7
votes
0answers
313 views

Why should the Laplacian in $\mathbb{C}^n$ act on a specific line bundle over the quadric $x^2=0$ in $\mathbb{P}^{n-1}$?

I recently encountered the following nice fact, and I'm wondering if it's part of a more general story. Let $x\in \mathbb{C}^n$ satisfy $$x^2:=\sum_i x_i^2 = 0,$$ and consider functions $f(x)$ ...
1
vote
0answers
82 views

Action of Landweber-Novikov algebra on infinite polynomial ring

Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...
4
votes
1answer
247 views

Poisson structure on the cotangent bundle

Let $X$ be a smooth variety over $\mathbb{C}$ and $\mathscr{A}$ a sheaf of twisted differential operators on $X$. The latter comes equipped with a natural filtration and the associated graded algebra ...
7
votes
3answers
446 views

Is there any general index theorem for manifold with boundary?

My understanding is Atiyah-Patodi-Singer solved the index theorem for manifold with boundary only for certain types of Dirac operators, correct? There is still no (or no hope to get) uniform theorem ...
1
vote
1answer
432 views

Lie bracket of algebraic vector fields

Let $X$ be an algebraic variety (over any field). The definition of the Lie bracket of two vector fields on $X$ (i.e. sections of the tangent sheaf) which I know characterizes vector fields as ...
7
votes
1answer
441 views

Is there a really big ring of differential operators in characteristic p?

$k$ is a field of characteristic $p$. $k[t]$ has canonical first-order differential operator $\partial$ As an endomorphism of $k[t]$, $\partial^p=0$. First way to fix it: Use the divided power ...
2
votes
1answer
395 views

What's an example of a commutative algebra over $\mathbb Q$ that fails to satisfy this version of the “PBW theorem”

In a recent question, I recalled the notion of differential operator, polyderivation, and principal symbol for a commutative algebra $A$ over some fixed commutative ring $k$. (I will not repeat those ...
4
votes
1answer
423 views

Localizability of differential operators a la Grothendieck

Hello, Maybe this question is trivial, so sorry Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1). Then we can define the module of differential operators $D^{\leq n} ...
6
votes
1answer
433 views

For which algebras does \{Differential Operators\} satisfy a PBW-like theorem?

Let $k$ be a commutative ring, $A$ a commutative $k$-algebra, and for some other part of why I'm asking this question I only care about the case when $k \supseteq \mathbb Q$. Recall the following ...
11
votes
2answers
1k views

The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...
8
votes
3answers
722 views

Is there a good account of D-affinity and localization theorem for partial flag varieties?

Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated ...
4
votes
0answers
271 views

Why should we consider D-module on flag variety of Lie algebra?

Why don't we stay at D-module on base affine space but go to study flag variety of Lie algebra? I remembered there are nice papers of Bernstein-Gelfand-Gelfand and Gelfand-Kirillov discussing the ...
18
votes
4answers
2k views

Have people successfully worked with the full ring of diferential operators in characteristic p?

This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...