtered Local System
- 3.4 Proof of Theorem 3.5
- 3.5 Proof of the Stability by Extension
- 3.6 More on the Level Structure
- 3.7 Comments
- 4 Stokes-Perverse Sheaves on Riemann Surfaces
- 4.1 Introduction
- 4.2 The Setting
- 4.3 The Category of Stokes-C-Constructible Sheaves on X
- 4.4 Derived Categories and Duality
- 4.5 The Category of Stokes-Perverse Sheaves on X
- 4.6 Direct Image to X
- 4.7 Stokes-Perverse Sheaves on X
- 4.8 Associated Perverse Sheaf on X
- 5 The Riemann-Hilbert Correspondence for Holonomic D-Modules on Curves
- 5.1 Introduction
- 5.2 Some Basic Sheaves
- 5.3 The Riemann-Hilbert Correspondence for Germs
- 5.4 The Riemann-Hilbert Correspondence in the Global Case
- 5.5 Compatibility with Duality for Meromorphic Connections
- 6 Applications of the Riemann-Hilbert Correspondence to Holonomic Distributions
- 6.1 Introduction
- 6.2 The Riemann-Hilbert Correspondence for Meromorphic Connections of Hukuhara-Turrittin Type
- 6.3 The Hermitian Dual of a Holonomic DX-Module
- 6.4 Asymptotic Expansions of Holonomic Distributions
- 6.5 Comments
- 7 Riemann-Hilbert and Laplace on the Affine Line (the Regular Case)
- 7.1 Introduction
- 7.2 Direct Image of the Moderate de Rham Complex
- 7.3 Topological Spaces
- 7.4 Topological Laplace Transform
- 7.5 Proof of Theorem 7.6 and Compatibility with Riemann-Hilbert
- 7.6 Compatibility of Laplace Transformation with Duality
- 7.7 Compatibility of Topological Laplace Transformation with Poincaré-Verdier Duality
- 7.8 Comparison of Both Duality Isomorphisms
- Part II Dimension Two and More
- 8 Real Blow-Up Spaces and Moderate de Rham Complexes
- 8.1 Introduction
- 8.2 Real Blow-Up
- 8.3 The Sheaf of Functions with Moderate Growth on the Real Blow-Up Space
- 8.4 The Moderate de Rham Complex
- 8.5 Examples of Moderate de Rham Complexes
- 9 Stokes-Filtered Local Systems Along a Divisor with Normal Crossings
- 9.1 Introduction
- 9.2 The Sheaf I on the Real Blow-Up (Smooth Divisor Case)
- 9.3 The Sheaf I on the Real Blow-Up (Normal Crossing Case)
- 9.4 Goodness
- 9.5 Stokes Filtrations on Local Systems
- 9.6 Behaviour by Pull-Back
- 9.7 Partially Regular Stokes-Filtered Local Systems
- 10 The Riemann-Hilbert Correspondence for Good Meromorphic Connections (Case of a Smooth Divisor)
- 10.1 Introduction
- 10.2 Good Formal Structure of a Meromorphic Connection
- 10.3 The Riemann-Hilbert Functor
- 10.4 Proof of the Full Faithfulness in Theorem 10.8
- 10.5 Elementary and Graded Equivalences
- 10.6 Proof of the Essential Surjectivity in Theorem 10.8
- 11 Good Meromorphic Connections (Formal Theory)
- 11.1 Introduction
- 11.2 Preliminary Notation
- 11.3 Good Formal Decomposition
- 11.4 Good Lattices
- 11.5 Proof of Theorem 11.18
- 11.6 Comments
- 12 Good Meromorphic Connections (Analytic Theory) and the Riemann-Hilbert Correspondence
- 12.1 Introduction
- 12.2 Notation
- 12.3 The Malgrange-Sibuya Theorem in Higher Dimension
- 12.4 The Higher Dimensional Hukuhara-Turrittin Theorem
- 12.5 The Riemann-Hilbert Correspondence
- 12.6 Application to Hermitian Duality of Holonomic D-Modules
- 12.7 Comments
- 13 Push-Forward of Stokes-Filtered Local Systems
- 13.1 Introduction
- 13.2 Preliminaries
- 13.3 Adjunction
- 13.4 Recent Advances on Push-Forward and Open Questions
- 13.5 An Example of Push-Forward Computation
- 13.6 The Topological Computation of the Stokes Filtration: Leaky Pipes
- 14 Irregular Nearby Cycles
- 14.1 Introduction
- 14.2 Moderate Nearby Cycles of Holonomic D-Modules
- 14.3 Irregular Nearby Cycles (After Deligne)
- 14.4 Another Proof of the Finiteness Theorem in Dimension Two
- 15 Nearby Cycles of Stokes-Filtered Local Systems
- 15.1 Introduction
- 15.2 Nearby Cycles Along a Function (the Good Case)
- 15.3 Nearby Cycles Along a Function (Dimension Two)
- 15.4 Comparison
- References
- Index of notation
- Index
