Simplification of exponential factors of irregular connections on $\mathbb P^1$

TL;DR

This paper develops a concrete simplification framework for exponential factors of irregular connections on $\mathbb{P}^1$, showing that a full simplification map $\widehat{S}$ yields a unique minimal ramification datum within each orbit under the basic operations. A local variant at infinity, $\widehat{S}_{\infty}$, yields a finite set of locally minimal level data for each even dimension $n$, including a unique $n=2$ case corresponding to the Painlevé I moduli space. The results organize all level data into infinite rooted trees (forests), one tree per dimension, and prove finiteness of minimal data for any fixed dimension, implying a finite (conjectural) set of isomorphism classes of elementary wild character varieties per dimension. These findings connect reduction procedures from irregular singularities to global moduli questions, and they suggest a structured, combinatorial picture for the landscape of elementary wild character varieties, with Painlevé I playing a central, unique role in dimension two.

Abstract

We give an explicit algorithm to reduce the ramification order of any exponential factor of an irregular connection on $\mathbb P^1$, using the same types of basic operations as in the Katz-Deligne-Arinkin algorithm for rigid irregular connections. The exponential factor reached when the algorithm terminates is, up to admissible deformations, the unique factor with minimal ramification order in the orbit of the initial factor under successive applications of basic operations. Furthermore, we show that for every even integer $n\geq 0$, there is up to admissible deformations a finite number of non-simplifiable exponential factors at infinity such that the corresponding elementary wild character variety has complex dimension $n$, which conjecturally implies that there is a finite number of isomorphism classes of elementary wild character varieties in any dimension. These results can be viewed as saying that the set of all possible level data of exponential factors has the structure of a disjoint union of an infinite number of infinite rooted trees, each tree being associated to a given dimension $n$ and with a finite number of trees for each $n$. In particular, in dimension 2 there is a unique tree, corresponding to the Painlevé I moduli space.

Figures (2)

• Figure 1: A small part of the tree corresponding to the connected component in $\Gamma_\infty^{\mathbb L}$ of the standard Lax representation $\mathbf L=\left\{\frac{5}{2}\right\}\in \mathbb L_\infty^{min}$ of the Painlevé I moduli space, which constitutes the root of the tree. The arrows indicate the local simplification map $S_\infty$. Every vertex of the tree actually has an infinite number of children. Every vertex is expected to be an elementary Lax representation of the Painlevé I moduli space, and Thm. \ref{['thm:intro_finite_number_minimal_level_data']} implies that there can be no other ones (but there will also exist non-elementary Lax representations, with several Stokes circles).
• Figure 2: Table of minimal level data at infinity, corresponding to one-vertex diagrams with up to 10 loops.

Theorems & Definitions (50)

• Definition 1.1
• Theorem 1.2: Thm. \ref{['thm:algorithm_gives_minimal_levels']}, Prop. \ref{['prop:characterization_minimal_level_data']}
• Theorem 1.3: Thm. \ref{['thm:algorithm_at_infinity']}
• Theorem 1.4
• Theorem 1.5: Thm. \ref{['thm:finite_number_minimal_factors']}, Prop. \ref{['prop:unique_non_simplifiable_small_dimension']}
• Conjecture 1.6
• Definition 2.1
• Proposition 2.2: see boalch2022twisted
• Example 2.1
• Lemma 2.3: doucot2024basic