# Finite braid group orbits on $\operatorname{SL}_{2}$-character varieties

Yeuk Hay Joshua Lam, Aaron Landesman, and Daniel Litt

###### Abstract.

Let $\Sigma_{0,n}$ be a $2$-sphere with $n$ punctures. We classify all
conjugacy classes of Zariski-dense representations
$\rho:\pi_{1}(\Sigma_{0,n})\to\operatorname{SL}_{2}(\mathbb{C})$ with finite
orbit under the mapping class group $\operatorname{Mod}_{0,n}$, such that the
local monodromy at one or more punctures has infinite order. We show that all
such representations are “of pullback type” or arise via middle convolution
from finite complex reflection groups. In particular, we classify all rank $2$
local systems of geometric origin on $\mathbb{P}^{1}\setminus D$, with $D$
generic, and with local monodromy of infinite order about at least one point
of $D$.

##  1\. Introduction

Fix $n>0$ and let
$(A_{1},\cdots,A_{n})\in\operatorname{SL}_{2}(\mathbb{C})^{n}$ be an $n$-tuple
of matrices such that

$\prod_{i=1}^{n}A_{i}=\operatorname{Id}.$

We say such an $n$-tuple is _non-degenerate_ if the $A_{i}$ generate a
Zariski-dense subgroup of $\operatorname{SL}_{2}(\mathbb{C})$. We take $X_{n}$
to be the set of such non-degenerate $n$-tuples, up to simultaneous
conjugation. That is,

$X_{n}:=\\{(A_{1},\cdots,A_{n})\in\operatorname{SL}_{2}(\mathbb{C})^{n}\mid\prod
A_{i}=\operatorname{Id}\text{ and $(A_{1},\cdots,A_{n})$ is non-
degenerate}\\}/\operatorname{SL}_{2}(\mathbb{C}).$

The set $X_{n}$ admits the action of a natural group of symmetries
$\operatorname{Mod}_{0,n}$, generated by $\sigma_{1},\cdots,\sigma_{n-1}$,

$\sigma_{i}:(A_{1},A_{2},\cdots,A_{n})\mapsto(A_{1},\cdots,A_{i}A_{i+1}A_{i}^{-1},A_{i},\cdots
A_{n}),$

called the _Hurwitz action_. There has been some interest over the last
decades in understanding the dynamics of the $\operatorname{Mod}_{0,n}$-action
on $X_{n}$, as we explain in § 1.2. Our main result is a complete
classification of the finite orbits of the $\operatorname{Mod}_{0,n}$-action
on $X_{n}$, at least when one of the $A_{i}$ has infinite order.

The dynamical system described above arises naturally from the study of the
character variety of $\Sigma_{0,n}$, the $2$-sphere with $n$ points removed.
The fundamental group $\pi_{1}(\Sigma_{0,n})$ has the presentation

$\pi_{1}(\Sigma_{0,n})=\langle\gamma_{1},\cdots,\gamma_{n}\mid\prod\gamma_{i}=\operatorname{id}\rangle,$

with $\gamma_{i}$ a loop around the $i$-th puncture. $X_{n}$ is the set of
complex points of the character variety parametrizing $2$-dimensional Zariski-
dense $\pi_{1}(\Sigma_{0,n})$-representations with trivial determinant, i.e.
the categorical quotient

$X_{n}:=\operatorname{Hom}(\pi_{1}(\Sigma_{0,n}),\operatorname{SL}_{2}(\mathbb{C}))^{\text{non-
deg}}\sslash\operatorname{SL}_{2}(\mathbb{C}),$

where the decoration “non-deg” indicates the image of the representation is
Zariski-dense. Explicitly, given a Zariski-dense representation
$\rho:\pi_{1}(\Sigma_{0,n})\to\operatorname{SL}_{2}(\mathbb{C})$, we set
$A_{i}=\rho(\gamma_{i})$.

The group $\operatorname{Mod}_{0,n}$ is precisely the mapping class group

$\operatorname{Mod}_{0,n}:=\pi_{0}(\operatorname{Homeo}^{+}(\Sigma_{0,n})),$

of $\Sigma_{0,n}$, acting on $X_{n}$ through its natural outer action on
$\pi_{1}(\Sigma_{0,n})$. ($\operatorname{Mod}_{0,n}$ is sometimes referred to
as “the spherical braid group on $n$ strands.”) We refer to conjugacy classes
of representations with finite orbit under this action as _MCG-finite_ , or
_canonical_ , representations of $\pi_{1}(\Sigma_{0,n})$.

###  1.1. Main results

Our main result is that such canonical representations with some $A_{i}$ of
infinite order are of one of two types:

  1. (1)

The “pullback” representations, classified by Diarra [Dia13, §3-§5], and

  2. (2)

Representations obtained via _middle convolution_ from representations of
finite complex reflection groups.

See 1.1.7 for a precise statement. These two families are not disjoint—many
“pullback” representations also arise from middle convolution. As the finite
complex reflection groups were classified by Shephard and Todd [ST54], this
amounts to a complete classification. We now explain the meaning of (1) and
(2) above.

Note that for $n<3$, there are no Zariski-dense representations
$\pi_{1}(\Sigma_{0,n})\to SL_{2}(\mathbb{C})$, as $\pi_{1}(\Sigma_{0,n})$ is
abelian. For $n\geq 3$, let $\mathscr{M}_{0,n}$ denote the moduli scheme
parametrizing genus zero curves with $n$ marked points, and let
$\pi_{n}:\mathscr{M}_{0,n+1}\to\mathscr{M}_{0,n}$ be the map forgetting the
last marked point. If

$\rho:\pi_{1}(\Sigma_{0,n})\to\operatorname{SL}_{2}(\mathbb{C})$

is an irreducible representation with finite mapping class group orbit, then
there exists, by [LL22a, Corollary 2.3.5], a dominant quasi-finite map
$B\to\mathscr{M}_{0,n}$, and an $\operatorname{SL}_{2}$-local system
$\mathbb{V}$ on

$\mathscr{C}^{\circ}:=\mathscr{M}_{0,n+1}\times_{\mathscr{M}_{0,n}}B,$

with the following property: if $X$ is a fiber of the projection
$\mathscr{C}^{\circ}\to B$, then $\mathbb{V}|_{X}$ has monodromy $\rho$. Thus
it suffices to classify rank $2$ local systems on such $\mathscr{C}^{\circ}$.

