# 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.

## Author: brianhepler

I'm a second-year math postdoc at the University of Wisconsin-Madison, and I think math is pretty neat. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best.