We prove that a holonomic binomial $D$--module $M_A (I,β)$ is regular if and only if certain associated primes of $I$ determined by the parameter vector $β\in \CC^d$ are homogeneous. We further describe the slopes of $M_A(I,β)$ along a coordinate subspace in terms of the known slopes of some related hypergeometric $D$--modules that also depend on $β$. When the parameter $β$ is generic, we also compute the dimension of the generic stalk of the irregularity of $M_A(I,β)$ along a coordinate hyperplane and provide some remarks about the construction of its Gevrey solutions.
We study binomial D-modules, which generalize A-hypergeometric systems. We determine explicitly their singular loci and provide three characterizations of their holonomicity. The first of these states that a binomial D-module is holonomic if and only if its corresponding singular locus is proper. The second characterization is an equivalence of holonomicity and L-holonomicity for these systems. The third refines the second by giving more detailed information about the L-characteristic variety of a non-holonomic binomial D-module.
We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators defined by the grading and a parameter in C^d. We determine the parameters for which these D-modules (i) are holonomic (equivalently, regular holonomic, when I is standard-graded); (ii) decompose as direct sums indexed by the primary components of I; and (iii) have holonomic rank greater than the generic rank. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arr…
In this paper we will present a way of examining the Stokes structure of certain irregular singular $\mathcal D$-modules, namely the direct image of exponentially twisted regular singular meromorphic connections, in a topological point of view. This topological description enables us to compute Stokes data for an explicit example concretely.
We study Fourier transforms of regular holonomic D-modules. By using the theory of Fourier-Sato transforms of enhanced ind-sheaves developed by Kashiwara-Schapira and D'Agnolo-Kashiwara, a formula for their enhanced solution complexes will be obtained. Moreover we show that some parts of their characteristic cycles and irregularities are expressed by the geometries of the original D-modules.
Let M be a complex of D-modules with bounded holonomic cohomology on a complex manifold. In this note, we prove that if the derived tensor product of M with itself is regular, then M is regular.
S.Block and H.Esnault constructed the local Fourier transform for D-modules. We present a different approach to the local Fourier transform, which makes its properties almost tautological. We apply the local Fourier transform to compute the local version of Katz's middle convolution.
For an algebraic vector bundle $E$ over a smooth algebraic variety $X$, a monodromic $D$-module on $E$ is decomposed into a direct sum of some $O$-modules on $X$. We show that the Hodge filtration of a monodromic mixed Hodge module is decomposed with respect to the decomposition of the underlying $D$-module. By using this result, we endow the Fourier-Laplace transform $M^{\wedge}$ of the underlying $D$-module $M$ of a monodromic mixed Hodge module with a mixed Hodge module structure. Moreover, we describe the irregular Hodge filtration on $M^{\wedge}$ concretely and show that it coincides with…
This is a survey paper on the Riemann-Hilbert correspondence on (irregular) holonomic D-modules, based on the 16-th Takagi lecture (2015/11/28). In this paper, we use subanalytic sheaves, an analogous notion to the one of indsheaves.
This is a survey paper based on a series of lectures given at the IHES in February/March 2015. In a first part, we recall the main results on the tempered holomorphic solutions of D-modules in the language of indsheaves and, as an application, the Riemann-Hilbert correspondence for regular holonomic modules. In a second part, we present the enhanced version of the first part, treating along the same lines the irregular holonomic case.
Given a mixed Hodge module and a meromorphic function f on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module, which extends the construction of arXiv:1302.4537. We show the strictness of the push-forward filtered D-module through any projective morphism, by using the theory of mixed twistor D-modules of T. Mochizuki. We consider the example of the rescaling of a regular function f, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss-Manin systems of f in ter…
We study some non-highest weight modules over an affine Kac-Moody algebra at non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight "vacuum" modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any module over the Kac-Moody algebra with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.
We study the irregularity of hypergeometric D-modules $\mathcal{M}_A (β)$ via the explicit construction of Gevrey series solutions along coordinate subspaces in $X =\mathbb{C}^n$. As a consequence, we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of $\mathcal{M}_A (β)$ given by M. Schulze and U. Walther in [21] still holds without any assumption on the matrix A. We also provide a lower bound for the dimensions of the spaces of Gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameter…
We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier-Laplace transformation for algebraic integrable twistor $\mathcal{D}$-modules.