By results of Corlette-Simpson [CS08, Theorem 2] (when $\mathbb{V}$ has quasi-
unipotent monodromy at infinity, see 2.2.2) and Loray-Pereira-Touzet [LPT16,
Theorem A] (otherwise), such local systems with Zariski-dense monodromy come
in two flavors: those pulled back through a map $\mathscr{C}^{\circ}\to Y$,
with $Y$ a Deligne-Mumford curve, and those pulled back from a so-called
polydisk Shimura variety. The former are, by definition, the “pullback”
representations classified by Diarra [Dia13, §3-§5], so all that remains is to
classify the latter—equivalently, those that underlie a rank $2$ integral
variation of Hodge structure, or are equivalently (in rank $2$, see 1.1.6) of
geometric origin, i.e. arise in the cohomology of a family of smooth proper
varieties over $\mathscr{C}^{\circ}$. A local system of geometric origin on a
_generic_ $n$-times punctured curve of genus zero automatically spreads out
over moduli space—thus it suffices to classify these. Here
$(\mathbb{P}^{1}_{\mathbb{C}},x_{1},\cdots,x_{n})$ is _generic_ if the
corresponding map
$\operatorname{Spec}(\mathbb{C})\to\mathscr{M}_{0,n,\mathbb{Q}}$ factors
through the generic point.

######  Definition 1.1.1.

An element $g\in\operatorname{GL}_{r}(\mathbb{C})$ is a _pseudoreflection_ if
$g-\operatorname{Id}$ has rank $1$. A finite subgroup of
$\operatorname{GL}_{r}(\mathbb{C})$ is a _finite complex reflection group_ if
it is generated by pseudoreflections.

This is our main technical result:

######  Theorem 1.1.2.

Let $D\subset\mathbb{P}^{1}$ be a generic reduced effective divisor of degree
$n$, containing $\infty$, and let $\mathbb{V}$ be a local system of rank $2$
of geometric origin on $X=\mathbb{P}^{1}\setminus D$ with non-scalar monodromy
at $\infty$. Suppose that $\mathbb{V}$ has Zariski-dense monodromy in
$\operatorname{SL}_{2}(\mathbb{C})$ and at least one local monodromy matrix of
$\mathbb{V}$ has infinite order. Then there exists a local system $\mathbb{U}$
on $X$ with monodromy group a finite complex reflection group, a rank one,
finite order local system $\mathbb{M}$ on $X$, and a rank one, finite order
local system $\chi$ on $\mathbb{G}_{m}$, such that
$\mathbb{V}=MC_{\chi}(\mathbb{U})\otimes\mathbb{M}.$

Here $MC_{\chi}$ denotes Katz’s middle convolution operation [Kat96], which
will be described in § 3. The proof, given in § 4.4, relies on a Hodge-
theoretic analysis of middle convolution.

######  Remark 1.1.3.

One may interpret the result as saying that $\mathbb{V}\otimes\mathbb{M}^{-1}$
appears in the cohomology of a family of curves over $X$, whose fiber over
$x\in X$ is a $(\mathbb{Z}/r\mathbb{Z}\times G)$-cover of $\mathbb{P}^{1}$,
branched over $\\{x\\}\cup D$, where $r$ is some positive integer and $G$ is a
finite complex reflection group. The cases with $G$ a dihedral group are
precisely those studied in [LL23], which gave us some hope that Theorem 1.1.2
could be true.

As we will see, Theorem 1.1.2 gives quite fine control of the local systems
$\mathbb{U},\chi,\mathbb{M}$; for example, the dimension and local monodromies
of $\mathbb{U}$ are given explicitly in 4.3.3, and their monodromy
representations may be computed algorithmically from that of $\mathbb{V}$ (and
vice versa) [DR07].

######  Remark 1.1.4.

In fact we prove a slightly more general result, Theorem 4.3.2, where the
condition that some local monodromy matrix have infinite order is relaxed in
favor of requiring that $\mathbb{V}$ has no unitary Galois conjugates. Note
that this latter condition is automatic if some local monodromy matrix has
infinite order. Indeed, local systems of geometric origin always have quasi-
unipotent local monodromy, i.e. all eigenvalues are roots of unity. Thus if
there is a local monodromy matrix of infinite order, it must not be
diagonalizable, and hence cannot be unitary.

The condition that $\mathbb{V}$ has no unitary Galois conjugates has attracted
some interest in the case of representations in $X_{3}$ with quasi-unipotent
monodromy at infinity, see [McM23b, McM23a].

######  Remark 1.1.5.

In the statement of Theorem 1.1.2, the condition of non-scalar monodromy at
infinity is just a notational convenience; as $\mathbb{V}$ has Zariski-dense
monodromy, at least one of the punctures must have non-scalar monodromy. So
this hypothesis can always be achieved by relabeling the points of $D$.

######  Remark 1.1.6.

From the point of view of local systems of geometric origin, the condition
that some $A_{i}$ have infinite order is natural. By Corlette-Simpson [CS08,
Theorem 2 and §9], rank $2$ local systems of geometric origin with Zariski-
dense monodromy always arise in the cohomology of a generically simple Abelian
scheme “of $\operatorname{GL}_{2}$-type.” The condition that some $A_{i}$ have
infinite order is precisely the condition that this Abelian scheme have some
point of potentially totally degenerate reduction, i.e. there is a point at
which the abelian scheme admits, after a finite extension of the corresponding
discretely valued field, a semistable model with toric special fiber.

In this way our main technical result, Theorem 1.1.2, classifies all non-
isotrivial $\operatorname{GL}_{2}$-type Abelian schemes over $\mathbb{P}^{1}$
with:

  1. (1)

at least one point of potentially totally degenerate reduction, and

  2. (2)

generic discriminant locus.

The following classification result for MCG-finite representations is deduced
in § 4.4:

######  Corollary 1.1.7.

Let $\rho:\pi_{1}(\Sigma_{0,n})\to SL_{2}(\mathbb{C})$ be a Zariski-dense MCG-
finite representation with at least one local monodromy matrix of infinite
order. Then either

  1. (1)

$\rho$ is a “pullback” representation (classified by Diarra [Dia13, §3-§5]),
or

  2. (2)

$\rho=MC_{\chi}(\gamma)\otimes\nu$, where
$\nu:\pi_{1}(\Sigma_{0,n})\to\mathbb{C}^{*}$ and
$\chi:\pi_{1}(\mathbb{G}_{m})\to\mathbb{C}^{*}$ are finite order characters,
and $\gamma:\pi_{1}(\Sigma_{0,n})\to\operatorname{GL}_{r}(\mathbb{C})$ is an
irreducible representation with image a finite complex reflection group.

