## Deligne’s regular solution in dimension 1

In this post, I want to recall the elements of the regular Riemann-Hilbert Correspondence (but only in dimension 1, on our small disk around the origin). We’ll talk more about the category of Meromorphic Connections on $X$ with singularities at 0, and how they’re just a different way of phrasing linear ODEs whose solutions have singularities at 0. From there, we examine the simplest class of solutions, those with regular singularities. These are, in general, multi-valued functions of $z$ that satisfy an analytic condition called moderate growth. From there, we can state Deligne’s solution to the regular R-H correspondence, and start to understand more about the failure in the irregular setting. We end with some emergent phenomena that occur only for irregular singularities, which add to the many difficulties in proving the irregular R-H correspondence. Lots of things from this post are from chapter 5 of “D-modules, perverse sheaves, and representation theory“.

Meromorphic Connections = Fancy ODE’s with singularities

Let $X$ be a small open ball around the origin in $\mathbb{C}$, $\mathscr{O} := \mathscr{O}_{X,0}$, and $\mathscr{O}(*0)$ the field of fractions of $\mathscr{O}$, representing holomorphic functions with possible poles at $0$. If $z$ is a local coordinate on $X$, then $\mathscr{O} \cong \mathbb{C}\{z\}$, and $\mathscr{O}(*0) \cong \mathbb{C}\{z\}[z^{-1}]$. Then, recall that a meromorphic connection consists of the data of a free $\mathscr{O}(*0)$-module $M$ and a $\mathbb{C}$-linear map $\nabla : M \to M$ satisfying the Leibniz rule $\nabla(fm) = \frac{df}{dz}u + f\nabla(u)$ for elements $f \in \mathscr{O}(*0), u \in M$. A morphism of meromorphic connections is just a map $\phi : (M,\nabla) \to (N,\nabla)$ that is $\mathscr{O}(*0)$-linear from $M$ to $N$ and commutes with the action of $\nabla$. This is actually a bit of a simplification of connections–normally we’d specify $\nabla$ as a map

$M \xrightarrow{\nabla} M \otimes_\mathscr{O} \Omega_{X/\mathbb{C}}^1$

Contracting this “universal” $\nabla$ with elements of $\mathscr{D}_X$ produces the original definition I gave.

These objects then form an Abelian category (since we’re really just restricting to a thick subcategory of Holonomic D-modules). Again, the point of all this is just to formalize both our fast-and-loose simplifications from last time, and to phrase the problem functorially. This category of meromorphic connections is just a simplified way of talking about (complex) linear ODE’s with (possible) singularities at $0$. Now, what does a general meromorphic connection look like?

On $X$ with our local coordinate $z$, choosing a system of generators of $M$ over $\mathscr{O}_X(*0)$, say $e_1(z),\cdots ,e_n(z)$ (these always exist globally on $X$ because $X$ is contractible). Then, looking at the action of $\nabla$ on this basis gives us a connection matrix with values in $\mathscr{O}(*0)$

$\nabla e_j =- \sum_{i=1}^n a_{ij}(z) e_i$,

the negative sign is just a convention, to make the next expression nicer. Thus, if $u(z) = \sum_i u_i(z)e_i(z)$ is a general element of $M$, then the above expression and the Leibniz rule gives us

$\nabla (\sum_{i=1}^n u_i e_i) = \sum_{i=1}^n \left (\frac{du_i}{dz}-\sum_{j=1}^n a_{ij}u_j \right )e_i$

Hence, the collection of flat sections of $\nabla$ (those $\vec{u} = \sum_i u_ie_i$ that satisfy $\nabla u = 0$) correspond to solutions of the system of linear ordinary differential equations

$\frac{d \vec{u}}{dz} = A(z)\vec{u}$

Likewise, any such differential equation gives rise to a connection–let $\widetilde M$ be the space of solutions to the above differential equation. Then, this equation is a rule for how the symbol $\frac{d}{dz}$ acts on the elements of $\widetilde{M}$, taken as the definition of $\nabla$.

The simplest case that arises from having meromorphic coefficients is when the connection matrix $A(z)$ has at worst poles of degree one, i.e., if $A(z) = \frac{A}{z}$ for some constant matrix $A \in M_n(\mathbb{C})$ (to compensate for maybe choosing a bad generating set for $M$ over $\mathcal{O}(*0)$, we only care if $A(z)$ is “gauge equivalent” to a matrix of the form $\frac{A}{z}$). In the rank one case, this is a differential equation of the form

$\frac{du}{dz} = \frac{\alpha}{z}u(z)$

for some $\alpha \in \mathbb{C}$, and has fundamental solution $u(z) = z^\alpha$. What sort of function is this? When $\alpha = 0,1,2,3,\cdots$, $z^\alpha$ is a globally defined holomorphic function (it’s just a monomial!), and when $\alpha = -1,-2,-3,\cdots$, $z^\alpha$ is a globally defined meromorphic function on $X$ that is holomorphic on $X^*=X-\{0\}$. When $\alpha \notin \mathbb{Z}$, we have to use the definition $z^\alpha := \exp(\alpha \log(z))$, where $\log(z)$ is only a well-defined function on the complement of a choice of branch cut. Moreover, while this is a perfectly valid solution on any open simply connected subset of $X^*$, as we travel around the origin and analytically continue $z^\alpha$, the value of this function jumps due to monodromy. Precisely, if we let $z \mapsto e^{2\pi \theta i}z$ as $0 \leq \theta \leq 1$ varies, we find

$z^\alpha \mapsto e^{2\pi i \alpha}z^\alpha := \exp\left \{\alpha(\log |z| +i(\arg(z)+2\pi)\right \}.$

By cutting up the domain of log, we can make it single-valued

This is an example of a multi-valued function on $X$, and doesn’t lie in $\mathcal{O}$ or $\mathcal{O}(*0)$, and naturally form a ring (after fixing a universal cover of $X^*$). We don’t need this entire ring quite yet (it contains other weird things like functions with essential singularities and Whitney functions), but we’ll come back to it.

Regular Singularities and Moderate Growth

Meromorphic connections which have connection matrices gauge equivalent to one of the form $\frac{A}{z}$ (we can even fudge a bit to allow $A(z) \in M_n(\mathcal{O})$) are said to have a regular singularity at 0. This is a bad choice of terminology, since sometimes “regular” means “non-singular” in algebraic geometry, so perhaps “tame” would’ve been a better name since they are classified by the rank of the connection and the monodromy matrix. These are the mildest sorts of singularities that appear in the theory of meromorphic connections, and are the the subject of the classical Riemann-Hilbert correspondence (in D-module language) of Kashiwara and Mebkhout. The solutions to these sorts of differential equation are very similar to our example of $z^\alpha$ given above; if $M$ has rank $n$, we can find by the Frobenius method $n$ linearly independent solutions of the form

$z^\alpha_i \phi_i(z)$

where $\alpha_i \in \mathbb{C}$ and $\phi_i(z)$ is a holomorphic function on $X$ with $f(0) \neq 0$ (the expression is slightly more complicated if an exponent $\alpha_i$ is repeated, or if some pair $\alpha_i$ and $\alpha_j$ differ by an integer). These functions may be only well-defined single-valued functions on certain arc neighborhoods of 0, i.e., open sets of the form

$S_\epsilon(a,b) := \{z \in X^* | 0<|z|<\epsilon, a < \arg(z) < b\}$

for some $\epsilon > 0$ and $0 \leq a < b \leq 2\pi$.

arc neighborhoods on which $z^{\frac{1}{3}}$ is single-valued

Solutions to ODE’s with regular singularities on $X$, even multi-valued ones, always behave “like” meromorphic functions in an appropriate arc neighborhood of any particular angle $\theta \in S^1$. When we say “behaves like a meromorphic function”, we mean that as we approach 0 from inside that arc neighborhood, the norm of the solution $u(z)$ grows only polynomially in $|z|^{-1}$, as if it were a Laurent series with only finitely many terms of negative degree.

More precisely, solutions $u(z)$ to ODEs with regular singularities are said to have moderate growth at $\theta \in S^1$, in the sense that there are $0< \epsilon,\delta \ll 1$, constant $C_\theta > 0$ and exponent $N_\theta \in \mathbb{N}$ for which

$|u(z)| \leq C_\theta|z|^{-N_\theta}$ (1)

on the arc neighborhood $S_\epsilon(\theta-\delta,\theta+\delta)$. As $\theta$ travels around $S^1$, the values of the constants $C_\theta$ and $N_\theta$ may change. We just say $u(z)$ has moderate growth at 0 if it has moderate growth at every $\theta \in S^1$.

For single-valued functions on $X$, moderate growth at 0 is equivalent to being meromorphic along 0.

This analytic characterization is actually a necessary and sufficient condition; a meromorphic connection $(M,\nabla)$ has a regular singularity at 0 if and only if all of its flat sections have moderate growth at 0. Algebraically, this can be characterized by Fuch’s Criterion, but we will not focus on this perspective.

Aside from this interesting growth condition, or the simplicity of their solutions, why do we care about regular singularities? Let $Conn^{reg}(X;0)$ be the category of Meromorphic Connections with regular singularities at 0, and $Conn(X^*)$ the category of flat connections on $X^*$.

Theorem (Deligne): The restriction functor $M \mapsto M_{|_{X^*}}$ induces an equivalence of categories

$Conn^{reg}(X;0) \xrightarrow{\thicksim} Conn(X^*)$

To show that we can always extend a given $(N,\nabla) \in Conn(X^*)$ of rank $n$ to an element of $Conn^{reg}(X;0)$, we note that $(N,\nabla)$ is uniquely determined by a monodromy representation $\rho : \pi_1(X^*) \to Gl_{n}(\mathbb{C})$ defined by the local system $\ker \nabla$. If we let $\gamma$ correspond to the element 1 in the identification $\pi_1(X^*) \cong \mathbb{Z}$, then $\rho$ is determined by the matrix $C = \rho(\gamma)$ . We can then always find a matrix $\Gamma \in M_{n}(\mathbb{C})$ such that $\exp(2 \pi i \Gamma)= C$, which we then use to define a connection matrix on $\widehat{N} := \mathscr{O}(*0)^{n}$ via

$\nabla e_q = -\sum_{1 \leq p \leq n} \frac{\Gamma_{pq}}{z} \otimes e_p$

where $\{e_1,\cdots,e_n\}$ is the standard basis for $\widehat{N}$. The resulting meromorphic connection $(\widehat{N},\nabla)$ clearly has a regular singularity at 0 and restricts to $N$ on $X^*$.

Essentially, the singularities of connections with regular singularities are so mild that they are completely determined by their monodromy around the singular point. Recalling the previous post, we obtain the following:

Corollary (Deligne): There is an equivalence of categories

$Conn^{reg}(X;0) \xrightarrow{\thicksim} Loc(X^*)$

$(M,\nabla) \mapsto (M_{|_{X^*}},\nabla_{|_{X^*}}) \mapsto \ker \nabla_{|_{X^*}}$

Where $Loc(X^*)$ is the category of finite-rank $\mathbb{C}$-local systems on $X^*$.

Irregular Singularities and Stokes Phenomena

We say a meromorphic connection $(M,\nabla)$ has an irregular singularity at 0 if it doesn’t have a regular singularity at 0; hence, flat sections $u$of $\nabla$ do not have moderate growth at 0. What does this mean? First off, the absence of regularity doesn’t necessarily mean $u$ doesn’t satisfy (1) everywhere, just that there are some $\theta \in S^1$ which have no arc neighborhood on which $u(z)$ has polynomial growth in $|z|^{-1}$.

For example, take $(\mathscr{O}(*0),d-\frac{1}{z^2})$, whose associated linear ODE has fundamental solution $u(z)=e^{\frac{1}{z}}$. Then, $u(z)$ has moderate growth at every $\frac{\pi}{2} < \theta < \frac{3\pi}{2}$. Why? On this region, $\textnormal{Re} \frac{1}{z} < 0$, and so $|u(z)| = e^{\textnormal{Re} \frac{1}{z}}$ decays to 0 exponentially as $z \to 0$. $u(z)$ does not have moderate growth on $(3\pi/2,\pi/2) (\mod 2\pi)$. When $\textnormal{Re} \frac{1}{z} > 0$, say $\theta = 0$ for simplicity, then

$e^{1/z} = \sum_{n =0}^\infty \frac{1}{z^n n!} > z^{-N}$

for every exponent $N > 0$.

Aside from no longer having moderate growth, one of the other reasons Deligne’s theorem no longer holds in the irregular setting is that we can’t uniquely specify a general meromorphic connection on $X$ by its restriction to $X^*$. We actually saw this in the last post–the Meromorphic Connections $(\mathscr{O}(*0),d-df)$ are all non-isomorphic (for $f \in \mathscr{O}(*0)/\mathscr{O}$), but their local systems of flat sections are all isomorphic to the constant local system $\mathbb{C}_{X^*}$. In essence, there are too many local systems with the same monodromy, if we have no restrictions on the singularities.

So, why do we care about these ODEs if they’re so difficult to deal with? Many interesting special functions arising in engineering and physics arise as solutions to ODEs with irregular singularities. Take, for example, the Airy equation

$\frac{d^2 u}{dz^2} = zu$

which has an irregular singularity at $z= \infty$, and its fundamental solutions $Ai(z)$ and $Bi(z)$ are approximated (in a rigorous sense, using asymptotic analysis) by linear combinations of the functions $u_\pm(z) = z^{-1/4}e^{\pm \frac{2}{3}z^{3/2}}$ for large values of $z$. Now, here is the interesting thing: a solution $Ai(z)$ of the Airy equation is an entire function of $z$, since the coefficient function (which is just $z$) is entire, but the functions $u_\pm$ are multi-valued functions. Hence, as we let $z \mapsto z e^{2\pi i}$, around $z=0$, the “true solution” $Ai(z)$ will return to its original value, but $u_+$ and $u_-$ will not. Hence, $Ai(z)$ and $A(z e^{2\pi I})$ cannot be represented by the same linear combination of $u_+$ and $u_-$.

This is known as the Stokes Phenomenon, where solutions to linear ODE’s can undergo drastic qualitative changes as $\theta \in S^1$ passes through certain angles. Specifically, these qualitative changes happen to the asymptotic behavior of the solutions in the different regions bounded by these special angles (sometimes called Stokes angles, or Stokes lines (or anti-Stokes lines, if you’re a physicist)). This is a phenomenon unique to the world of irregular singularities. We go through all of this extra work because sometimes it can be exceedingly difficult to express exact solutions in terms of elementary functions (or even special functions!), and the approximations in terms of exponential factors can often tell us much more about the “physical” properties the true solutions that are difficult to see from their formulas. We recommend Meyer’s “A Simple Explanation of the Stokes Phenomenon” for this perspective.

In the next post, I’ll talk some more about the formalism of approximation and asymptotic analysis, and how it naturally leads to a way of characterizing solutions of with irregular singularities via the Stokes filtration.

## Irregular Riemann-Hilbert Correspondence: introduction to the problem

One of the most successful bridges between analysis and algebraic geometry is the classical Riemann-Hilbert (R-H) correspondence between regular holonomic D-modules and perverse sheaves on complex manifolds, where $\mathscr{D}$ is the sheaf of differential operators with holomorphic coefficients (proved independently by Kashiwara and Mebkhout in 1984). This correspondence is a far-reaching generalization of Hilbert’s 21st Problem asking about the existence of ordinary differential equations (ODE’s) with regular singularities on a Riemann surface with prescribed monodromy groups. A great introduction to this problem (prior to the work of Kashiwara and Mebkhout) is Katz’s 1976 paper “An overview of Deligne’s work on Hilbert’s 21st problem“. Loosely, it is the “correct” generalization of the correspondence between vector bundles with flat connection and local systems to the singular setting, which is itself a massive generalization of the local existence and uniqueness of ordinary differential equations.

The problem of extending the R-H correspondence to cover holonomic $\mathscr{D}$-modules with irregular singularities has recently been settled by Kashiwara-D’Agnolo in general, and many others in special cases (e.g., Deligne, Malgrange, Mochizuki, Sabbah, and Kedlaya to name a few). These objects correspond topologically to enhanced perverse ind-sheaves (and several other equivalent Abelian categories, following Deligne‘s Stokes-$\mathbb{C}$-perverse sheaves, Kuwagaki’s irregular perverse sheaves, and Ito’s $\mathbb{C}$-constructible enhanced ind-sheaves). I’ll refer to any of these equivalent Abelian categories as irregular perverse sheaves. A great short intro is Sabbah’s 2019 article “What are irregular perverse sheaves?“, and I’m basing much of this first post on his notes.

I plan to write several posts on this topic, things I wish I had when I first started trying to learn this subject. Since I’m coming from the purely topological/perverse sheaf side of the old (hah) R-H correspondence for regular holonomic $\mathscr{D}$-modules, I will assume you are also just as ignorant as I was coming in–it won’t be obvious to you why we suddenly need new topics from functional analysis or asymptotic analysis, like it wasn’t obvious to me. Neither will be the jump to ind-sheaves, instead of usual sheaves, or what exactly the extra variable is doing in enhanced ind-sheaves. These are all things I hope to talk about.

The easiest possible situation in which to understand the irregular R-H correspondence is when $X$ is a open complex disk around the origin in $\mathbb{C}$. This is Deligne’s version–meromorphic connections ($\mathscr{D}$-module side) and Stokes-filtered local systems (topological side). Before we get into details, you should know that, loosely, the difficulty will be in distinguishing exponential factors that pop up in solutions to these differential equations. Everything that follows will be in the hope that we can fix this problem.

Holonomic D-modules and Meromorphic Connections

In the local analytic setting in $\mathbb{C}$, holonomicity of a $\mathscr{D}_X$-module $M$ just translates into saying there is a finite set of points $X_0$ in $X$ off of which $M_{|_{X -X_0}}$ is an integrable connection of finite rank. The only thing that changes from the regular singularity case is that we are now making no assumptions about what $M$ looks like at the points of $X_0$. Let’s just assume $M$ has only one “singularity”, so $X_0 =\{0\}$ and $X$ is a connected open neighborhood of the origin. Then, holonomicity is equivalent to giving a finite dimensional $\mathbb{C}$-vector space $M$ and a $\mathbb{C}$-linear map $\nabla : M \to M$ that satisfies, for all $m \in M$ and $f \in \mathscr{O}_X(*0)$, the Leibniz rule:

$\nabla(fm) = f\nabla(m) + \frac{df}{dz} m$

(where $z$ is a local coordinate on $X$ with $z(0) = 0$).

The Problem

Let $f \in \mathscr{O}_X(*0)/\mathscr{O}_X$ be a meromorphic function on $X$, and consider the meromorphic connection

$(M,\nabla) = (\mathscr{O}_X(*0),d+df)$

Then, the flat sections of $M$ correspond to solutions of the differential equation $\nabla u =0$, i.e., $du = -udf$. These flat sections form a rank one $\mathbb{C}$-local system $\ker \nabla$ with stalk $\mathbb{C}\cdot e^{-f}$ at every point of $X^*:= X-\{0\}$. The monodromy action is trivial, we find $\textnormal{DR}_{X^*}(M) \cong \ker \nabla \cong \mathbb{C}_{X^*}$, where $\textnormal{DR}_{X^*}$ is the de Rham functor.

Now, here is the problem: $f \in \mathscr{O}_X(*0)/\mathscr{O}_X$ was arbitrary, but the local system we end up with doesn’t depend on the function we started with. That is, the de Rham functor is no longer faithful when we extend to holonomic $\mathscr{D}_X$-modules with possibly irregular singularities. The question, then, is what is a natural category of objects that correspond to solutions of differential equations with irregular singularities?

Deligne’s answer to this question is to keep the local systems away from the singularities, and add a filtration at the “boundary” that keeps track of how fast these solutions grow as they approach the singularity from different directions. These objects are called Stokes-filtered local systems. Easy enough, right? They are perhaps the simplest to understand solution of the problem, but they only really work well in dimension one (although there is some work in higher dimensions, especially Sabbah’s work Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2), and we need to introduce some notions from asymptotic analysis to work with the filtration. These will be the focus of the next post.