Introduction to stokes structures by Claude Sabbah

By Claude Sabbah

This examine monograph offers a geometrical description of holonomic differential platforms in a single or extra variables. Stokes matrices shape the prolonged monodromy facts for a linear differential equation of 1 complicated variable close to an abnormal singular aspect. the current quantity offers the process when it comes to Stokes filtrations. For linear differential equations on a Riemann floor, it additionally develops the comparable thought of a Stokes-perverse sheaf.This standpoint is generalized to holonomic structures of linear differential equations within the complicated area, and a normal Riemann-Hilbert correspondence is proved for vector bundles with meromorphic connections on a posh manifold. functions to the distributions options to such platforms also are mentioned, and diverse operations on Stokes-filtered neighborhood structures are analyzed. learn more... I-Filtrations -- Stokes-Filtered neighborhood structures in measurement One -- Abelianity and Strictness -- Stokes-Perverse Sheaves on Riemann Surfaces -- The Riemann-Hilbert Correspondence for Holonomic -Modules on Curves -- purposes of the Riemann-Hilbert Correspondence to Holonomic Distributions -- Riemann-Hilbert and Laplace at the Affine Line (the commonplace Case) -- genuine Blow-Up areas and average de Rham Complexes -- Stokes-Filtered neighborhood platforms alongside a Divisor with basic Crossings -- The Riemann-Hilbert Correspondence for stable Meromorphic Connections (Case of a gentle Divisor) -- reliable Meromorphic Connections (Formal idea) -- stable Meromorphic Connections (Analytic conception) and the Riemann-Hilbert Correspondence -- Push-Forward of Stokes-Filtered neighborhood platforms -- abnormal within sight Cycles -- close by Cycles of Stokes-Filtered neighborhood platforms

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7. The natural morphism I1 ! `/ is compatible with the order. Proof. 4 ` ), we have Á 6 0 ) ŒÁ` 6 0. t u We will now introduce a reduction procedure with respect to the level. 3, L6Œ'` D X ˇŒ where the sum is taken in L . Then X L<Œ'` WD ˇŒ ` <Œ'` L6Œ Œ ` ` 6Œ'` L6 ` D X ˇŒ ; ` <Œ'` L6 : 42 3 Abelianity and Strictness Indeed, L<Œ'` D XX ˇŒ ` <Œ'` ˇŒÁ` 6Œ'` L6Á ; Á Œ ` and for a fixed Á, the set of  2 S 1 for which there exists Œ ` satisfying ŒÁ` 6 Œ ` < Œ'` is equal to the set of  such that ŒÁ` < Œ'` .

KXb/. kIe´t ;6;Db / is an abelian category. kXb/ are exact. kIe´t ;6;Db /. 7 Stokes-Perverse Sheaves on X e Fig. 1 The space X near Sx1 (x 2 D) 61 Xx X x S 1x = S 1x e). 3), defining a Stokes1. L ; L / on each Sx1o , xo 2 D. F b is a C-constructible sheaf on bxo with singularity at b 2. F xo (xo 2 D) at xo 2D Xb most. b. 3. b is an isomorphism gr0 L ! b { 1 jb F Morphisms between such triples consist of pairs . ; b/ of morphisms in the respective categories which are compatible with b. kIe´t ;6;Db /.

In such a case, the de Rham complex DR M has cohomology in degree 0 at most and H 0 DR M D Ker r D M r is the sheaf of horizontal local sections of M . It is a locally constant sheaf of finite dimensional C-vector spaces. Conversely, M can be recovered from M r as M ' OX ˝C M r with the standard connection on OX . This defines a perfect correspondence (equivalence C. 1007/978-3-642-31695-1 5, © Springer-Verlag Berlin Heidelberg 2013 65 66 5 The Riemann–Hilbert Correspondence for Holonomic D -Modules on Curves of categories) between holomorphic vector bundles with connection and locally constant sheaves of finite-dimensional C-vector spaces.

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