In § 5 we analyze all finite complex reflection groups of rank at least $5$,
and in particular we show in 5.1.4 that local systems as in Theorem 1.1.2
cannot have non-scalar monodromy at more than $6$ points. Combining this
analysis with Diarra’s result [Dia13, Théorème 5.1], which rules out
“pullback” representations in this regime, we deduce the following corollary
in § 5.7.

######  Corollary 1.1.8.

Let

$\rho:\pi_{1}(\Sigma_{0,n})\to\operatorname{SL}_{2}(\mathbb{C})$

be a representation with Zariski-dense image and non-scalar local monodromy at
each puncture. Suppose the local monodromy at one or more punctures has
infinite order, and that $[\rho]\in X_{n}$ has finite orbit under the action
of $\operatorname{Mod}_{0,n}$. Then $n\leq 6$.

In § 5.8, we give a number of examples of such local systems on $\Sigma_{0,n}$
with non-scalar monodromy at $n\leq 6$ points, arising via middle convolution
from finite complex reflection groups, which show that the bound of 1.1.8 is
sharp. 1.1.8 proves part of the conjecture of [Tyk22, §11] in the case under
consideration, i.e. when there is a local monodromy matrix of infinite order
and $n>6$.

###  1.2. Previous work

The action of $\text{Mod}_{0,n}$ on $X_{n}$ has been heavily studied, in
several quite distinct contexts. While we cannot summarize all of this work
here, we do our best to indicate the various perspectives motivating research
in the subject.

####  1.2.1. Dynamics of character varieties

Fixing conjugacy classes
$C_{1},\cdots,C_{n}\subset\operatorname{SL}_{2}(\mathbb{C})$ and, setting
$\mathcal{Z}=(C_{1},\cdots,C_{n})$, we define the _relative character variety_
$X_{n}(\mathcal{Z})$ of $\pi_{1}(\Sigma_{0,n})$ to be the subset of $X_{n}$
consisting of those conjugacy classes of $(A_{1},\cdots,A_{n})$ such that
$A_{i}\in C_{i}$ for all $i$. As the action of $\operatorname{Mod}_{0,n}$
simply permutes the $C_{i}$, there is an index $n!$ subgroup
$\operatorname{PMod}_{0,n}\subset\operatorname{Mod}_{0,n}$ whose action
preserves $X_{n}(\mathcal{Z})$. The dynamics of this action, and its higher
genus analogues, are extremely well-studied, though still far from completely
understood. Our results amount to a classification of finite orbits in
$X_{n}(\mathcal{Z})$ when some $C_{i}$ has infinite order.

The study of $X_{4}$ goes back to Fricke and Klein [FK65], and the study of
its dynamics dates to work of Markoff [Mar79, Mar80]. In this case
$X_{4}(\mathcal{Z})$ is an open subset of the affine cubic surface

$x^{2}+y^{2}+z^{2}+xyz=ax+by+cz+d;$

the dependence of $a,b,c,d$ on $\mathcal{Z}$ may be found in [CL09, Equation
(1.5)], where Cantat and Loray studied the holomorphic dynamics of the
$\operatorname{Mod}_{0,4}$-action on $X_{4}$ in some detail. Goldman [Gol97,
Gol06], Previte-Xia [PX00, PX02b], Pickrell-Xia [PX02a, PX03] and others
studied the action of the mapping class group of a surface $\Sigma_{g,n}$ on
its character variety (valued in a compact Lie group) with the aim of
understanding ergodicity and density properties of this action. Brown, Eskin,
Filip, and Rodriguez-Hertz have ongoing work studying this action with the
goal of understanding orbit closures and invariant measures. Bourgain,
Gamburd, and Sarnak, [BGS16b, BGS16a] as well as Chen [Che23], used the
dynamics of the $\operatorname{Mod}_{0,4}$-action on $X_{4}$ to prove strong
approximation results for $X_{4}$; Whang [Wha20a, Wha20b, Wha20c] began the
process of generalizing these results to higher genus and $n$ large. See also
Michel’s work [Mic06], which amounts to an analysis of the finite orbits of
the action of $\operatorname{Mod}_{0,n}$ on the real points of certain higher
rank relative character varieties, parametrizing representations into (not
necessarily finite) reflection groups.

Finally, two of the authors of this paper studied the action of the mapping
class group $\operatorname{Mod}_{g,n}$ of a higher genus surface
$\Sigma_{g,n}$ on the character variety parametrizing $r$-dimensional
representations of $\pi_{1}(\Sigma_{g,n})$, and showed that if $g>r^{2}-1$,
all finite orbits correspond to representations with finite image [LL22a],
after an analogous result was proved by Biswas-Gupta-Mj-Whang in rank $2$
[BGMW22].

####  1.2.2. Isomonodromy

Much previous work on the question considered here has been done in the
context of _isomonodromy_. In particular classifying finite orbits of the
$\operatorname{Mod}_{0,n}$-action on $X_{n}$ when $n=4$ is equivalent to
classifying algebraic solutions to the Painlevé VI equation [LT14]; this is
the primary source of historical interest in the question. Our work in this
paper may be considered analogously—our main result classifies algebraic
solutions to the Schlesinger system

$\begin{dcases}\frac{\partial B_{i}}{\partial\lambda
j}=\frac{[B_{i},B_{j}]}{\lambda_{i}-\lambda_{j}}&i\neq j\\\ \frac{\partial
B_{i}}{\partial\lambda_{i}}=-\sum_{j\neq
i}\frac{[B_{i},B_{j}]}{\lambda_{i}-\lambda_{j}}\\\ \sum B_{i}=0\end{dcases}$

where the $B_{i}\in\mathfrak{sl}_{2}(\mathbb{C})$, $i=1,\cdots,n$, with
$\operatorname{exp}(2\pi iB_{j})$ of infinite order for some $j$; algebraic
solutions to this system correspond to finite orbits of
$\operatorname{Mod}_{0,n}$ on $X_{n}$ (see [Cou17, Theorem A] or [LL22a,
Remark 2.3.6]).

