Riemann-Hilbert Correspondence Research Articles

Recovery of vanishing cycles by log geometry

We first construct compatible actions of the product of the unit interval and the unit circle as a monoid on a semi-stable degeneration of pairs and on the associated log topological spaces. Then we show that the log topological family is locally trivial in piecewise smooth category over the base, i.e., the associated log topological family recovers the vanishing cycles of the original semi-stable degeneration in the most naive sense. Using this result together with the log Riemann-Hilbert correspondence, we introduce two types of integral structure of the variation of mixed Hodge structure associated to a semi-stable degeneration of pairs.

An analytic Riemann-Hilbert correspondence for semi-simple Lie groups

Geometric Representation Theory for semi-simple Lie groups has two main sheaf theoretic models. Namely, through Beilinson-Bernstein localization theory, Harish-Chandra modules are related to holonomic sheaves of D \mathcal D modules on the flag variety. Then the (algebraic) Riemann-Hilbert correspondence relates these sheaves to constructible sheaves of complex vector spaces. On the other hand, there is a parallel localization theory for globalized Harish-Chandra modules—i.e., modules over the full semi-simple group which are completions of Harish-Chandra modules. In particular, Hecht-Taylor and Smithies have developed a localization theory relating minimal globalizations of Harish-Chandra modules to group equivariant sheaves of D \mathcal D modules on the flag variety. The main purpose of this paper is to develop an analytic Riemann-Hilbert correspondence relating these sheaves to constructible sheaves of complex vector spaces and to discuss the relationship between this “analytic" study of global modules and the preceding “algebraic" study of the underlying Harish-Chandra modules.

Moduli of regular holonomic 𝒟-modules with normal crossing singularities

This paper solves the global moduli problem for regular holonomic Dmodules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to “pre-D-modules”), and then introducing a notion of (semi-)stability and applying Geometric Invariant Theory to construct a coarse moduli scheme for semistable pre-D-modules. A moduli is constructed also for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism between these moduli spaces.

On the Reconstruction Theorem of Holonomlc Modules In the Gevrey Classes

The notion of holonomic systems entails a natural generalization of ordinary differential equations to higher dimension. A holonomic system is, by definition, a left coherent 8 (or 3)) module whose characteristic variety is Lagrangian. It enjoys many good properties (see Kashiwara [Kl], [K2] and Kashiwara-Kawai [K-K]): for example, all cohomology groups associated with its solution sheaf are constructible. As special functions satisfy systems of ordinary differential equations with regular singularities, a holonomic system with regular singularities introduced by Kashiwara-Oshima [K-O] and [K-K] is particularly important. It is well known that the category of holonomic modules with regular singularities is equivalent to that of perverse sheaves through Riemann-Hilbert correspondence (Kashiwara [K3] and Mebkout [Me]). Moreover the regularity of holonomic modules are stable under many operations (integration, restriction, etc.). For holonomic modules with irregular singularities, Kashiwara-Kawai obtained the following remarkable theorem.

The Orders of Invariant Eigendistributions

In a series of papers, Harish-Chandra studied the invariant eigendistributions and in particular established a fundamental theorem which states that any invariant eigendistribution on a connected semisimple Lie group is locally L (cf. [H]). It may be available to reconsider this result from the view point of microlocal analysis. On the other hand, recently Hotta and Kashiwara [HK] have shown that the system of differential equations which governs an invariant eigendistribution on a semisimple Lie algebra is regular holonomic (=a holonomic system with regular singularities in the sense of [KK]). Among other things they showed, by using this result, that the holonomic system in question corresponds to the intersection cohomology complex defining Springer's representations of the Weyl group through the Riemann-Hilbert correspondence. In this paper we examine a microlocal property of the invariant eigendistributions. The results of this paper is quite unsatisfactory in comparison with those mentioned above. But the author hopes that our attempts will be developed in future. We now explain the contents shortly. In the first half we consider the homonomic system M^ which governs an invariant eigendistribution on a connected linear semisimple Lie group. An invariant of a holonomic system is the set ordA(u) of the orders along each irreducible component A of the characteristic variety of the system in question. Here u is a section of the system on the generic points of A. We attempt to determine ord^(w) for the system MI. Unfortunately, we cannot do it for every irreducible component A of the characteristic Ch(*SKx) but if an irreducible component A of Ch(^) satisfies the condition (A) in (3.1), we can calculate the orders along A. In this case, though c5^z is not a simple holonomic system in the sense of Sato-Kashiwara, ord^w) along such an irreducible component A consists of only one element 0. This is the main result of the first half (Theorem (3.4)). It rarely occurs that

A Riemann–Hilbert correspondence in positive characteristic

We explain a version of the Riemann-Hilbert correspondence for $p$-torsion \'etale sheaves on an arbitrary $\mathbf{F}_p$-scheme.