Holonomic D-modules Research Articles

Computing Holonomic D-Modules Associated to a Family of Non-isolated Hypersurface Singularities via Comprehensive Gröbner Systems of PBW Algebra

Computing Holonomic D-Modules Associated to a Family of Non-isolated Hypersurface Singularities via Comprehensive Gröbner Systems of PBW Algebra

Holonomic <i>D</i>-Modules Associated with a Simple Line Singularity and the Vertical Monodromy

Holonomic D-modules associated with certain kind of non-isolated hypersurface singularities, introduced by T. de Jong, are considered. Structures of these holonomic D-modules such as characteristic variety, monodromy are explicitly determined. The key of our approach is the use of the notion of algebraic local cohomology. An application to Betti numbers of local Milnor fibers is also discussed.

D-modules of pure Gaussian type and enhanced ind-sheaves

Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier–Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of Sabbah.

METHODS FOR COMPUTING <i>b</i>-FUNCTIONS ASSOCIATED WITH <i>µ</i>-CONSTANT DEFORMATIONS: CASE OF INNER MODALITY TWO

New methods for computing parametric local b-functions are introduced for µ-constant deformations of semi-weighted homogeneous singularities. The keys of the methods are comprehensive Gröbner systems in Poincaré-Birkhoff-Witt algebra and holonomic D-modules. It is shown that the use of semi-weighted homogeneity reduces the computational complexity of b-functions associated with µ-constant deformations. In the case of inner modality two, local b-functions associated with µ-constant deformations are obtained by the resulting method and given the list of parametric local b-functions.

On a topological counterpart of regularization for holonomic 𝒟-modules

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 correspondence. Here, we study some of their properties. In particular, we provide a germ formula for the sheafification of enhanced specialization and microlocalization.

The de Rham functor for logarithmic D-modules

In the first part we deepen the six-functor theory of (holonomic) logarithmic D-modules, in particular with respect to duality and pushforward along projective morphisms. Then, inspired by work of Ogus, we define a logarithmic analogue of the de Rham functor, sending logarithmic D-modules to certain graded sheaves on the so-called Kato–Nakayama space. For holonomic modules we show that the associated sheaves have finitely generated stalks and that the de Rham functor intertwines duality for D-modules with a version of Poincaré–Verdier duality on the Kato–Nakayama space. Finally, we explain how the grading on the Kato–Nakayama space is related to the classical Kashiwara–Malgrange V-filtration for holonomic D-modules.

A prime-characteristic analogue of a theorem of Hartshorne-Polini

Let R be an F-finite Noetherian regular ring containing an algebraically closed field k of positive characteristic, and let M be an F-finite F-module over R in the sense of Lyubeznik (for example, any local cohomology module of R). We prove that the Fp-dimension of the space of F-module morphisms M→E(R/m) (where m is any maximal ideal of R and E(R/m) is the R-injective hull of R/m) is equal to the k-dimension of the Frobenius stable part of HomR(M,E(R/m)). This is a positive-characteristic analogue of a recent result of Hartshorne and Polini for holonomic D-modules in characteristic zero.

Iterated Extensions and Uniserial Length Categories

In this paper, we study length categories using iterated extensions. We fix a field k, and for any family S of orthogonal k-rational points in an Abelian k-category mathcal {A}, we consider the category Ext(S) of iterated extensions of S in mathcal {A}, equipped with the natural forgetful functor mathbf {Ext}(mathsf {S}) to mathbf {mathcal {A}}(mathsf {S}) into the length category mathbf {mathcal {A}}(mathsf {S}). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel’s criterion, we give a complete classification of the indecomposable objects in mathbf {mathcal {A}}(mathsf {S}) when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family S in mathcal {A}. As an application, we classify all graded holonomic D-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when D is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish.

On irregularities of Fourier transforms of regular holonomic [formula omitted]-modules

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.

An Algorithm for Computing Grothendieck Local Residues II: General Case

Grothendieck local residue is considered in the context of symbolic computation. Based on the theory of holonomic D-modules, an effective method is proposed for computing Grothendieck local residues. The key is the notion of Noether operator associated to a local cohomology class. The resulting algorithm and an implementation are described with illustrations.

An Algorithm for Computing Grothendieck Local Residues I: Shape Basis Case

Grothendieck local residue is considered in the context of symbolic computation. Basic ideas of our approach are the use of local cohomology, holonomic D-modules, and Noether operators. An effective method is introduced for computing Grothendieck local residues of rational n-forms under shape basis condition. Resulting algorithms that avoid the use of Grobner bases on the Weyl algebra and an implementation are described. Some examples are also given for illustration.

