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First of all, let me warn that my knowledge of the correspondence is rather superficial, and I apologize for any technical inaccuracies below.

Setting

Let $X$ be a smooth complex algebraic variety, and $D_X$ the sheaf of differential operators on $X$. A $D$-module on $X$ is a sheaf of $D_X$-modules.

The Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic $D_X$-modules and that of constructible sheaves. ("Recently'' extended to the irregular case by D'Agnolo and Kashiwara (link).)

Holonomic $D_X$-modules are in particular coherent over $D_X$. Furthermore, $O_X$-coherent $D_X$-coherent $D$-modules are locally free of finite rank, that is algebraic vector bundles on $X$ with flat (or integrable) connection.

Equivalently, the Riemann-Hilbert correspondence gives an equivalence from the category of flat connections on algebraic vector bundles on $X$ with (regular) singularities to the category of local systems of finite-dimensional complex vector spaces on $X$.

Questions

  • General: Can such a correspondence be generalized to vector bundles with not necessarily flat connections? If so, what would we expect local systems to be replaced with? If not, what are the obstacles to lifting flatness?

  • Specific: Where is the flatness/non-flatness of the connection being encoded in the correspondence on both sides? Namely, where can one exactly read flatness/non-flatness?

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    $\begingroup$ Holonomic $D_X$-modules need not be coherent as $O_X$ modules, but the ones that are will be vector bundles over $X$. Perhaps this is what you meant to say. $\endgroup$ – Keerthi Madapusi Pera Aug 18 '17 at 22:02
  • $\begingroup$ Right, holonomic $D_X$-modules are coherent as $D_X$-modules but not necessarily as $O_X$-modules. As a $O_X$-module, $D_X$-modules are generally quasi-coherent (locally free, albeit of infinite rank). Right? Is there then an analog/interpretation of these $D_X$-coherent but not necessarily $O_X$-coherent modules in terms of vector bundles (possibly with non-flat connection)? Or, geometrically, is something completely different? $\endgroup$ – Carlos Aug 19 '17 at 1:37
  • $\begingroup$ There are certainly results about non-flat connections, but of some very special kinds. The best known may be the Kobayashi-Hitchin correspondence, proven by Donaldson and Uhlenbeck-Yau, which associates an hermitian Kähler-Einstein vector bundle to a stable vector bundle. $\endgroup$ – Wille Liou Aug 19 '17 at 10:07
  • $\begingroup$ Although the Kobayashi-Hitchin correspondence does indeed deal with certain non-flat connections (the Hermite-Einstein connection), it is entirely within the same geometrical side: semi-stable vector bundles and Einstein-Hermitian vector bundles. $\endgroup$ – Carlos Aug 19 '17 at 15:51
  • $\begingroup$ I don't understand your question, but maybe you're interested in the Atiyah groupoid, ncatlab.org/nlab/show/Atiyah+Lie+groupoid and the fact that $Rep (At (P)) \cong Rep (G)$ for $P$ a principal $G$-bundle.This isomorphism is a generalization for the smooth case by using $P$ equals to the universal covering space of the base space. For non smooth stuff you probably need some stratified version (using the exit path category maybe instead of the path groupoid). In any case, its also true that the holonomy completely determines a vector bundle with connection for the smooth case. $\endgroup$ – user40276 Aug 23 '17 at 22:13
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I'm a little bit familiar with the smooth category version. Fixing a basepoint $b$, you can parallel transport the fiber from the base point to any other point $p$. The transfer is path-independent if and only if going around the loop $b \to_1 p \to_2 b$ has trivial holonomy. Now according to the Ambrose-Singer theorem if the curvature of the connection is zero, the only nontrivial holonomy can come from topologically nontrivial loops and thus you get a local system. If the curvature doesn't vanish you may get a nontrivial holonomy even along homotopically trivial loops and thus you can try attaching the restricted holonomy groups to each fiber which would capture this behaviour.

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  • $\begingroup$ Good, thank you. This is indeed in the direction I'm roughly aiming to: the holonomy group is controlling the curvature of the connection. The question now is: where and how can I read the holonomy at the level of the triangulated category of $D$-coherent modules? $\endgroup$ – Carlos Aug 19 '17 at 15:48
  • $\begingroup$ Also, what kind of object do we get by "attaching the restricted holonomy groups to each fiber"? $\endgroup$ – Carlos Aug 19 '17 at 15:48
  • $\begingroup$ @Carlos I haven't seen this done anywhere but then again I wasn't looking for it. Sorry. If you do find something, you can always answer your own question. (And please do! I'd be delighted to read it.) $\endgroup$ – Vít Tuček Aug 21 '17 at 7:32
  • $\begingroup$ If you have a flat connection, you can still have nontrivial global holonomy, but you can see it only when you consider parallel transport along homotopically nontrivial loops which is pretty much what the $\mathrm{Sol}$ functor is doing. The main problem here is that once your connection has nontrivial curvature you don't have a $\mathcal{D}$-module! So either there is some cunning way to modify or extend the connection to get a $\mathcal{D}$-module or you get a Riemann-Hilbert correspondence outside of the theory of $\mathcal{D}$-modules. $\endgroup$ – Vít Tuček Aug 21 '17 at 7:34
  • $\begingroup$ It was already extremely useful, thank you. So it doesn't seem to hurt to understand properly what the $\mathrm{Sol}$ functor is really doing. $\endgroup$ – Carlos Aug 21 '17 at 21:10
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I'm not sure there is a full answer to the "General" question, at least, within the $D$-module setting. I'll leave some remarks about non-flatness partially answering the question, and perhaps someone will tune in and complete this to a proper answer.


