Regular and Irregular Holonomic D-Modules (London Mathematical Society Lecture Note Series)

Description

D-module theory is essentially the algebraic study of systems of linear partial differential equations. This book, the first devoted specifically to holonomic D-modules, provides a unified treatment of both regular and irregular D-modules. The authors begin by recalling the main results of the theory of indsheaves and subanalytic sheaves, explaining in detail the operations on D-modules and their tempered holomorphic solutions. As an application, they obtain the Riemann-Hilbert correspondence for regular holonomic D-modules. In the second part of the book the authors do the same for the sheaf of enhanced tempered solutions of (not necessarily regular) holonomic D-modules. Originating from a series of lectures given at the Institut des Hautes Études Scientifiques in Paris, this book is addressed to graduate students and researchers familiar with the language of sheaves and D-modules, in the derived sense. Contents Introduction 1 A review on sheaves and D-modules 1.1 Sheaves 1.2 D-modules 2 Indsheaves 2.1 Ind-objects 2.2 Indsheaves 2.3 Ring action 2.4 Sheaves on the subanalytic site 2.5 Some classical sheaves on the subanalytic site 3 Tempered solutions of D-modules 3.1 Tempered de Rham and Sol functors 3.2 Localization along a hypersurface 4 Regular holonomic D-modules 4.1 Regular normal form for holonomic modules 4.2 Real blow up 4.3 Regular Riemann–Hilbert correspondence 4.4 Integral transforms with regular kernels 4.5 Irregular D-modules: an example 5 Indsheaves on bordered spaces 5.1 Bordered spaces 5.2 Operations 6 Enhanced indsheaves 6.1 Tamarkin’s construction 6.2 Convolution products 6.3 Enhanced indsheaves 6.4 Operations on enhanced indsheaves 6.5 Stable objects 6.6 Constructible enhanced indsheaves 6.7 Enhanced indsheaves with ring action 7 Holonomic D-modules 7.1 Exponential D-modules 7.2 Enhanced tempered holomorphic functions 7.3 Enhanced de Rham and Sol functors 7.4 Ordinary linear differential equations and Stokes phenomena 7.5 Normal form 7.6 Enhanced de Rham functor on the real blow up 7.7 De Rham functor: constructibility and duality 7.8 Enhanced Riemann–Hilbert correspondence 8 Integral transforms 8.1 Integral transforms with irregular kernels 8.2 Enhanced Fourier–Sato transform 8.3 Laplace transform References Notations Index