The history of the case of $X_{4}$—the Painlevé VI equation—is summarized
nicely by Boalch [Boa10]. Finite orbits (equivalently, algebraic solutions to
the Painlevé VI equation) were discovered by many people, including Andreev-
Kitaev [AK02], Boalch [Boa07, Boa06a, Boa06b, Boa05, Boa10], Doran [Dor01],
Kitaev [Kit06, Kit05], Dubrovin-Mazzocco [DM00], Hitchin [Hit95], and others.
The list of finite orbits they found was eventually proven to be complete by
Lisovyy & Tykhyy, using a computer-aided proof [LT14].

Finite orbits of the $\operatorname{Mod}_{0,5}$-action on $X_{5}$ were studied
by Calligaris-Mazzocco [CM18a] and Tykhyy [Tyk22], who gave a complete,
computer-aided classification in this case. Beyond this, no general
classification was known prior to this work, as far as we are aware.

####  1.2.3. Middle convolution

The use of middle convolution and finite complex reflection groups to produce
solutions to the Painlevé VI equation, where middle convolution is closely
related to the “Okamoto transform” [LSS13, §9], has been known for some time;
it appears implicitly in Boalch’s work [Boa03, Boa05] and explicitly in work
of Dettweiler-Reiter [DR07]. Girand [Gir16, §5] used middle convolution to
produce some finite braid group orbits in $X_{5}$. Our Hodge-theoretic
analysis of middle convolution has precursors in [DS13, DS18, Mar21]; in
principle we expect we could have used their results to prove 4.2.1, though we
chose to give a direct argument.

Kitaev conjectured [Kit02, Kit05] that all finite orbits in the case of
$X_{4}$ (namely, algebraic solutions to the Painlevé VI equation) are, up to
Okamoto transformation/middle convolution, either “pullback representations”
or have finite monodromy; this conjecture was confirmed by Lisovyy-Tykhyy’s
classification [LT14]. It is interesting to consider our results here in light
of this conjecture: our work gives a computer-free proof in the case some
local monodromy matrix has infinite order, and a generalization to the case
that $n>4$.

###  1.3. Sketch of proof

Our proof of Theorem 1.1.2 is closely related to Katz’s classification of
rigid local systems [Kat96]. Given an irreducible rigid local system
$\mathbb{V}$ of rank at least $2$ on $\mathbb{P}^{1}\setminus D$, Katz gives
an algorithm for choosing a rank one local system $\mathbb{L}$ on
$\mathbb{P}^{1}\setminus D$, and a rank one local system $\chi$ on
$\mathbb{G}_{m}$, such that $MC_{\chi}(\mathbb{V}\otimes\mathbb{L})$ is rigid
and has rank less than that of $\mathbb{V}$. Hence iterating this procedure
eventually yields a local system of rank one. As middle convolution is an
invertible operation, this shows that all rigid local systems can be
constructed out of local systems of rank one.

We show in § 4 that a variant of Katz’s algorithm, applied to a local system
satisfying the hypotheses of Theorem 1.1.2, necessarily yields a local system
with all of its Galois conjugates _unitary_. Combined with the fact that
Katz’s algorithm preserves integrality of a local system, this implies that
the local system thus produced has finite monodromy. We conclude that its
monodromy is given by a finite complex reflection group by analyzing the
effect of middle convolution on local monodromy. Unlike in Katz’s situation,
our application of middle convolution typically _increases_ the rank of the
local system in question.

It is natural to ask: to what extent does middle convolution explain MCG-
finite local systems in higher rank? Rigid local systems are evidently MCG-
finite (as they are isolated points of relative character varieties, and hence
are permuted by $\operatorname{Mod}_{0,n}$). The fact that a variant of Katz’s
algorithm extends beyond the rigid case to classify some MCG-finite local
systems in our setting is, we think, suggestive.

###  1.4. Notation

A _family of $n$-pointed projective lines_ is a map
$\pi:\mathbb{P}^{1}\times\mathscr{M}\to\mathscr{M}$ equipped with $n$ disjoint
sections $s_{1},\cdots,s_{n}$. The _associated family of $n$-punctured
projective lines_ $\pi^{\circ}:\mathscr{C}^{\circ}\to\mathscr{M}^{\circ}$ is
the restriction of $\pi$ to the complement of the images of the $s_{i}$. For
$n\geq 3$ we say that $\pi$ is a _versal family_ if the induced map
$\mathscr{M}\to\mathscr{M}_{0,n}$ is dominant, and in this case we refer to
$\pi^{\circ}$ as the associated _punctured versal family_. We work over the
complex numbers throughout.

###  1.5. Acknowledgments

We are grateful for many useful conversations with Philip Boalch, Serge
Cantat, Bertrand Deroin, Alex Eskin, Bruno Klingler, Oleg Lisovyy, Frank
Loray, Claude Sabbah, Will Sawin, Carlos Simpson, and Nicolas Tholozan. Lam
was supported by a Dirichlet Fellowship at Humboldt University, Berlin during
the course of this work. Litt was supported by the NSERC Discovery Grant,
”Anabelian methods in arithmetic and algebraic geometry.”

##  2\. Parabolic Higgs bundles, variations of Hodge structure, and
isomonodromy

In this section we recall some preliminaries on parabolic Higgs bundles, non-
abelian Hodge theory for non-compact Riemann surfaces, and isomonodromy.

###  2.1. Preliminaries on parabolic bundles

Let $C$ be a smooth proper curve, and let $D=x_{1}+\cdots+x_{n}$ be a reduced
divisor on $C$. A parabolic bundle $E_{\star}$ on $(C,D)$ is a vector bundle
$E$ on $C$, equipped with a decreasing filtration
$E_{x_{j}}=E^{1}_{j}\supsetneq E^{2}_{j}\supsetneq\cdots\supsetneq
E^{n_{j}+1}_{j}=0$ at each $x_{j}$, with attached weights
$0\leq\alpha_{j}^{1}<\cdots<\alpha_{j}^{n_{j}}<1$. See [LL22b, §2] for further
background, notions of homomorphisms of parabolic bundles, quotients and
subbundles, slope, stability, etc. We will need the following specialization
of the definition of parabolic Hom sheaves in [LL22b, §2.2]:

If $E_{\star},F_{\star}$ are parabolic line bundles on $(C,D)$, the sheaf of
parabolic homomorphisms is

(2.1)
$\displaystyle\underline{\operatorname{Hom}}(E_{\star},F_{\star})_{0}=\underline{\operatorname{Hom}}(E,F)(-D^{\prime}),$

where $D^{\prime}\subset D$ is the set of points $x_{j}$ at which the weight
of $E_{x_{j}}$ is greater than that of $F_{x_{j}}$.

If $(E,\nabla:E\to E\otimes\Omega^{1}_{C}(\log D))$ is a flat vector bundle on
$C$ with logarithmic singularities along $D$, then $E$ naturally obtains the
structure of a parabolic bundle as in [LL22b, Definition 3.3.1]. If $x$ is a
point of $D$, let $M$ be the residue matrix of $\nabla$ at $x$. Then the
weights of the parabolic structure of $E$ at $x$ are

$\operatorname{Re}(\lambda)-\lfloor\operatorname{Re}(\lambda)\rfloor,$

where $\lambda$ runs over the eigenvalues of $M$.

###  2.2. Preliminaries on local systems and variations of Hodge structure

Recall that a complex variation of Hodge structure on $C\setminus D$ is a
triple $(V,V^{p,q},\mathbf{D})$ where

  * •

$V$ is a $C^{\infty}$ complex vector bundle on $C\setminus D$, equipped with a
decomposition $V=\oplus V^{p,q}$,

  * •

$\mathbf{D}$ is a flat connection on $V$ satisfying Griffiths transversality,
i.e. $\mathbf{D}(V^{p,q})\subset A^{1,0}(V^{p,q}\oplus V^{p-1,q+1})\oplus
A^{0,1}(V^{p,1}\oplus V^{p+1,q-1}),$ where $A^{i,j}(E)$ denotes the sheaf of
$E$-valued $(i,j)$-forms.

It is moreover polarizable if $V$ can be equipped with a Hermitian form $\psi$
flat with respect to $\mathbf{D}$, and such that $V^{p,q}$ and
$V^{p^{\prime},q^{\prime}}$ are orthogonal to each other for
$(p,q)\neq(p^{\prime},q^{\prime})$, and $(-1)^{p}\psi$ is positive definite on
$V^{p,q}$.

Mehta and Seshadri [MS80] famously give a correspondence between polystable
parabolic bundles of slope zero on a pair $(C,D)$ and unitary local systems on
$C\setminus D$. The following is a standard generalization of their theorem to
polarizable complex variations of Hodge structure $(V,V^{p,q},\mathbf{D})$ on
$C\setminus D$.

######  Proposition 2.2.1.

  1. (1)

$(E,\nabla)\vcentcolon=(\ker(\mathbf{D})\otimes\mathcal{O},\mathrm{id}\otimes
d)$ is a holomorphic flat vector bundle, and is moreover equipped with the
Hodge filtration $F^{p}V\vcentcolon=\oplus V_{j\geq p}^{j,q}$ by holomorphic
sub-bundles. We refer to it as the _holomorphic flat vector bundle_ associated
to $(V,V^{p,q},\mathbf{D})$.

  2. (2)

There exists an unique flat bundle $(\overline{E},\overline{\nabla})$ on $C$
with logarithmic singularities along $D$ extending $(E,\nabla)$, known as the
Deligne canonical extension, such that the eigenvalues of the residues of
$\overline{\nabla}$ have real parts in $[0,1)$. Moreover,
$(\overline{E},\overline{\nabla})$ has a Hodge filtration extending that on
$E$, which we also denote by $F^{\bullet}$.

  3. (3)

$\overline{E}$ may be given the structure of a parabolic bundle as in [LL22b,
Definition 3.3.1], which we denote by $\overline{E}_{\star}$. The associated
graded
$(\oplus_{i}\operatorname{gr}^{i}_{F^{\bullet}}\overline{E}_{\star},\theta)$
of $(\overline{E}_{\star},F^{\bullet},\overline{\nabla})$ is a Higgs bundle,
and is moreover parabolically polystable of slope zero.

###### Proof.

Point (1) is standard and straightforward to check. The rest of the
proposition is a form of [Sim90, Theorem 8]. The construction of the Deligne
canonical extension is given for example in [LL22b, Definition 4.1.2]. For the
construction of the parabolic structure on $\overline{E}$, see [LL22b,
Definition 3.3.1]. For the claim in (2) that the Hodge filtration extends, as
well as the semistability statement in (3), we refer the reader to [LL22b,
Proposition 4.1.4] and the references therein. ∎

######  Definition 2.2.2.

Let $X$ be a smooth quasiprojective variety and $\mathbb{V}$ a local system on
$X$. We say that $\mathbb{V}$ has _quasi-unipotent monodromy at infinity_ if
there exists a simple normal crossings compactification $X\subset\overline{X}$
so that the local monodromy about each component of the boundary
$\overline{X}\setminus X$ is _quasi-unipotent_ , i.e. all eigenvalues are
roots of unity.

###  2.3. Isomonodromy

We let $D$ be a reduced effective divisor of degree $n$ on $\mathbb{P}^{1}$.
Let $(\mathscr{E},\nabla:\mathscr{E}\to\mathscr{E}\otimes\Omega^{1}(\log D))$
be a logarithmic flat vector bundle on $(\mathbb{P}^{1},D)$.

Suppose we are given a contractible complex manifold $B$, $o\in B$ a point,
and a divisor $\mathscr{D}\subset\mathbb{P}^{1}\times B$, étale over $B$.
Suppose moreover we have a fixed isomorphism
$(\mathbb{P}^{1}_{o},\mathscr{D}_{o})\simeq(\mathbb{P}^{1},D)$. Then there is
a canonical flat vector bundle

$(\widetilde{\mathscr{E}},\widetilde{\nabla}:\widetilde{\mathscr{E}}\to\widetilde{\mathscr{E}}\otimes\Omega^{1}_{\mathbb{P}^{1}\times
B}(\log\mathscr{D}))$

equipped with an isomorphism
$(\widetilde{\mathscr{E}},\widetilde{\nabla})|_{\mathbb{P}^{1}_{o}}\overset{\sim}{\to}(\mathscr{E},\nabla)$,
see [LL22b, §3.4] or [Mal83, Théorème 2.1], referred to as the _isomonodromic
deformation_ of $(\mathscr{E},\nabla)$. This is etymologically justified
because the monodromy representation associated to
$(\widetilde{\mathscr{E}},\widetilde{\nabla})|_{\mathbb{P}^{1}\setminus\mathscr{D}_{b}})$
is independent of $b\in B$.

