Examples of Stokes data

Question

I'm trying to learn Stokes data but can't find an example to get my teeth into it.

Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence $$\text{regular holonomic D modules}\ \stackrel{\sim}{\longrightarrow} \ \text{perverse sheaves}$$ which for instance sends the regular linear ODE $Pf=0$ to its sheaf of solutions, which forms a local system. As I understand it the point of Stokes data is to give something like $$\text{holonomic D modules}\ \stackrel{\sim?}{\longrightarrow} \ \text{perverse sheaves + Stokes data}$$ and it should send the linear ODE $Pf=0$ to its sheaf of solutions (plus extra data).

For instance, take $X=\mathbf{P}^1$. Then the above equivalence should send (ignoring shifts) $$\mathscr{D}_X1 \ \longrightarrow \ \mathbf{C}$$ $$\mathscr{D}_Xe^{1/x} \ \longrightarrow \ \mathbf{C}.$$ These D modules are given by the ODEs $y'=0$ and $y'+y/x^2=0$. So the fact that they are sent to the same local system is not a counterexample of RH, since the second is irregular. I gather that under the $?$ map, $\mathscr{D}_Xe^{1/x}$ is sent to $\mathbf{C}$ along with some extra data at the irregular point $x=0$.

Question. What explicitly is the Stokes data of $\mathscr{D}_Xe^{1/x}$ (and in similar cases)? Is there an obvious relation to Stokes lines of the associated ODE?

Answer 1

It helps to understand irregular singularities as the merging of regular singular points, say $$(x^2-a^2)y'+y=0$$ as $a\to0$. For nonzero $a$ the data is encoded as monodromy (constant) matrices acting on your local solutions, given by the analytic continuation along loops generating the fundamental group.

Parts of the monodromy data makes it to the limit, namely the loops that are not cut by the merging. In the example that would be a loop encircling once both points $\pm a$, its limit encircling once the singularity $0$ and providing the monodromy part.

Yet part of the data won't make it to the limit, and something is lost if one looks only at the monodromy of the multivalued solutions. In the example that would be the monodromy associated to any one of the loops encircling only one singularity $\pm a$.

In the scalar case, the monodromy $y\mapsto c^\pm y$ around $\pm a$ is given by $$c^\pm=\exp\frac{\pm1}{2a}$$ which gets wild as $a\to 0$. Observe though that $c^+c^-=1$ makes it to the limit as the monodromy of $x\mapsto \exp \frac{1}{x}$.

So, where has the lost data gone? And, what is the link with Stokes lines? In the above example the Stokes data is trivial, but simply considering the modified ODE $$(x^2-a^2)y'+y=x$$ gives a non-trivial example. For $a=0$ the so-called Euler's equation has a unique power-series solution $$\hat y(x) = \sum_n (n!)x^{n+1} $$ which doesn't sum as an analytic object in the usual way. By Borel-Laplace-summing this series you obtain two analytic solutions, each one defined on a sector containing a half-plane, from which you deduce two sectorial systems of solutions. The Stokes data comes from the comparison between these two systems where the sectors overlap.The overlapping location is determined by the bissecting lines of the sectors, i.e. the Stokes lines. In the example you can obtain a Liouvillian representation of the solutions by explicit integration, therefore providing an integral representation for the Stokes data. You end up with formulas with coefficients given by values of the Gamma function (more details are linked at the end).

The discussion above streeses the fact that monodromy data is not a good presentation since it doesn't pass to the limit when a regular system degenerates onto an irregular one. Moreover the distinction monodromy/Stokes data is rather artificial, since Stokes data has also a meaning as gluing of local solutions. I prefer the view where everything is "Stokes data": one can always subsdivide $\mathbb P_1$ into "sectors" attached to the singular points, on which you have a trivial system, and the sectorial systems get compared in the pairwise intersections of said generalized sectors. In the case of a singularity (regular or not) you can form a neighborhood around it by tiling contiguous sectors: the composition of the Stokes operators coming from crossing the corresponding overlaps attached at the singularity gives you the monondromy operator. All the Stokes data passes to the limit in cases of merging.

As the construction shows, the Stokes data is not attached to an element of the fundamental group of $X\setminus sing$, like the monodromy, but rather to the "dual" groupoid of paths linking singular points (with an explicit representation as a path-integral operator).

A rich combinatorics comes from these considerations in the case of higher-Poincaré-rank systems (merging of $>2$ singular points).

To read more about the above topics, look for papers by Christiane Rousseau (Montréal)

1. (Linear systems) Jacques Hurtubise, Caroline Lambert, and Christiane Rousseau. Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k. Mosc. Math. J., 14(2):309–338, 427, 2014.

2. (Non-linear, but with a detailed analysis of generalizations of the examples above) a book chapter of mine https://hal-cnrs.archives-ouvertes.fr/hal-01170840

3. (Non-linear, detailed construction and study of the generalized sectors) with Christiane Rousseau https://hal-cnrs.archives-ouvertes.fr/hal-01890315

4. (Slightly non-linear, confluence of Stokes data in the Painlevé family) by Martin Klimes https://arxiv.org/abs/1609.05185

Answer 2

Let $\mathcal{L}$ be a local system on $X\setminus x$.

Take $S^1$ (the sphere bundle of $X$ at $x$) and the sheaf $\mathcal{V}$ of ``section germs", i.e. $$\mathcal{V}(\theta,\theta')\ =\ \lim_{\epsilon\to 0}\mathcal{L}(\text{sector with radius }\epsilon\text{ and angle }\theta\text{ to }\theta').$$ So a section of $\mathcal{V}$ is a flat section of $\mathcal{L}$ defined on a small sector.

