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.
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abelian abelian groups Amod0 AmodDeX associated assume assumption blowing-up Chap cMreg coefficients cohomology in degree compatible complex manifold components consider constant sheaf coordinates Corollary defined Definition derived category dimension direct image duality equivalent exists exponential factors finite set follows formal decomposition germ global graded Hausdorff holomorphic function holonomic D-modules homeomorphism I-filtration Iét inclusion induces isomorphism k-vector spaces Laplace transform lattice Lemma Let us set locally constant matrix meromorphic connection Mochizuki moderate growth module Moreover natural morphism non-ramified nonzero normal crossing divisor normal crossings notation NP(a open set perverse sheaf pre-I-filtration Proof of Proposition prove pull-back purely monomial push-forward ramified real blow-up space regular singularities Remark resp restriction Rham complex Riemann surface Riemann–Hilbert correspondence Riemann–Hilbert functor satisfies Sect set of exponential similarly singular points Stokes filtration Stokes-filtered local system Stokes-perverse sheaves stratified I-covering subset subsheaf t-structure topological zero