Given $(\mathscr{E},\nabla)$ as above, with $n\geq 3$, we may always take $B$
to be the universal cover of the moduli space $\mathscr{M}_{0,n}$ of
$n$-pointed curves of genus zero. In this case we refer to the fiber of
$(\widetilde{\mathscr{E}},\widetilde{\nabla})$ over a general $b\in B$ as _the
isomonodromic deformation of $(\mathscr{E},\nabla)$ to a general nearby
$n$-pointed curve_. Here, by _general_ we mean “on the complement of a nowhere
dense analytic subset.” (See [LL22b, Definition 3.4.4] and the surrounding
text for a discussion of this notion.)

######  Proposition 2.3.1.

Let
$(\mathscr{E},\nabla:\mathscr{E}\to\mathscr{E}\otimes\Omega^{1}_{\mathbb{P}^{1}}(\log
D))$ be a logarithmic flat vector bundle of rank $2$ on $(\mathbb{P}^{1},D)$
with irreducible monodromy. Suppose that the isomonodromic deformation
$(\mathscr{E}^{\prime},\nabla^{\prime})$ of $(\mathscr{E},\nabla)$ to a
general nearby $n$-pointed curve $(\mathbb{P}^{1},D^{\prime})$ (with parabolic
structure as in [LL22b, Definition 3.3.1]) is not parabolically semistable,
with maximal destabilizing sub-bundle $F_{\star}$. Then
$\underline{\operatorname{Hom}}(F_{\star},\mathscr{E}^{\prime}/F_{\star})_{0}=\mathscr{O}(-1)$.

###### Proof.

This follows from a deformation-theoretic argument, as in the proof of [LL22b,
Theorem 6.1.1]. The connection $\nabla^{\prime}$ yields a nonzero
$\mathscr{O}$-linear map
$T_{\mathbb{P}^{1}}(-D)\to\underline{\operatorname{Hom}}(F_{\star},\mathscr{E}^{\prime}/F_{\star})_{0}$
(that the map is non-zero follows as the monodromy of
$(\mathscr{E}^{\prime},\nabla^{\prime})$ is irreducible). As this is a map of
line bundles, the cokernel is torsion, and hence the map is surjective on
$H^{1}$.

As $D^{\prime}$ is general, [LL22b, Lemma 6.4.2] implies that this map induces
the zero map

$H^{1}(T_{\mathbb{P}^{1}}(-D^{\prime}))\to
H^{1}(\underline{\operatorname{Hom}}(F_{\star},\mathscr{E}^{\prime}/F_{\star})_{0}).$

Thus in particular we must have
$H^{1}(\underline{\operatorname{Hom}}(F_{\star},\mathscr{E}^{\prime}/F_{\star})_{0})=0$.
But as $F$ is maximal destabilizing, the parabolic degree of
$\underline{\operatorname{Hom}}(F_{\star},\mathscr{E}^{\prime}/F_{\star})$
(and hence the honest degree of
$\underline{\operatorname{Hom}}(F_{\star},\mathscr{E}^{\prime}/F_{\star})_{0}$)
is negative; thus
$\underline{\operatorname{Hom}}(F_{\star},\mathscr{E}^{\prime}/F_{\star})_{0}$
must be the unique-up-to-isomorphism line bundle on $\mathbb{P}^{1}$ with
negative degree and vanishing $H^{1}$, namely $\mathscr{O}(-1)$. ∎

######  Proposition 2.3.2.

Let $\mathbb{V}$ be an irreducible complex local system of rank $2$ on
$\mathbb{P}^{1}\setminus D$ and let
$(\mathscr{E},\nabla:\mathscr{E}\to\mathscr{E}\otimes\Omega^{1}_{\mathbb{P}^{1}}(\log
D))$ be its Deligne canonical extension. Suppose that the isomonodromic
deformation $(\mathscr{E}^{\prime},\nabla^{\prime})$ of $(\mathscr{E},\nabla)$
to a general nearby $n$-pointed curve $(\mathbb{P}^{1},D^{\prime})$ underlies
a polarizable complex variation of Hodge structure. Then there exists a point
$x$ of $D$ such that $\mathbb{V}$ does not have unipotent monodromy at $x$.

###### Proof.

Assume to the contrary that $\mathbb{V}$ has unipotent monodromy at all points
$x$ of $D$. If $\mathbb{V}$ is unitary, then in fact $\mathbb{V}$ has trivial
monodromy at all points $x$ of $D$, contradicting its irreducibility. So we
may assume $\mathbb{V}$ is not unitary, i.e. after shifting indices, it
underlies a variation of Hodge structure with $h^{1,0}=h^{0,1}=1$, and
$h^{i,j}=0$ otherwise. Let $F\subset\mathscr{E}$ be the non-trivial piece of
the Hodge filtration.

As $\mathbb{V}$ has unipotent monodromy along $D$, the induced parabolic
structure on $\mathscr{E}$ is trivial, so the Higgs bundle
$\oplus_{i}\operatorname{gr}^{i}_{F^{\bullet}}(\mathscr{E})=F\oplus\mathscr{E}/F$
with the Higgs field induced by $\nabla$ is stable as a Higgs bundle of slope
zero, by 2.2.1(3). As $F$ is a quotient Higgs bundle, it must have positive
degree, so we have $\deg F\geq 1,\deg\mathscr{E}/F\leq-1$. Thus
$\deg\underline{\operatorname{Hom}}(F,\mathscr{E}/F)\leq-2$, contradicting
2.3.1. ∎

##  3\. Middle convolution

We now recall the definition and basic properties of Katz’s middle convolution
operation. Aside from Katz’s book [Kat96], the reader may find [DR07, Sim09]
to be useful references.

###  3.1. Definitions

As before, we let $D\subset\mathbb{P}^{1}$ be a reduced effective divisor
containing $\\{\infty\\}$, and set $X=\mathbb{P}^{1}\setminus D$. We let
$U=X\times X\setminus\Delta$, where $\Delta$ is the diagonal, and let
$j:U\hookrightarrow\mathbb{P}^{1}\times X$ be the natural inclusion. Let
$\pi_{1}:U\to X$ be the projection onto the first coordinate, and let
$\pi_{2}:\mathbb{P}^{1}\times X\to X$ be the projection onto the second
coordinate. Let $\alpha:U\to\mathbb{G}_{m}$ be the map $(x,y)\mapsto x-y$.
Some of this data is depicted in Figure 1.

