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 arrangement in C^d. In the special case of Horn hypergeometric D-modules, when I is a lattice basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated A-hypergeometric functions. This study relies fundamentally on the explicit lattice point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article arxiv:0803.3846.
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 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. This allows us to transfer some of the formalism from the classical setting and in particular show that every holonomic module on an open subscheme can be extended to a…
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.
We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely beyond an analytic subset of codimension equal to or larger than three in a Lagrangian subvariety containing the support of the DQ-module.
In this manuscript we prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic D-modules, in this context, have finite length. We obtain these results using a more general version of Bernstein filtrations.
We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology support loci of a holonomic D-module are finite unions of linear subvarieties, which go through points of finite order for objects of geometric origin; that the standard t-structure on the derived category of holonomic complexes corresponds, under the Fourier-Mukai transform, to a certain perverse coherent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov; and that Fourier-Mukai transforms of simple holonomic D-modules are intersection complex…
We study relative and logarithmic characteristic cycles associated to holonomic $\mathscr D$-modules. As applications, we obtain: (1) an alternative proof of Ginsburg's log characteristic cycle formula for lattices of regular holonomic $\mathscr D$-modules following ideas of Sabbah and Briancon-Maisonobe-Merle, and (2) the constructibility of the log de Rham complexes for lattices of holonomic $\mathscr D$-modules, which is a natural generalization of Kashiwara's constructibility theorem.
We explain a formalism of regular holonomic $D$-modules for algebraic geometers using the distinguished triangles associated with algebraic local cohomology together with meromorphic Deligne extensions of local systems as well as the dual functor.
This paper shows algebraically that the Fourier transform preserves the rigidity index of irreducible regular holonomic $\mathcal{D}_{\mathbb{P}^1}[*\{\infty\}]$-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.
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.
We define the notion of Betti structure for holonomic D-modules which are not necessarily regular singular. We establish the fundamental functorial properties. We also give auxiliary analysis of holomorphic functions of various types on the real blow up.
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.
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.
In this paper we prove the constructibility on the subanalytic sites of the sheaves of tempered holomorphic solutions of holonomic D-modules on complex analytic manifolds. Such a result solves a conjecture of M. Kashiwara and P. Schapira (2003).
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.
Recently, the Riemann-Hilbert correspondence was generalized in the context of general holonomic D-modules by A. D'Agnolo and M. Kashiwara. Namely, they proved that their enhanced de Rham functor gives a fully faithfully embedding of the derived category of cohomologically holonomic complexes of D-modules into the derived category of complexes of real constructible enhanced ind-sheaves. In this paper, we study a condition when a complex of real constructible enhanced ind-sheaves $K$ is induced by a cohomologically holonomic complex of D-modules. We characterize such $K$ in terms of the restric…
We compute formal invariants associated with the cohomology sheaves of the direct image of holonomic D-modules of exponential type. We also prove that every formal C[[t]]<\partial_t>-modules is isomorphic, after a ramification, to a germ of formalized direct image of analytic D-module of exponential type.
We describe the category of regular holonomic modules over the ring D[[h]] of linear differential operators with a formal parameter h. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional t-structure related to h-torsion.
Let $X$ be a complex manifold. In "Microlocal study of Ind-sheaves I: microsupport and regularity", M. Kashiwara e P. Schapira made the conjecture that a holonomic D-module $\shm$ is regular holonomic if and only if $R\mathcal{I}{hom}_{β_X\shd_X}(β_X\shm,\sho_X^t)$ is regular (in the sense of "Microlocal study of Ind-sheaves I: microsupport and regularity"), the "only if" part of this conjecture following immediately from "Microlocal study of Ind-sheaves I: microsupport and regularity". Our aim is to prove this conjecture in dimension one.
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.
In this paper we prove the preconstructibility of the complex of tempered holomorphic solutions of holonomic D-modules on complex analytic manifolds. This implies the finiteness of such complex on any relatively compact open subanalytic subset of a complex analytic manifold. Such a result is an essential step for proving a conjecture of M. Kashiwara and P. Schapira (2003) on the constructibility of such complex.
Consider a complex symplectic manifold $X$ and the algebroid $W_X$ of quantization-deformation. For two regular holonomic modules $L_i$ ($i=0,1$) supported by smooth Lagrangian manifolds, we prove that the complex $Rhom_{W_X}(L_1,L_0)$ is constructible and perverse and dual to the complex $Rhom_{W_X}(L_0,L_1)$.
