Archive for May 2025

The Quill 17 ~ D-Modules

$\mathcal{D}$-modules are very important objects in algebraic geometry and mathematical physics (for example, in string theory). In a previous post, we saw that Hecke eigensheaves on a $Bun_G$ (a set of isomorphisms of vector bundles) are $\mathcal{D}$-modules. They are a quasi-coherent sheaf of the scheme of differential operators.


As the name suggests, $\mathcal{D}$-modules are just modules over the ring of differential operators on some variety $X$. 
Let $X$ be a smooth variety over a field $k$ (of char 0), then $O_X$ is the structure sheaf over it. Let $\mathcal{F}$ be a quasi-coherent sheaf over $O_X$. Let us denote $D$ as a differential operator and for open affine subset $U\in X$, $\mathcal{D}(U)$ denotes the ring of differential operators on $X$. Then $U \to D(U)$ is a quasi-coherent sheaf of $O_X$-modules. And we call this sheaf of differentials operators $\mathcal{D}$-modules.

$\mathcal{D}$ is sheaf of non-commutative algebra. A very good example is Weyl Algebra $A_n(k)$. Locally, they are generated by the Heisenberg commutation relations here. Globally, we are interested in seeing if a solution exists by gluing them. So, $\mathcal{D}$-modules help us to understand local-global pictures of linear differential equations.

There is a specific class of D-modules that were of interest to Kashiwara - holonomic modules in the Riemann-Hilbert correspondence. Anyway, the central concept to understand in $\mathcal{D}$ is that they are a sheaf of modules over the ring of differential operators and they are quasi-coherent to $O_X$-modules. For physicists, geometric Langlands has been putting the duality between the category of D-modules on $Bun_G$ and the Fukaya category of Lagrangians in its cotangent bundle. Moreover, $\mathcal{D}$-modules give a geometric meaning to the automorphic forms.

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The Quill 16 ~ What is $Bun_n(\mathbb{F}_q)$?

We take a finite field $\mathbb{F}_q$ and define a curve (smooth, projective) over it, $X$. Now $X$ can also be taken on a Riemann surface when doing geometric Langlands (that is, switching from the curves defined over the finite fields to curves defined over the Riemann surface). Anyway, we define 

$$Bun_n = \{\text{set of isomorphism classes of $n$ rank (holomorphic) vector bundles on curve } X \}$$
$Bun_n$ is a moduli space and an algebraic stack, so a moduli stack! The set would be countable, as there are finitely many vector bundles over each finite field extension. One is then interested in studying a restricted (cuspoidal) space of functions on $Bun_n$.

Well, the Hecke operators (which correspond to modifications at points) from classical Langlands can be geometrized as well. They can be projected to the $Bun_n$ as well. We can define Hecke eigensheaves on $Bun_n$ which correspond to moduli of the rank $n$ local systems on the curve $X$. This is the geometric Langlands correspondence. There is the Hecke correspondence between the Hecke stack and the moduli stack.
Moreover, these Hecke eigensheaves are D-modules on $Bun_n$ satisfying a certain property set by the vector bundle $E$.

Now, a very short remark on why $Bun_n$ is an algebraic stack but not a scheme. Because the moduli space has non-trivial automorphisms. Moreover, a general scheme can not track the automorphisms, so if you were to construct a scheme $Bun_n$, you would fail to account for the automorphism groups. Moreover, the stack structure on $Bun_n$ is important for Langlands correspondence as well, even for defining D-modules on it or for the Hecke correspondence to work. We will look into the Hecke eigenvsheaves and what comes before all of these geometrizations in some later post.

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