######  Definition 3.1.1.

Given a local system $\mathbb{W}$ on $X$, and a rank one local system $\chi$
on $\mathbb{G}_{m}$, the middle convolution $MC_{\chi}(\mathbb{W})$ is defined
to be

$MC_{\chi}(\mathbb{W}):=R^{1}\pi_{2*}j_{*}(\pi_{1}^{*}\mathbb{W}\otimes\alpha^{*}\chi).$

This definition is a translation of Katz’s [Kat96, §2.8], avoiding the
language of perverse sheaves. See [DR03, Theorem 1.1] for a proof these two
definitions agree.

\begin{overpic}[trim=0.0pt 341.43306pt 0.0pt 0.0pt,clip]{middle-convolution-
diagram.pdf} \put(30.0,8.0){$x_{1}$} \put(39.0,8.0){$x_{2}$}
\put(48.0,8.0){$x_{3}$} \put(60.0,8.0){$\infty$} \put(45.0,19.0){$\pi_{2}\circ
j$} \put(86.0,37.0){$x_{1}$} \put(86.0,44.0){$x_{2}$} \put(86.0,52.0){$x_{3}$}
\put(86.0,61.0){$\infty$} \put(76.0,49.0){$\pi_{1}$} \put(20.0,63.0){$U$}
\put(15.0,9.0){$X$} \put(84.0,69.0){$X$} \put(25.0,31.0){$\Delta$}
\end{overpic}

Figure 1. A depiction of $U$ with its two projections. The region enclosed in
a dashed box is depicted in Figure 2. In general, the local system
$\alpha^{*}\chi$ has nontrivial local monodromy about the diagonal $\Delta$,
$\pi_{1}^{-1}(\infty),(\pi_{2}\circ j)^{-1}(\infty)$, and the local system
$\pi_{1}^{*}\mathbb{W}$ has non-trivial local monodromy about
$\pi_{1}^{-1}(x_{i})$, $\pi_{1}^{-1}(\infty)$.

###  3.2. Basic properties

######  Proposition 3.2.1.

If $\mathbb{W}$ is irreducible and $\chi$ is non-trivial, then

  1. (1)

$MC_{\chi}(\mathbb{W})$ is irreducible, and

  2. (2)

$MC_{\chi}(MC_{\chi^{-1}}(\mathbb{W}))=\mathbb{W}$.

If $\mathbb{W}$ underlies a complex variation of Hodge structure and $\chi$ is
unitary, then $MC_{\chi}(W)$ underlies a complex variation of Hodge structure
as well.

###### Proof.

(1) and (2) are [Kat96, Theorem 2.9.7]. The statement about complex variations
is part of [DS13, Proposition 3.1.1]. ∎

We recall the effect of middle convolution on local monodromies. By
$J(\lambda,\ell)$ we mean the $\ell\times\ell$ Jordan block with generalized
eigenvalue $\lambda$, so, e.g., $J(\lambda,2)$ is the matrix

$\begin{pmatrix}\lambda&1\\\ 0&\lambda\end{pmatrix}.$

######  Proposition 3.2.2 ([Kat96, Corollary 3.3.6], [DR07, Lemma 5.1]).

Suppose $\chi$ is a rank one local system on $\mathbb{G}_{m}$ with monodromy
$\lambda\neq 1$ around $0$. The functor $MC_{\chi}$ changes the local
monodromies of a local system $\mathbb{W}$ as follows:

  1. (1)

At each finite puncture $x_{i}\in D\setminus\\{\infty\\}$, each Jordan block
$J(\beta,\ell)$ of the local monodromy of $\mathbb{W}$ contributes
$J(\beta\lambda,\ell^{\prime})$ to the Jordan canonical form of the local
monodromy of $MC_{\chi}(\mathbb{W})$ at $x_{i}$ where

$\ell^{\prime}=\begin{cases}\ell\ \ \ \ \ \ \ \ \ \text{if}\ \beta\neq
1,\lambda^{-1}\\\ \ell-1\ \ \text{if}\ \beta=1\\\ \ell+1\ \ \text{if}\
\beta=\lambda^{-1}.\end{cases}$

All other Jordan blocks are of the form $J(1,1)$.

  2. (2)

At $\infty$, each $J(\beta,\ell)$ contributes
$J(\beta\lambda^{-1},\ell^{\prime})$ where

$\ell^{\prime}=\begin{cases}\ell\ \ \ \ \ \ \ \ \ \text{if}\ \beta\neq
1,\lambda\\\ \ell+1\ \ \text{if}\ \beta=1\\\ \ell-1\ \ \text{if}\
\beta=\lambda.\end{cases}$

All other Jordan blocks are of the form $J(\lambda^{-1},1)$.

##  4\. A modification of Katz’s algorithm

We now present a slight variant of Katz’s middle convolution algorithm,
specialized to the rank $2$ case. Katz was interested in classifying rigid
local systems on $\mathbb{P}^{1}\setminus D$; he showed that his algorithm,
applied to an irreducible rigid local system of rank $r>1$, will always
produce a rigid local system of rank $r^{\prime}<r$. By contrast, in our
setting of rank $2$ MCG-finite local systems, the algorithm will not reduce
the rank of our local system in general—this would in fact imply the local
system is rigid. Rather, it will produce a local system with finite monodromy,
possibly with much larger rank.

###  4.1. The algorithm

Let $X=\mathbb{P}^{1}\setminus D$, for $D=\\{x_{1},\cdots,x_{n-1},\infty\\}$
some reduced effective divisor, and let $\mathbb{V}$ be an irreducible rank
$2$ local system on $X$. We assume the isomonodromic deformation of
$\mathbb{V}$ to a nearby $n$-pointed curve underlies a non-unitary polarizable
complex variation of Hodge structure, as in 2.3.2. Following [Kat96, §5.2],
our variant of Katz’s algorithm proceeds as follows:

  1. (1)

Tensor with a rank one local system $\mathbb{L}_{1}$ so that
$\mathbb{V}^{\prime}=\mathbb{V}\otimes\mathbb{L}_{1}$ does not have non-
trivial scalar monodromy at any of the $x_{i}$.

  2. (2)

Tensor with a rank one local system $\mathbb{L}_{2}$ so that the local
monodromy of
$\mathbb{V}^{\prime\prime}=\mathbb{V}^{\prime}\otimes\mathbb{L}_{2}$ at
$x_{i}$ has $1$ as an eigenvalue for $i=1,\cdots,n-1$.