On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.
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…
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.
The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic D-modules and that of constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.
This paper solves the global moduli problem for regular holonomic D-modules 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.
We introduce the notion of strong regular holonomic ${\mathcal{D}}_{{X\times S}/S}$-module and we prove that the functor ${\mathrm{RH}}^S$ introduced by T. Monteiro Fernandes and C. Sabbah in [14] takes image in ${\mathsf{D}}^{\mathrm{b}}_{\mathrm{srhol}}({\mathcal{D}}_{{X\times S}/S})$ (complexes of ${\mathcal{D}}_{{X\times S}/S}$-module whose cohomologies are strongly regular). We prove that for $\dim X=\dim S=1$ the functor solution functor ${}^\mathrm{p}{\mathrm{Sol}}$ restricted to ${\mathsf{D}}^{\mathrm{b}}_{\mathrm{srhol}}({\mathcal{D}}_{{X\times S}/S})$ is an equivalence of categories …
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…
In algebraic geometry, one studies the solutions to polynomial equations, or, equivalently, to linear partial differential equations with constant coefficients. These lecture notes address the more general case when the coefficients are polynomials. The letter D stands for the Weyl algebra, and a D-module is a left module over D. We focus on left ideals, or D-ideals. We represent holonomic functions in several variables by the linear differential equations they satisfy. This encoding by a D-ideal is useful for many problems, e.g., in geometry, physics and statistics. We explain how to work wit…
In this paper we study a correspondence between cyclic modules over the first Weyl algebra and planar algebraic curves in positive characteristic. In particular, we show that any such curve has a preimage under a morphism of certain ind-schemes. This property might pave the way for an indirect proof of existence of a canonical isomorphism between the group of algebra automorphisms of the first Weyl algebra over the field complex numbers and the group of polynomial symplectomorphisms of $\mathbb{C}^2$.
Beilinson--Bernstein localisation relates representations of a Lie algebra $\mathfrak{g}$ to certain $\mathcal{D}$-modules on the flag variety of $\mathfrak{g}$. In [arXiv:2002.01540], examples of $\mathfrak{sl}_2$-representations which correspond to $\mathcal{D}$-modules on $\mathbb{CP}^1$ were computed. In this expository article, we give a topological description of these and extended examples via the Riemann-Hilbert correspondence. We generalise this to a full characterisation of $\mathfrak{sl}_2$-representations which correspond to holonomic $\mathcal{D}$-modules on $\mathbb{CP}^1$ with a…
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）
For a smooth variety $Y$ over a perfect field of positive characteristic, the sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf of $PD$-differential operators) is known to be an Azumaya algebra over $T^*_{Y'},$ the cotangent space of the Frobenius twist $Y'$ of $Y.$ Thus to a sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of $T^*_{Y'},$ called the $p$-support, namely the support of $M$ seen as a sheaf on $T^*_{Y'}.$ We study here the family of $p$-supports assigned to the reductions modulo primes $p$ of a holonomic $\mathcal{D}$-module. We prov…
A polynomial endomorphism $σ\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $σ$ is extended to an endomorphism $σ$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $σ$) $P_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. It is shown that the $A_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. Furthermore, the $A_n$-module ${}^σ P_n$ is holonomic and as a result h…
Let $K$ be an {\em arbitrary} field of characteristic $p>0$ and $\CD (P_n)$ be the ring of differential operators on a polynomial algebra $P_n$ in $n$ variables. A long anticipated {\em analogue of the inequality of Bernstein} is proved for the ring $\CD (P_n)$. On the way, analogues of the concepts of (Gelfand-Kirillov) {\em dimension, multiplicity, holonomic modules} are found in prime characteristic (giving answers to old questions of finding such analogs).An analogue of the {\em Quillen's Lemma} is proved for simple {\em finitely presented} $\CD (P_n)$-modules. In contrast to the character…
We present a method to compute the holonomic extension of a $D$-module from a Zariski open set in affine space to the whole space. A particular application is the localization of coherent $D$-modules which are holonomic on the complement of an affine variety.
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…
We establish some cohomological bounds in D-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the b-function lemma for non-holonomic D-modules.
