Ebook: Introduction to stokes structures

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. Read more... I-Filtrations -- Stokes-Filtered Local Systems in Dimension One -- Abelianity and Strictness -- Stokes-Perverse Sheaves on Riemann Surfaces -- The Riemann-Hilbert Correspondence for Holonomic -Modules on Curves -- Applications of the Riemann-Hilbert Correspondence to Holonomic Distributions -- Riemann-Hilbert and Laplace on the Affine Line (the Regular Case) -- Real Blow-Up Spaces and Moderate de Rham Complexes -- Stokes-Filtered Local Systems Along a Divisor with Normal Crossings -- The Riemann-Hilbert Correspondence for Good Meromorphic Connections (Case of a Smooth Divisor) -- Good Meromorphic Connections (Formal Theory) -- Good Meromorphic Connections (Analytic Theory) and the Riemann-Hilbert Correspondence -- Push-Forward of Stokes-Filtered Local Systems -- Irregular Nearby Cycles -- Nearby Cycles of Stokes-Filtered Local Systems