First off, to understand how one can lift flatness we have to know where and how it is being encoded on each side of the correspondence:

-on the $D$-module side it comes straight from the tangent sheaf $\Theta_X$ being a Lie subalgebra of the algebra of differential operators $D_X$. It is, thus, how $\Theta_X$ sits inside $D_X$ that determines the flatness of the underlying connection;

(Aside: We may consider deformations of the algebra $\Theta_X$ where we still preserve the Lie algebra structure, generally leading to twisted $D$-modules. These seem to correspond to connections with fixed scalar curvature (mathoverflow), which under the correspondence give constructible sheaves on a gerbe over the base (mathoverflow). $D$-module theory proper doesn't seem to be the natural ground to explore generic non-flat vector bundles and, therefore, below we will make do with the subset of $O_X$-coherent $D$-modules, i.e. plain vector bundles with flat connection.)

-on the local systems side, the understanding of flatness comes from viewing them as representations of the fundamental groupoid. The fundamental groupoid encodes the notion of paths up to homotopy, which then assembles to a connection where any curvature is washed out. So, it is homotopy invariance that kills curvature. There's still holonomy in the form of monodromy, which is discrete due to flatness. Thus, if we want to lift flatness we need a weaker notion of homotopy invariance.


Let us summarize the web of correspondences at play:

  1. vector bundles with flat connection

  2. local systems

  3. representation of the fundamental groupoid

We go from 1. to 2. via the parallel sections functor $\ker\nabla$, with flatness translating into Frobenius integrability condition on the existence of (unique local) solutions to the first-order ODEs $\ker\nabla$ given arbitrary initial data. These are finite-dimensional vector spaces. The correspondence 2. to 3. arises via standard results from covering spaces (already mentioned above; see mathoverflow). We may go directly from 1. to 3. by calculating the parallel transport of the (flat) connection, which descends to a representation of the fundamental group as result of flatness (equivalently, it is the monodromy of the solutions to $\ker\nabla$). The inverse direction goes via the associated bundle construction.

Hence, there are two places where one may explore "non-flatness": the $\ker\nabla$-functor and the parallel transport functor.

$\ker\nabla$-functor

I've partially addressed this recently here Kernel of a non-integrable connection, but let me go over it again for the sake of completeness.

For not necessarily flat connections, $\ker\nabla$ is still a well-defined subsheaf of the sheaf of sections of the vector bundle. In particular, it is a sheaf of finite-dimensional vector spaces with the stalk-rank $\dim(\ker\nabla)_x$ bounded by the vector bundle rank. However, $\ker\nabla$ is no longer a local system if $\nabla$ is not flat. That happens if and only if $$ x\mapsto \dim(\ker\nabla)_x $$ is a locally constant function on $X$ (see Lemma 1.6 Conrad). Furthermore, given the stratification of $X$ by stalk-dimension $$ X^{\leq d}:=\{x\in X | \dim(\ker\nabla)_x\leq d\}\,, $$ we have that the restriction of $\ker\nabla$ to the subsets $X^{\leq d}-X^{\leq d-1}$ is locally constant for all $d$. Hence, cleary $\ker\nabla$ is a constructible sheaf.

This map from vector bundles with connection to constructible sheaves is certainly not invertible. As already mentioned by @user40276 the exit-path category is relevant here. In fact, a theorem due to MacPherson says that the category of constructible sheaves on $X$ is equivalent to the category of representations of the exit-path category (see Treumann). The exit-path category encodes the notion of exit-paths again up to homotopy. Therefore, on each stratum the curvature of the connection is being killed again, and what we are able to recover is a constructible vector bundle with flat connection (see this thesis of a Block's student; and also mathoverflow). Basically, it is a flat vector bundle on each stratum.

The question remains on how to better characterize the image of $\ker\nabla$ such that we recover a correspondence. See related questions mathoverflow, math.stackexchange.

Parallel transport functor

In the smooth setting, the parallel transport of a connection can be assembled into a functor $\text{tra}$ by sending each point $x\in X$ to the vector space $E_x$, the fiber of $E$ over $x$, and a path $\gamma: x\to y$ to the parallel transport map $\text{tra}(\gamma):E_x\to E_y$. This notion is invariant under thin-homotopy, thus defining a functor from the path-groupoid of $X$ to $\text{Vect}$. Thin-homotopy is indeed the proper refinement of full homotopy to be able to accommodate (non-zero) curvature. In fact, as shown by (Schreiber & Waldorf), and also by (Berwick-Evans & Pavlov), the category of vector bundles with (not necessarily flat) connection is equivalent to the category of representations of the path-groupoid.

Remark: Notice that there's a canonical functor from the path-groupoid to the fundamental groupoid, by sending thin-homotopies to full homotopies. Therefore, a vector bundle with connection is flat if, as a representation of the path-groupoid, it factors through the fundamental groupoid.


Therefore, to close this to a circle of correspondences, as in 1.-2.-3.-1. above, and answer the question being asked we would need a sheaf-theoretic characterization of the category of representations of the path-groupoid. That is, the generalization of local systems (or constructible sheaves) for the non-flat/thin-homotopy case. I've asked about this here Category of representations of the path-groupoid.

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