We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\expφ$ for any meromorphic function $φ$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also…
The subject of this paper is an algebraic version of the irregular Riemann-Hilbert correspondence which was mentioned in [arXiv:1910.09954] by the author. In particular, we prove an equivalence of categories between the triangulated category of algebraic holonomic D-modules on a smooth algebraic variety and the one of algebraic C-constructible enhanced ind-sheaves. Moreover we show that there exists a t-structure on the triangulated category of algebraic C-constructible enhanced ind-sheaves whose heart is equivalent to the abelian category of algebraic holonomic D-modules. Furthermore we shall…
We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric $\mathcal{D}$-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations.
Using twisted nearby cycles, we define a new notion of slopes for complex holonomic D-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections E\_t parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the E\_t. This generalizes the regularity of the Gauss-Manin connection proved by Katz and Deligne. Finally, we address some questions about an…
The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commutin…
We study the rescalability of integrable mixed twistor $D$-modules. We prove some basic functoriality of the rescalability and the associated irregular Hodge filtration. We also observe that rescalable integrable mixed twistor $D$-modules are equivalent to exponential Hodge modules.
Given a (not necessarily regular) holonomic D-module defined on the product of two complex manifolds, we prove that the associated correspondence commutes (in some sense) with the De Rham functor. We apply this result to the study of the classical Laplace transform. The main tools used here are the theory of ind-sheaves and its enhanced version.
We propose a definition of (polarized) wild twistor D-modules, generalizing to objects with irregular singularities that of (polarized) regular twistor D-modules. We give a precise analysis in dimension one.
We study Fourier transforms of regular holonomic D-modules. In particular we show that their solution complexes are monodromic. An application to direct images of some irregular holonomic D-modules will be given. Moreover we give a new proof to the classical theorem of Brylinski and improve it by showing its converse.
In this paper, we reprove the Riemann-Hilbert correspondence for regular holonomic D-modules of [M. Kashiwara, Publ. Res. Inst. Math. Sci., 1984] (see also [Z. Mebkhout, Compositio Math., 1984.]) by using the irregular Riemann-Hilbert correspondence of [A. D'Agnolo and M. Kashiwara, Publ. Math. Inst. Hautes Etudes Sci., 2016]. Moreover, we also prove the algebraic one by the same argument. For this purpose, we study C-constructible enhanced ind-sheaves of [Y. Ito, Tsukuba journal of Mathematics, 2020, Rend. Sem. Mat. Univ. Padova., 2021.] in more detail.
On a complex manifold, the embedding of the category of regular holonomic D-modules into that of holonomic D-modules has a left quasi-inverse functor $\mathcal{M}\mapsto\mathcal{M}_{\mathrm{reg}}$, called regularization. Recall that $\mathcal{M}_{\mathrm{reg}}$ is reconstructed from the de Rham complex of $\mathcal{M}$ by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert c…
We study the irregularity sheaves attached to the $A$-hypergeometric $D$-module $M_A(β)$ introduced by Gel'fand et al., where $A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and $β\in\mathbb{C}^d$. More precisely, we investigate the slopes of this module along coordinate subspaces. In the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector $L$ on torus equivariant generators. To this end we introduce the $(A,L)$-umbrella, a simplicial complex determined by $A$ and $L$, and identify its facets with the component…
These three lectures present some fundamental and classical aspects of microlocal analysis. Starting with the Sato's microlocalization functor and the microsupport of sheaves, we then construct a microlocal analogue of the Hochschild homology for sheaves and apply it to recover index theorems for D-modules and elliptic pairs. In the third lecture, we construct the ind-sheaves of temperate and Whitney holomorphic functions and give some applications to the study of irregular holonomic D-modules.
Centre de Mathématiques Laurent Schwartz (CMLS), CNRS, École polytechnique, Institut Polytechnique de Paris（洛朗·施瓦茨数学中心（CMLS），法国国家科学研究中心，巴黎综合理工学院，巴黎理工学院）
Let $\mathcal{M}$ be a holonomic algebraic $\mathcal{D}$-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to $\mathcal{M}$ at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of $\mathcal{M}$ are obtained by Legendre transform from the exponential factors of $\mathcal{M}$. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.
