Related papers: On the D-module of an isolated singularity

We develop the formalism of derived divided power algebras, and revisit the theory of derived De Rham and derived crystalline cohomology in this framework. We characterize derived De Rham cohomology of a derived commutative algebra $A$ over…

For a germ $f$ on a complex manifold $X$, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the…

In the simplicial theory of hypercoverings, we replace the indexing category $\Delta$ by the \emph{symmetric simplicial category} $\Delta S$ and study (a class of) $\Delta S$-hypercoverings, which we call \emph{spaces admitting symmetric…

This is now an expository note about the following classical problem. Let $(X, \bf 0)$ be the germ of a hypersurface in $(\mathbb C^n,\bf 0)$ with an ordinary singularity of multiplicity $m$ at the origin $\bf 0$. A natural question to ask…

We show that for a noetherian algebra $A$ whose bounded dg derived category is smooth, the singular Hochschild cohomology (=Tate--Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity…

If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve…

We study the ring of differential operators D(X) on the basic affine space X=G/U of a complex semisimple group G with maximal unipotent subgroup U. One of the main results shows that the cohomology group H^*(X,O_X) decomposes as a finite…

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…

Based on the recent progress in the irregular Riemann-Hilbert correspondence, we study the monodromies at infinity of the holomorphic solutions of Fourier transforms of holonomic D-modules in some situations. Formulas for their eigenvalues…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of…

We calculate the category of D-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham…

Let R be either a polynomial or a formal power series ring in a finite number of variables over a field k of characteristic p > 0 and let D be the ring of klinear differential operators of R. In this paper we prove that if f is a non-zero…

Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There…

We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a…

In this article we study the exterior power structure of the algebraic de Rham cohomology group associated with the Gelfand hypergeometric function and its confluent family. The hypergeometric function F(z) is a function on the Zariski open…

Let $R$ be a polynomial or power series ring over a field $k$. We study the length of local cohomology modules $H^j_I(R)$ in the category of $D$-modules and $F$-modules. We show that the $D$-module length of $H^j_I(R)$ is bounded by a…

In this paper, we study minimal free resolutions for modules over rings of linear differential operators. The resolutions we are interested in are adapted to a given filtration, in particular to the so-called V-filtrations. We are…

In this work we consider a family of function classes constructed by means of the Gauss hypergeometric function $_2F_1(1,1;2;z) =-\frac{\log(1-z)}{z}$. We demonstrate that this family, in fact, constitutes classes of analytic functions…

We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions, and derive applications regarding the local cohomological dimension, the Du Bois…

We compute the moduli of endomorphisms of the de Rham and crystalline cohomology functors, viewed as a cohomology theory on smooth schemes over truncated Witt vectors. As applications of our result, we deduce Drinfeld's refinement of the…