TL;DR The Stokes data of $\mathcal{L}$ is the filtration $\mathcal{V}_\alpha\subseteq \mathcal{V}$, consisting of section germs $f$ such that $f e^{-\int \alpha}$ has at worst Laurent poles near zero.

Here $\alpha=\sum_{n\ge -N} a_n z^{n/k}dz$ ranges over all etale germs of meromorphic one forms.

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Old, overly wordy answer

To complement Loïc Teyssier's excellent answer, this is the algebro-geometric interpretation of Stokes data, first in the case of $e^{1/x}$.

0. A zeroeth approximation: Stokes data is the information that as $x\to 0$, $$e^{1/x}\ \longrightarrow \ \begin{cases} 0& \text{if }\text{arg}x\in (-\pi/2,\pi/2)\\ \infty & \text{if }\text{arg}x\in (\pi/2,3\pi/2) \end{cases}.$$ Here $x\to 0$ along rays (lines to the origin of constant argument). So Stokes data remembers how the limiting behavoir of the solution approaching the singular point depends on the argument.

Let's turn this into sheaf language. Take ODE on the disk $X=\Delta$ with singular point $0$, and local system of solutions $\mathscr{L}$ on $\Delta\setminus 0$. To be able to talk about the limiting behavior of solutions as $x\to 0 $ along rays, take the real oriented blowup at $0$ $$\pi \ :\ \widetilde{X}\ \longrightarrow \ X,$$ then $\pi^{-1}\mathscr{L}$ is a local system containing this information. Identify the fibre above $0$ with $S^1$. Write $\mathscr{V}$ for the restriction of $\mathscr{L}$ to $S^1$; this is where that information is stored.

1. A first approximation: Stokes data is a subsheaf $$\mathscr{V}^0\ \subseteq \ \mathscr{V}$$ given by the solutions with at worst a finite order pole in the given direction. Thus a germ $f$ lies in $\mathscr{V}^0_\theta$ if the size of $f(re^{i\theta})$ is bounded by $r^{-n}$ for some $n$ (this is not quite true, this needs to hold for a sector containing $\theta\in S^1$). In the $e^{1/x}$ example, this is $$\mathbf{C}_{(-\pi/2,\pi/2)}e^{1/x} \ \subseteq \ \mathbf{C}_{S^1}e^{1/x}.$$

The actual definition asks for (a little) more information about the limiting behavior.

2. A second approximation: Stokes data is a collection of subsheaves $$\mathscr{V}^\alpha\ \subseteq\ \mathscr{V}$$ for every $\alpha\in \Omega^1_\Delta(\star 0)$ a meromorphic one form on $\Delta$ with poles only at $0$. A germ $f$ lies in $\mathscr{V}^\alpha_\theta$ iff $$f(re^{i\theta}) e^{-\int \alpha}$$ is bounded by $r^{-n}$ in a small sector containing $\theta$.

These subsheaves fit together to form a filtration, in that $$\mathscr{V}^\alpha_\theta\ \subseteq \ \mathscr{V}^\beta_\theta$$ whenever $e^{\int\alpha}e^{-\int \beta}$ has aforementioned boundedness property on a sector containing $\theta$. This gives a partial order on $\Omega^1(\star 0)_\theta$, for which the above is a filtration (a lie: you need to replace $\Omega^1(\star 0)$ by its quotient by the forms with at worst simple poles). Moreover, there's a grading on $\mathscr{V}_\theta$ for which this is the associated filtration.

$\infty$. Stokes data as in $2$ a filtration of $\mathscr{V}$ by a partially ordered sheaf, but using a slightly different indexing poset: you replace the Zariski fibre $\Omega^1(\star 0)_\theta$ with the etale fibre. In practice this means that you consider $\alpha=\sum_{n\ge n_0} a_n x^{n/k}dx$ for all $k\in\mathbf{N}$ and instead of just $k=1$.

So e.g. it contains information that $$e^{1/x}e^{\int \frac{dx}{\sqrt{x}^5}}\ =\ e^{1/x-2/3\sqrt{x}^3} \ \longrightarrow\ \begin{cases} 0&\text{if }\theta\in \pm(\pi,2\pi/3)\\ \infty&\text{if }\theta\in (-2\pi/3,2\pi/3) \end{cases}$$ where $\sqrt{x}$ is the positive square root defined off the negative reals.

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In this language, Stokes lines are just the phenomenon that $f e^{-\int\alpha}$ flips between satisfying and not satisfying the boundedness condition for only finitely many angles $\theta$, so you can see the Stokes lines directly in the sheaves $\mathscr{V}^\alpha$.

e.g. in the $e^{1/x}$ example, $\theta=\pm \pi/2$ are the two Stokes lines

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Everything in this answer comes from

1. La classification des connexions irrégulières à une variable, by Malgrange. http://www.numdam.org/item/CIF_1982__17__A1_0/
2. Twisted wild character varieties, by Boalch and Yamakawa. https://arxiv.org/abs/1512.08091

The definition of a Stokes structure on a sheaf is $4.1$ of the first reference (it's the same as I've written above), how to give a Stokes structure in the ODE case is the top of page $7$. A Riemann Hilbert correspondence (which justifies the above definition of Stokes data) is theorem $4.2$.