Introduction to Stokes structures

Abstract

The purpose of these lectures is to introduce the notion of a Stokes-perverse sheaf as a receptacle for the Riemann-Hilbert correspondence for holonomic D-modules. They develop the original idea of P. Deligne in dimension one, and make it enter the frame of perverse sheaves. They also give a first step for a general definition in higher dimension, and make explicit particular cases of the Riemann-Hilbert correspondence, relying on recent results of T. Mochizuki.

Expanded Lecture notes of lectures given in Lisboa, jan. 2009. V2: 175 pages, 5 figures. Some corrections and improvements and new Lecture 6 added. V3: 191 pages, Lecture 14 added. V4: 202 pages, Duality properties added in Lectures 5 & 7. V5: revised and corrected version: 242 pages

References in corpus (5)

• Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l'irrégularité
• Asymptotic behaviour of variation of pure polarized TERP structures
• Good formal structure for meromorphic flat connections on smooth projective surfaces
• Holonomic D-modules with Betti structure
• Formal structure of direct image of holonomic D-modules of exponential type

Cited by in corpus (26)

• Riemann-Hilbert correspondence for holonomic D-modules
• On the irregular Hodge filtration of exponentially twisted mixed Hodge modules
• An analogue of Dubrovin's conjecture
• Holonomic D-modules with Betti structure
• A Log PSS morphism with applications to Lagrangian embeddings
• A microlocal approach to the enhanced Fourier-Sato transform in dimension one
• Topological computation of some Stokes phenomena on the affine line
• Quadratic relations between periods of connections
• Differential systems of pure Gaussian type
• A remark on the irregularity complex
• D-modules of pure Gaussian type and enhanced ind-sheaves
• Gevrey expansions of hypergeometric integrals I
• Perverse schobers and the Algebra of the Infrared
• On the rapid decay homology of F.Pham
• Irregular perverse sheaves
• CFNS Ad-Hoc meeting on Radiative Corrections Whitepaper
• Remarks on rigid irreducible meromorphic connections on the projective line
• Integrable deformations and degenerations of some irregular singularities
• The Stokes phenomenon for an irregular Gelfand-Kapranov-Zelevinsky system associated with the rank one lattice
• Categorification of Legendrian knots
• La construction d'Abbes et Saito pour les connexions méromorphes: aspect formel en dimension 1
• The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles
• Kashiwara conjugation and the enhanced Riemann-Hilbert correspondence
• Singularities of functions: a global point of view
• Homotopy theory of Stokes data
• A Riemann-Hilbert correspondence for cohomology theories of closed 1-forms