24
$\begingroup$

D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic geometry, real analytic geometry, and algebraic geometry. However, it seems to me (as an outsider) that the literature on D-modules does not treat the case of smooth manifolds.

I have a few related questions:

  1. Are D-modules a useful notion in the smooth manifold setting?
  2. If so, is there a good reference that discusses D-modules in the smooth manifold setting?
  3. Is the theory of D-modules useful for studying flat real vector bundles and their corresponding local systems on smooth manifolds?

I assume that since smooth manifolds can be defined as locally ringed spaces, then at least some of the theory of $D$-modules must carry over. I realize that the sheaf of smooth functions on a manifold is soft, but I would hope that doesn't make D-modules over smooth manifolds uninteresting.

Improve this question
$\endgroup$
3
  • 4
    $\begingroup$ Well, one of the early results which generated a lot of interest in D-modules was Bernshtein's proof that a linear differential operator on $\mathbb{R}^n$ with constant coefficients has a fundamental solution. The argument has generalizations at least to certain kinds of operators and certain kinds of manifolds. math1.tau.ac.il/~bernstei/Publication_list/publication_texts/… $\endgroup$
    – Paul Siegel
    Commented Mar 7, 2017 at 13:05
  • $\begingroup$ What is your definition of D-modules in this context? A natural definition would be, I guess, quasicoherent sheaves on the "smooth de Rham space". These quasicoherent sheaves should encode much more information than derivatives. $\endgroup$
    – Z. M
    Commented Jul 28 at 21:39
  • 1
    $\begingroup$ See the introduction of arxiv.org/abs/1111.2087 $\endgroup$
    – coLaideronnette
    Commented Jul 29 at 10:47

1 Answer 1

Reset to default
3
+25
$\begingroup$

Your « outsider » hunch is correct. Basically, the fact that there exist $C^\infty$ functions which are nowhere analytic is ruining the D-modules approach. It is in fact even worse: there exist $C^\infty$ functions whose analytic wave-front-set is the full sphere bundle.

No D-modules approach could have predicted the non-uniqueness result of Tychonoff for the linear heat equation: there exist non-zero smooth solutions $u$ of $$ \frac{\partial u}{\partial t}-\Delta_x u=0 \text{ on $\mathbb R_t\times \mathbb R_x$},\qquad \forall x\in \mathbb R, u(0,x)=0. $$

Improve this answer
$\endgroup$
4
  • 3
    $\begingroup$ Can you give more detail as to what aspects of the theory of smooth D-modules this will ruin? There are analytic D-modules whose singular support is the whole cotangent/sphere bundle, e.g. $\mathcal{D}_X$ itself. $\endgroup$
    – Pulcinella
    Commented Jul 29 at 9:15
  • $\begingroup$ @Pulcinella What is your expected definition of D-modules in the smooth context? $\endgroup$
    – Z. M
    Commented Jul 29 at 9:54
  • $\begingroup$ Probably just $\mathcal{O}$-modules on the de Rham stack of $X$, e.g. mathoverflow.net/questions/415565/de-rham-via-topoi $\endgroup$
    – Pulcinella
    Commented Jul 29 at 10:00
  • 2
    $\begingroup$ @Pulcinella The Tychonoff counterexample is using the existence of non-zero smooth functions which are flat at $t=0$ and also of non-temperate distributions. Again, I am very doubtful that this type of counterexample could have been produced via an algebraic approach. $\endgroup$
    – Bazin
    Commented Jul 29 at 17:14

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .