Claude Sabbah

Abelianity and Strictness

We prove that the category of (k)-Stokes-filtered local systems on S 1 is abelian. The main ingredient, together with vanishing properties of the cohomology, is the introduction of the level structure. Abelianity is also a consequence of the Riemann–Hilbert correspondence considered in Chap. 5, but it is instructive to prove it over the base field (k).

Stokes-Filtered Local Systems Along a Divisor with Normal Crossings

We construct the sheaf (mathcal{I}) to be considered as the index sheaf for Stokes filtrations. This is a sheaf on the real blow-up space of a complex manifold along a family of divisors. We will consider only divisors with normal crossings. The global construction of (mathcal{I}) needs some care, as the trick of considering a ramified covering cannot be used globally. The important new notion is that of goodness. It is needed to prove abelianity and strictness in this setting, generalizing the results of Chap. 3.

Stokes-Filtered Local Systems in Dimension One

We consider Stokes filtrations on local systems on S 1. We review some of the definitions of the previous chapter in this case and make explicit the supplementary properties coming from this particular case. This chapter can be read independently of Chap. 1.

Applications of the Riemann–Hilbert Correspondence to Holonomic Distributions

To any holonomic (mathcal{D})-module on a Riemann surface X is associated its Hermitian dual, according to M. Kashiwara. We give a proof that the Hermitian dual is also holonomic. As an application, we make explicit the local expression of a holonomic distribution, that is, a distribution on X (in Schwartz’ sense) which is solution to a nonzero holomorphic differential equation on X. The conclusion is that working with ({C}^{infty }) objects hides the Stokes phenomenon.

The Riemann–Hilbert Correspondence for Good Meromorphic Connections (Case of a Smooth Divisor)

This chapter is similar to Chap. 5, but we add holomorphic parameters. Moreover, we assume that no jump occurs in the exponential factors, with respect to the parameters. This is the meaning of the goodness condition in the present setting. We will have to treat the Riemann–Hilbert functor in a more invariant way, and more arguments will be needed in the proof of the main result (equivalence of categories) in order to make it global with respect to the divisor. For the sake of simplicity, we will only consider the case of germs of meromorphic connections along a smooth divisor.

Good Meromorphic Connections (Analytic Theory) and the Riemann–Hilbert Correspondence

This chapter is similar to Chap. 10, but we now assume that D is a divisor with normal crossings. We start by proving the many-variable version of the Hukuhara–Turrittin theorem, that we have already encountered in the case of a smooth divisor. It will be instrumental for making the link between formal and holomorphic aspects of the theory. The new point in the proof of the Riemann–Hilbert correspondence is the presence of non-Hausdorff eale spaces, and we need to use the level structure to prove the local essential surjectivity of the Riemann–Hilbert functor. As an application of the Riemann–Hilbert correspondence in the good case and of the fundamental results of K. Kedlaya and T. Mochizuki on the elimination of turning points by complex blowing-ups, we prove a conjecture of M. Kashiwara asserting that the Hermitian dual of a holonomic (mathcal{D})-module is holonomic, generalizing the original result of M. Kashiwara for regular holonomic (mathcal{D})-modules to possibly irregular holonomic (mathcal{D})-modules and the result of Chap. 6 to higher dimensions.

Good Meromorphic Connections (Formal Theory)

This chapter is a prelude to the Riemann–Hilbert correspondence in higher dimensions, treated in Chap. 12. We explain the notion of a good formal structure for germs of meromorphic connections having poles along a divisor with normal crossings.

Real Blow-Up Spaces and Moderate de Rham Complexes

The purpose of this chapter is to give a global construction of the real blow-up space of a complex manifold along a family of divisors. On this space is defined the sheaf of holomorphic functions with moderate growth, whose basic properties are analyzed. The moderate de Rham complex of a meromorphic connection is introduced, and its behaviour under the direct image by a proper modification is explained. This chapter ends with an example of a moderate de Rham complex having cohomology in degree (geq 1), making a possible definition of Stokes-perverse sheaves more complicated than in dimension one.

Riemann–Hilbert and Laplace on the Affine Line (the Regular Case)

The Laplace transform ({}^{mathrm{F}}!M) of a holonomic (mathcal{D})-module M on the affine line ({mathbb{A}}^{!1}) is also holonomic. If M has only regular singularities (included at infinity), ({}^{mathrm{F}}!M) provides the simplest example of an irregular singularity (at infinity). We will describe the Stokes-filtered local system attached to ({}^{mathrm{F}}!M) at infinity in terms of data of M. More precisely, we define the topological Laplace transform of the perverse sheaf ({{}^{mathrm{p}}! DR }^{mathrm{an}}M) as a perverse sheaf on (widehat{{mathbb{A}}}^{!1}) equipped with a Stokes structure at infinity. We make explicit this topological Laplace transform. As a consequence, if (k) is a subfield of (mathbb{C}) and if we have a (k)-structure on ({{}^{mathrm{p}}! DR }^{mathrm{an}}M), we find a natural (k)-structure on ({{}^{mathrm{p}}! DR {}^{mathrm{an}}}^{mathrm{F}}!M) which extends to the Stokes filtration at infinity. In other words, the Stokes matrices can be defined over (k). We end this chapter by analyzing the behaviour of duality by Laplace and topological Laplace transformation, and the relations between them.

Irregular Nearby Cycles

In this chapter, we review Deligne’s definition of irregular nearby cycles for holonomic (mathcal{D})-modules and recall Deligne’s finiteness theorem in the algebraic case. We give a new proof of this theorem when the support of the holonomic (mathcal{D})-module has dimension two, which holds in the complex analytic setting and which makes more precise the non-vanishing nearby cycles.