D Modules Bernstein

Monday, December 28, 2020

Lecture 3 020702 14 4. Preservation of holonomicity under direct images 136 4.

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In m y lecture I will discuss the theory of mo dules o v er rings di eren tial op erators for short D-mo dules.

D modules bernstein. Perhaps the most conceptual though not the easiest way is to deduce it from the involutivity of the annihilator Ga. We begin with deﬁning some basic functors on D-modules introduce the notion of characteristic variety and of a holonomic D-module. Gröbner Bases in D-Modules.

Organization and references 2 13. Bernstein inequality 119 8. Base change 128 iii.

De nition and examples 3 22. Pullback and pushforward of D-modules 8 24. D-modules on smooth algebraic varieties 3 21.

October 27 2020 300 pm - 400 pm. Works of Beilinson-Bernstein and Kashiwara and since then had a number of spectacular applications in Algebraic Geometry Representation theory and Topology of singular spaces. This is a useful reference that lists most of the relevant D-module theory without proof.

However the category DDy is more geometric and in particular provides a geometric interpretation of the Ext-groups. Closed immersions and Kashiwaras theorem 120 9. We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system SingularWe discuss new approaches to the computation of Bernstein operators of logarithmic annihilator of a polynomial of annihilators of rational functions as well as complex powers of polynomials.

To end the seminar we want to give the classical application of D-modules given by BeilinsonBernstein and Kashiwara to the KazhdanLusztig conjecture. As applications we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. Holonomic D-modules 133 2.

D-modules on general a ne. Classification of simple holonomic D-modules. Left and right D-modules 6 23.

D-modules and Bernstein-Sato polynomials. Part of the Algorithms and Computation in Mathematics book series AACIM volume 28 Abstract. We study log D -modules on smooth log pairs and construct a comparison theorem of log de Rham complexes.

About Daniel Bath is a PhD candidate at Purdue University. We discuss b-functions and study the Riemann-Hilbert correspondence between. D-mo dules and functors.

D-modules Bernstein-Sato ideals and topology of rank 1 local systems. Analytic continuation of distributions with respect to a parameter and D-modules 013102 5 2. Bernsteins inequality and its applications 020502 10 3.

This would require from the beginning to use more sophisticated homological technique derived categories. 1 Introduction to functional equations. In this part we will study D-modules on general algebraic varieties.

Consider sdi erential operators P 1P s2D An. Bernstein Lectures on D-modules. Application to Bernstein-Sato Polynomials.

Holonomic D-modules and duality. The BeilinsonBernsteinDeligne definition of a perverse sheaf proceeds through the machinery of triangulated categories in homological algebra and has a very strong algebraic flavour although the main examples arising from GoreskyMacPherson theory are topological in nature because the simple objects in the category of perverse sheaves are the intersection cohomology complexes. Lectures 13 and 14 updated 325.

NOTES ON D-MODULES AND CONNECTIONS WITH HODGE THEORY 5 Theorem 118 Bernsteins inequality. D-MODULES BERNSTEIN-SATO POLYNOMIALS F-INVARIANTS 5 that is the free R-module generated by the diﬀerential operators 1 t. Via Zoom Video Conferencing Speaker.

D-modules and their applications Joseph Bernstein Nov 04 2007 D-modules and their applications is the second part of a year long advanced course. Lei Wu - University of Utah. The singular support 14 25.

Akhil Mathews Notes on. Granger Universit e dAngers LAREMA UMR 6093 du CNRS 2 Bd Lavoisier 49045 Angers France Meeting of GDR singularit es. A right D-module is de ned similarly.

For any nonzero ﬁnitely generated D n-module we have dimChM n. Local cohomology of D-modules 124 10. This means that this ideal is closed under the Poisson bracket.

X n where Kis a eld of characteristic zero or to KX 1. Conventions and notation 3 14. Iv CONTENTS Chapter V.

Week of Mar 30. In case of general. We denote by ModD X the collection of left D-modules and ModD op X the collection of right D-modules.

D-modules Bernstein--Sato ideals and polynomials cohomology support loci the logarithmic de Rham complex resonance varieties. O-coherent D-modules vs. Compared to DDy the category D LDy G has the advan-tage of being defined in terms of D-modules on the space Y itself whereas to define DDy one has to use free resolutions P -t Y of the G-space Y.

He does research in algebraic geometry and commutative algebra often related to D-modules. We will spend some time discussing this technique. D-modules on singular varieties.

Mar 4 Lecture 12. The proof uses Sabbahs generalized b -functions. A useful reference for any technical details or proofs of theorems stated in Bernsteins notes.

A left D-module is a sheaf Mon Xsuch that MU is a left D XU-module for each open U X. D-modules Perverse Sheaves and Representation theory by Hotta Takeuchi Tanisaki. D-MODULES AND REPRESENTATION THEORY Contents 1.

Bernsteins notes on Algebraic D-modules. There are a number of ways to prove this. We relate properties of D-modules over R to D-modules over SWe show that the localization R f and the local cohomology module H I i R have finite length as D-modules over RFurthermore we show the existence of the BernsteinSato polynomial.

Here we will have to be brief but the outline given in Bernsteins notes explains this in a short way. A classi cation of irreducible holonomic modules 138 5. Sato Kashiw ara Ka w ai Bernstein Ro os Bj ork Malgrange Beilinson.

Local cohomology of holonomic modules 140 6. Let us consider a ring Requal to KX 1. This book also contains a good exposition of t-structures.

Mar 2 Lecture 11. JBernstein Algebraic theory of D-modules J-PSchneiders An introduction to D-modules PMaisonobe CSabbah Aspects of the theory of D-modules DArapura Notes on D-modules and connections with Hodge theory Geometric representation theory Geometric Langlands seminar webpage VGinzburg Geometric methods in representation theory of Hecke algebras and quantum groups VGinzburg. D-modules on a ne varieties 5 1.

The log index theorem naturally generalizes the. Functional dimension and homological algebra 21 5. Hodge ideals and mixed Hodge modules 1-5 avril 2019 Angers France Preliminary version.

In this chapter we introduce Gröbner bases in a particular non-commutative ring and we show how they can be. Lecture 5 021402 24 6. Then consider M D An.

We study the structure of D-modules over a ring R which is a direct summand of a polynomial or a power series ring S with coefficients over a field. The generic vanishing theorem of Green-Lazarsfeld says that for general elements in the Picard variety of a projective manifold their cohomology groups. Holonomic D-modules 133 1.

Lectures 15 and 16 updated 41. This theory started ab out 15 y ears ago and no w it is clear that has v ery aluable applications in man y elds of mathematics. Week of Mar 23.

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