Holonomic and perverse logarithmic D-modules

We introduce the notion of a holonomic D-module on a smooth (idealized) logarithmic scheme and show that Verdier duality can be extended to this context. In contrast to the classical case, the pushforward of a holonomic module along an open immersion is in general not holonomic. We introduce a “perverse” t-structure on the category of coherent logarithmic D-modules which makes the dualizing functor t-exact on holonomic modules. Conversely this t-exactness characterizes holonomic modules among all coherent logarithmic D-modules. We also introduce logarithmic versions of the Gabber and Kashiwara–Malgrange filtrations.

A dichotomy for the injective dimension of F-finite F-modules and holonomic D-modules

Let M be either a holonomic D-module over a formal power series ring with coefficients in a field of characteristic zero, or an F-finite F-module over a noetherian regular ring of characteristic p > 0. We prove that enjoys a dichotomy property: it has only two possible values, or .

Simple [formula omitted]-module components of local cohomology modules

For a projective variety V⊂Pkn over a field of characteristic zero, with homogeneous ideal I in A=k[x0,…,xn], we consider the local cohomology modules HIi(A). These have a structure of holonomic D-module over A, and we investigate their filtration by simple D-modules. In case V is nonsingular, we can describe completely the simple D-module components of HIi(A) for all i, in terms of the Betti numbers of V.

T-Structures for relative [formula omitted]-modules and t-exactness of the de Rham functor

This paper is a contribution to the study of relative holonomic D-modules. Contrary to the absolute case, the standard t-structure on holonomic D-modules is not preserved by duality and hence the solution functor is no longer t-exact with respect to the canonical, resp. middle-perverse, t-structure. We provide an explicit description of these dual t-structures. We use this description to prove that the solution functor as well as the relative Riemann–Hilbert functor are t-exact with respect to the dual t-structure and to the middle-perverse one while the de Rham functor is t-exact for the canonical, resp. middle-perverse, t-structure and their duals.

Irregular Hodge Theory

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 $\exp\varphi$ for any meromorphic function $\varphi$. 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 show that those rigid irreducible holonomic D-modules on the complex projective line whose local formal monodromies have eigenvalues of absolute value one, are equipped with such an irregular Hodge filtration in a canonical way, up to a shift of the filtration. In a chapter written jointly with Jeng-Daw~Yu, we make explicit the case of irregular mixed Hodge structures, for which we prove in particular a Thom-Sebastiani formula.

Comprehensive Gröbner systems in PBW algebras, Bernstein–Sato ideals and holonomic D-modules

A computation method of comprehensive Gröbner systems is introduced in Poincaré–Birkhoff–Witt (PBW) algebras and applications to Bernstein–Sato ideals, holonomic D-modules for parametric cases are considered. The proposed method provides in particular holonomic D-modules associated with primary components of Bernstein–Sato ideals. Furthermore, with our implementation, effective methods are illustrated for computing b-functions of μ-constant deformations of hypersurface isolated singularities by utilizing holonomic D-modules. High versatility of the proposed method is also illustrated by several examples.

Cohomology support loci of local systems

The support S of Sabbah's specialization complex is a simultaneous generalization of the set of eigenvalues of the monodromy on Deligne's nearby cycles complex, of the support of the Alexander modules of an algebraic knot, and of certain cohomology support loci. Moreover, it equals conjecturally the image under the exponential map of the zero locus of the Bernstein-Sato ideal. Sabbah showed that S is contained in a union of translated subtori of codimension one in a complex affine torus. Budur-Wang showed recently that S is a union of torsion-translated subtori. We show here that S is always a hypersurface, and that it admits a formula in terms of log resolutions. As an application, we give a criterion in terms of log resolutions for the (semi-)simplicity as perverse sheaves, or as regular holonomic D-modules, of the direct images of rank one local systems under an open embedding. For hyperplane arrangements, this criterion is combinatorial.

Finiteness properties of local cohomology modules over differentiable admissible algebras

Using the theory of D-modules, we prove some finiteness properties for local cohomology modules with respect to a family of supports on the spectrum of a k-algebra R which satisfies certain conditions introduced by L. Núñez-Betancourt in [15]. These results generalize in a sense most of the finiteness properties established by G. Lyubeznik in [9]. We also introduce the class of quasi-holonomic D-modules which properly contains the class of holonomic D-modules and is closed by taking submodules, quotients, extensions, direct limits, localizations at multiplicative subsets of R and local cohomology.

Van den Essen’s theorem on the de Rham cohomology of a holonomic [formula omitted]-module over a formal power series ring

In this expository paper, we give a complete proof of van den Essen’s theorem that the de Rham cohomology spaces of a holonomic D-module are finite-dimensional in the case of a formal power series ring over a field of characteristic zero. This proof requires results from at least five of van den Essen’s papers as well as his unpublished thesis, and until now has not been available in a self-contained document.