Related papers: A Localization Algorithm for $D$-modules
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…
In this paper, we study the holonomic $D$-modules when $D$ is the ring of $k$-linear differential operators on $A = k[\Gamma]$, the coordinate ring of an affine monomial curve over the complex numbers $k = \mathbb C$. In particular, we…
Given a $D$-module $M$ generated by a single element, and a polynomial $f$, one can construct several $D$-modules attached to $M$ and $f$ and can define the notion of the (generalized) $b$-function following M. Kashiwara. These modules are…
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…
The aim of this note is a combinatorial description of a category of $D$-modules over an affine space, smooth along the stratification defined by an arrangement of hyperplanes. These $D$-modules are assumed to satisfy certain non-resonance…
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.
An algorithm computing the restriction of a holonomic D-module to a linear subspace was given by T.Oaku in 1997. We consider a problem of computing the restriction for a given holonomic D-module with parameters. We will give a partial…
In this paper, we treat $\mathscr{D}$-modules on the basic affine space $G/U$ and their global sections for a semisimple complex algebraic group $G$. Our aim is to prepare basic results about large non-irreducible modules for the branching…
Let $f$ be a real polynomial of $x = (x_1,\dots,x_n)$ and $\varphi$ be a locally integrable function of $x$ which satisfies a holonomic system of linear differential equations. We study the distribution $f_+^\lambda\varphi$ with a…
Let K be a subfield of the complex numbers, and let D be the Weyl algebra of K-linear differential operators on K[x_1,...,x_n]. If M and N are holonomic left D-modules we present an algorithm that computes explicit generators for the finite…
We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex…
Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the…
We generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of graph(F)\cup(\partial D\times D) -- where D is the open unit disc and graph(F) denotes the graph of a continuous D-valued function F -- to the…
The goal of this paper is to generalize several basic results from the theory of $\cal{D}$-modules to the representation theory of rational Cherednik algebras. We relate characterizations of holonomic modules in terms of singular support…
We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an l-dimensional coordinate…
We consider the D-module defined as the push-forward of a rank one linear system on the complement of a central plane hyperplane arrangement, and calculate its decomposition series, using algebraic calculations in the Weyl algebra.
In this text, we illustrate the use of local methods in the theory of (irregular) holonomic D-modules. I. (The Euler characteristic of the de~Rham complex) We show the invariance of the global or local Euler characteristic of the de~Rham…
We study the growth of representations of the Lie algebra of vector fields on the affine space that admit a compatible action of the polynomial algebra. We establish the Bernstein inequality for these representations, enabling us to focus…
Let $k$ be a field of characteristic zero and I an ideal defining an arrangement of linear subspaces in the affine space $A^n_k$. We compute the D-module theoretic characteristic cycle of the local cohomology modules $H^r_I(k[x_1,...,x_n])$…
We use localization technique to construct new families of irreducible modules of affine Kac-Moody algebras. In particular, localization is applied to the first free field realization of the affine Lie algebra A_1^{(1)} or, equivalently, to…