Suppose the base field algebraically closed and of zero characteristic.
There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion is Y. Berest, G. Wilson, "Differential isomorphism and equivalence of algebraic varieties" MR2079372, whose terminology I use here).
(Differential isomorphism) Let $X$ and $Y$ be two irreducible affine varieties. If they have isomorphic rings of differential operators $\mathcal{D}(X) \simeq \mathcal{D}(Y)$, are they isomorphic? In general, what this says about the relation between $X$ and $Y$?
(Differential equivalence) Let $X$ and $Y$ be two irreducible affine varieties. If $\mathcal{D}(X)$ and $\mathcal{D}(Y)$ are Morita equivalent, what this says about the relation between $X$ and $Y$?
It can be shown without much difficulty that for an affine irreducible $X$, $\mathcal{D}(X)$ is an Ore domain, and hence admits a skew-field of fractions, $\mathbb{D}(X)$.
- Question 1: If $X$ and $Y$ are two irreducible affine varieties and $\mathbb{D}(X) \simeq \mathbb{D}(Y)$, are they birationally equivalent? In general, what this implies about the relation between $X$ and $Y$?
There are only two facts I know regarding this question.
The first one is that if $\mathbb{D}(X) \simeq \mathbb{D}(Y)$, then $X$ and $Y$ must have the same dimension, as expected. For smooth varieties this is a well known result with different proofs in the literature, but if we do not assume the varieties smooth, the only proof I know uses some rather non-trivial facts about the lower transcendece degree by J. J. Zhang, which is a noncommutative analogue of the commutative transcendence degree.
The only other work I know of in this direction is J. P. Bell and A. Smoktunowicz, "Rings of differential operators on curves", MR3004084, for $X$ and $Y$ smooth projective curves. In the paper it is shown that if $\mathbb{D}(X) \simeq \mathbb{D}(Y)$, then $X$ and $Y$ should have the same genus. However, if $X$ and $Y$ are non-isomorphic curves of the same genus, it is unkown if $\mathbb{D}(X)$ should be different of $\mathbb{D}(Y)$.
Edit: adding more context The above question about curves is particularly relevant in view of Artin's conjecture about the birational classification of noncommutative projective surfaces. Namely, Artin has given an finite and small list of, conjecturally, all division algebras that can appear as "noncommutative funtion fields" of such surfaces. In the list we have the division algebras $\mathbb{D}(X)$, where $X$ is a smooth projective curve. Hence, the problem of determining when, give two curves $X$ and $Y$, $\mathbb{D}(X)$ and $\mathbb{D}(Y)$ are isomorphic is a very important one.
- Question 2: How we can distinguish $\mathbb{D}(X)$ from $\mathbb{D}(\mathbb{A}^n)$; i.e., show that it is not a Weyl field (the skew-field of fractions of the Weyl algebra)?
I know of something about Question 2 related to the famous Gelfand-Kirillov Conjecture (cf. J. Alev, A. Ooms, M. Van den Bergh, "A class of counterexamples to the Gel'fand-Kirillov conjecture", MR1321564).
