Introduction to Stokes Structures

1. Front Matter

   Pages i-xiv

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2. Dimension one

   1. Front Matter

      Pages 21-21

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   2. \(\mathcal{I}\)-Filtrations

      • Claude Sabbah

      Pages 1-19

   3. Stokes-Filtered Local Systems in Dimension One

      • Claude Sabbah

      Pages 23-38

   4. Abelianity and Strictness

      • Claude Sabbah

      Pages 39-50

   5. Stokes-Perverse Sheaves on Riemann Surfaces

      • Claude Sabbah

      Pages 51-64

   6. The Riemann–Hilbert Correspondence for Holonomic \(\mathcal{D}\)-Modules on Curves

      • Claude Sabbah

      Pages 65-78

   7. Applications of the Riemann–Hilbert Correspondence to Holonomic Distributions

      • Claude Sabbah

      Pages 79-88

   8. Riemann–Hilbert and Laplace on the Affine Line (the Regular Case)

      • Claude Sabbah

      Pages 89-111

3. Dimension two and more

   1. Front Matter

      Pages 113-113

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   2. Real Blow-Up Spaces and Moderate de Rham Complexes

      • Claude Sabbah

      Pages 115-129

   3. Stokes-Filtered Local Systems Along a Divisor with Normal Crossings

      • Claude Sabbah

      Pages 131-146

   4. The Riemann–Hilbert Correspondence for Good Meromorphic Connections (Case of a Smooth Divisor)

      • Claude Sabbah

      Pages 147-157

   5. Good Meromorphic Connections (Formal Theory)

      • Claude Sabbah

      Pages 159-175

   6. Good Meromorphic Connections (Analytic Theory) and the Riemann–Hilbert Correspondence

      • Claude Sabbah

      Pages 177-193

   7. Push-Forward of Stokes-Filtered Local Systems

      • Claude Sabbah

      Pages 195-206

   8. Irregular Nearby Cycles

      • Claude Sabbah

      Pages 207-225

   9. Nearby Cycles of Stokes-Filtered Local Systems

      • Claude Sabbah

      Pages 227-238

4. Back Matter

   Pages 239-249

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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.

• 34M40, 32C38, 35A27
• Meromorphic connection
• Stokes filtration
• Stokes-perverse sheaf
• real blowing-up
• ordinary differential equations
• partial differential equations

• Centre de mathématiques, CNRS Ecole polytechnique, Palaiseau, France

  Claude Sabbah

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