Differential Isomorphism and Equivalence of Algebraic Varieties

@article{Berest2003DifferentialIA,
  title={Differential Isomorphism and Equivalence of Algebraic Varieties},
  author={Yuri Yu. Berest and George V. Wilson},
  journal={arXiv: Algebraic Geometry},
  year={2003}
}
In this survey article we discuss the question: to what extent is an algebraic variety determined by its ring of differential operators? In the case of affine curves, this question leads to a variety of mathematical notions such as the Weyl algebra, Calogero-Moser spaces and the adelic Grassmannian. We give a fairly detailed overview of this material. 
View PDF on arXiv
18 Citations
Background Citations
5

18 Citations

On the isomorphism problem for the rings of differential operators on smooth affine varieties

Differential operators on toric varieties and Fourier transform

. We show that Fourier transforms on the Weyl algebras have a geo-metric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential…

Differential operators on toric varieties and Fourier transform

Abstract.We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential…

A Remark on Letzter-Makar-Limanov Invariants

We give a natural cohomological interpretation of Letzter-Makar-Limanov invariants for rings of differential operators on algebraic curves.

$A_{\infty}$-modules and Calogero-Moser Spaces

We re-examine the bijective correspondence between the set of isomorphism classes of ideals of the first Weyl algebra and associated quiver varieties (Calogero-Moser spaces) \cite{BW1, BW2}. We give…

On subgroups of the Dixmier group and Calogero-Moser spaces

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1(k) $. In particular, we give a geometric presentation for these groups in terms of…

Cusps and D-modules

1.1. V-modules on singular varieties. Let Y denote a variety over a field and let Vy denote the full sheaf of differential operators on Y (in characteristic zero Vy is the familiar sheaf of…

Quasi-invariants of complex reflection groups

Abstract We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in…

Trees, Amalgams and Calogero-Moser Spaces

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1 $. In particular, we give a geometric presentation for these groups in terms of…

A∞-modules and Calogero-Moser spaces

The relations between these objects are well known and almost immediate. Thus, (1) is essentially the definition of (closed) points of HilbðCÞ. The bijection (1)! (2) is given by taking the quotient…

References

SHOWING 1-10 OF 53 REFERENCES

Differential operators on semi-invariants for tori and weighted projective spaces

In this paper we study rings of differential operators for modules of covariants for one-dimensional tori. In particular we analyze when they are Morita equivalent, when they are simple, and when…

Noncommutative Projective Schemes

An analogue of the concept of projective scheme is defined for noncommutative N-graded algebras using the quotient category C of graded right A-modules modulo its full subcategory of torsion modules.…

Nakai’s conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi’s extension of Nakai’s conjecture to arbitrary characteristic. This gives a new…

Differential operators on rational projective curves

When rings of differential operators are maximal orders

Abstract Let A be a commutative domain, finitely generated as an algebra over a field k of characteristic zero and write (A) for the ring of k -linear differential operators. Then A is an Ore domain…

Some Morita Equivalences of Rings of Differential Operators

Abstract Let N be a free Z -module of rank r and M its Z -module dual. Let σ be a simplicial cone in N ⊗ Z Q and σ∨ its dual cone in M ⊗ Z Q . Let Λ be a subsemigroup of the additive semigroup σ ∨ ∩…

Rings of differential operators over rational affine curves

— Let X be an irreducible algebraic curve over the complex numbers such that its normalization is the affine line, and the normalization map is injective. Let D(X) denote its ring of differential…

The differential operator ring of an affine curve

The purpose of this paper is to investigate the structure of the ring D(R) of all linear differential operators on the coordinate ring of an affine algebraic variety X (possibly reducible) over a…

Differential Operators on an Affine Curve

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by 3)x the sheaf of differential operators on X, and 2)(X) = F(X, 3)x), the ring…

Limits of Compactified Jacobians and D-Modules on Smooth Projective Curves

Abstract LetSbe a smooth projective curve and D Sits sheaf of differential operators. This paper classifies the rank one torsion-free D S-modules up to isomorphism. Such a module E has a degree which…
...