7

After a search, I did not find much that has been said about $D$-modules on higher genus curve. I am curious if there is much known about them. Let me provide a summary of what is known first. I am focused on the algebraic setting of nonsingular projective curves over $\mathbb{C}$.

  1. For $\mathbb{P}^1_{\mathbb{C}}$, the $D$-modules here are well understood due to the Beilinson-Bernstein Localization Theorem. In particular, $D$-modules are related to representations of the Lie algebra $\mathfrak{sl}_2$.

  2. For elliptic curves $E$, the derived category of $D$-modules can be understood via the Laumon-Rothstein Fourier transform. In this case, $D$-modules on $E$ are equivalent to coherent sheaves on $E^\natural$, the moduli space of line bundles with integrable connection.

  3. For higher genus curves $C$, there are no references I can find. Even in the case of hyperelliptic curves $C\to \mathbb{P}^1_{\mathbb{C}}$, there isn't anything out there from my search.

Question: What can be said about $D$-modules on higher genus curves $g\geq 2$? In the case of hyperelliptic curves $f:C\to\mathbb{P}^1_{\mathbb{C}}$, can anything be said about holonomic $D$-modules (recall a $D$-module $M$ on $C$ is holonomic iff $f_*M$ is holonomic and so maybe an answer can be given in this way?).

Edit (12/23/24): One thing I wanted to point out for future answers is what I'm not looking for. Recall that there is the classification of $D$-modules on a formal disk (and its punctured version) due to Levelt and Turritin. If one admits this and combines this with Beauville-Laszlo descent, then there is perhaps a way to describe $D$-modules on higher genus curves. The issue is then (a) how descent would work and (b) how $D$-modules on $C$ minus some set of points would behave. I've never seen a precise statement of how this works, but since the description is more "local" as opposed to "global" like the case of genus $0,1$, this is not a description I find as interesting.

hm2020
  • 10,015
user1515097
  • 361
  • 9
  • If $C$ is a smooth projective curve over a field $k$ and if $E$ is a finite rank locally trivial sheaf on $C$, there is the Atiyah class $c(E)$ which is trivial iff $E$ has a connection. The Atiyah class $c(E)$ can be used to define the Chern classes of $E$, hence if $a(E)=0$ is trivial it follows "all Chern classes of $E$ are trivial". Hence having a connection $\nabla$ puts strong conditions on $E$. Are you able to produce non trivial examples of such? – hm2020 Feb 01 '25 at 16:09
  • Note: If $U \subseteq C$ is an affine open subscheme and $E_U$ is the restriction of $E$ to $U$, there will always be a connection $\nabla_U$ on $E_U$. – hm2020 Feb 01 '25 at 16:25
  • Over the complex number field $k$ there is the Riemann-Hilbert correspondence which gives a flat connection $(E(\rho), \nabla)$ from any finite dimensional complex representation $\rho$ of the topological fundamental group $\pi_1(C)$. – hm2020 Feb 02 '25 at 10:10
  • I'm not sure how this helps me -- the Riemann-Hilbert correspondence already ensures I have many regular holonomic $D$-modules I could think about so long as I provide examples of perverse sheaves w.r.t. an algebraic stratification. For example, https://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf gives examples of perverse sheaves on (hyperelliptic) curves. – user1515097 Feb 04 '25 at 01:26
  • you are stating that the push forward of a holonomic D-module $E$ is a holonomic D-module $f_E$. If $E$ comes from a finite dimensional complex representation of the fundamental group, it should follow that $f_E$ is a finite rank vector bundle on $\mathbb{P}^1$ with a flat connection, but $\mathbb{P}^1$ is simply connected, hence there are no such nontrivial examples. Are you sure this statement is correct? – hm2020 Feb 04 '25 at 09:38
  • maybe I'm confused on your claims. $E$ is only a holonomic $D$-module -- which does not guarantee that $E$ is itself a connection or comes from a finite dimensional complex rep of $\pi_1$. That is only true if $E$ is an integrable connection. – user1515097 Feb 04 '25 at 14:21

0 Answers0