  3. (3)

If the the local monodromy of $\mathbb{V}^{\prime\prime}$ at infinity is
unipotent, relabel the points of $D$ so that this is no longer the case; this
is always possible by 2.3.2. Let $\nu\neq 1$ be one of the eigenvalues of the
local monodromy of $\mathbb{V}^{\prime\prime}$ at $\infty$. Let $\chi$ be the
rank one local system on $\mathbb{G}_{m}$ with local monodromy $\nu$ at $0$
and local monodromy $\nu^{-1}$ at $\infty$.

  4. (4)

Set $\mathbb{U}=MC_{\chi}(\mathbb{V}^{\prime\prime})$.

Note that as $\mathbb{V}$ underlies a polarizable complex variation of Hodge
structure, the rank one local systems $\mathbb{L}_{1},\mathbb{L}_{2},\chi$ are
all unitary. If $\mathbb{V}$ has monodromy conjugate to a representation
valued $GL_{r}(\mathscr{O}_{K})$ for some number field $K$, then
$\mathbb{L}_{1},\mathbb{L}_{2},\chi$ are valued in
$\mathscr{O}_{K^{\prime}}^{\times}$ for some finite extension $K^{\prime}/K$,
and $MC_{\chi}(\mathbb{V}^{\prime\prime})$ has monodromy conjugate to a
subgroup of $GL_{r^{\prime}}(\mathscr{O}_{K^{\prime}})$.

In the rigid local systems setting, Katz iterates his procedure; we will halt
after a single pass. Otherwise our variant of Katz’s algorithm differs from
his only in the relabeling in step (3), which is unnecessary in the rigid
setting; we also have opted to avoid his use of the language of perverse
sheaves.

###  4.2. Unitarity of monodromy

Throughout this section we assume $n\geq 3$. We will now begin preparations to
show that Katz’s algorithm, applied to a motivic rank $2$ local system on
$\mathbb{P}^{1}\setminus D$ with $D$ generic and with infinite order at one or
more punctures (or more generally, with no unitary Galois conjugates),
produces a local system with finite monodromy.

With notation as in § 1.4, let
$(\pi:\mathscr{C}=\mathbb{P}^{1}\times\mathscr{M}\to\mathscr{M},s_{1},\cdots,s_{n+1})$
be a versal family of $(n+1)$-pointed projective lines, let
$\pi^{\circ}:\mathscr{C}^{\circ}\to\mathscr{M}$ be the associated family of
punctured projective lines, and let
$j:\mathscr{C}^{\circ}\hookrightarrow\mathscr{C}$ be the natural inclusion.
Let $X$ be a general fiber of $\pi^{\circ}$, say over $m$,
$\overline{X}=\mathbb{P}^{1}$ the smooth compactification, and
$D=\overline{X}\setminus X$, $D=\\{x_{1},\cdots,x_{n+1}\\}$.

We now perform the key computation in the proof:

######  Proposition 4.2.1 (Key computation).

Let $\mathbb{V}$ be a rank $2$ $\mathbb{C}$-local system on
$\mathscr{C}^{\circ}$ such that $\mathbb{V}|_{X}$ is irreducible. Suppose that
$\mathbb{V}$ underlies a polarizable complex variation of Hodge structure with
$h^{1,0}=h^{0,1}=1$, and $h^{p,q}=0$ for $(p,q)\neq(1,0),(0,1)$. Suppose
further that for $i=1,\cdots,n$, the local monodromy of
$\mathbb{V}|_{C^{\circ}}$ at $x_{i}$ has $1$ as an eigenvalue. Then
$R^{1}\pi_{*}j_{*}\mathbb{V}$ has unitary monodromy.

###### Proof.

Let $(\mathscr{E},\nabla)$ be the logarithmic flat vector bundle on
$\overline{X}$ obtained as the Deligne canonical extension of
$\mathbb{V}|_{X}\otimes\mathscr{O}_{X}$ to $\overline{X}$. Let
$F\subset\mathscr{E}$ be the non-trivial piece of the Hodge filtration; $F$ is
a line bundle by hypothesis. Set $\overline{\mathbb{V}}$ to be the complex
conjugate of $\mathbb{V}$, and
$(\overline{\mathscr{E}},\overline{F},\overline{\nabla})$ the corresponding
filtered flat vector bundle on $\overline{X}$. Note that
$\overline{\mathbb{V}}$ also satisfies the hypotheses of the Proposition.

It is enough to show that the Hodge filtration on
$R^{1}\pi_{*}j_{*}\mathbb{V}$, which is a priori of length three, in fact has
length one. Indeed, in this case the polarization is a flat unitary metric.

We claim it suffices to show that $\mathscr{E}/{F}=\mathscr{O}(-1)$. Indexing
the Hodge filtration $F^{\bullet}$ on $R^{1}\pi_{*}j_{*}\mathbb{V}$ so that it
is supported in degrees $-1,0,1$, we have (see [DS13, §2.3] for example) that

$H^{1}(\mathscr{E}/{F})=\operatorname{gr}^{-1}_{F^{\bullet}}R^{1}\pi_{*}j_{*}\mathbb{V}|_{m}.$

Now we have that

$\overline{\operatorname{gr}^{-1}_{F^{\bullet}}R^{1}\pi_{*}j_{*}\mathbb{V}}|_{m}=\operatorname{gr}^{1}_{F^{\bullet}}R^{1}\pi_{*}j_{*}\overline{\mathbb{V}}|_{m}.$

In particular, if
$\mathscr{E}/{F}=\overline{\mathscr{E}}/\overline{F}=\mathscr{O}(-1)$ and
hence
$\operatorname{gr}^{-1}_{F^{\bullet}}R^{1}\pi_{*}j_{*}\mathbb{V}|_{m}=\operatorname{gr}^{-1}_{F^{\bullet}}R^{1}\pi_{*}j_{*}\overline{\mathbb{V}}|_{m}=0$,
the same is true for
$\operatorname{gr}^{1}_{F^{\bullet}}R^{1}\pi_{*}j_{*}\mathbb{V}|_{m}$, and we
may conclude.

We now show, via a calculation with parabolic bundles, that
$\mathscr{E}/{F}=\mathscr{O}(-1)$. Let

$\theta:F\to\mathscr{E}/F\otimes\Omega^{1}(\log D)$

be the $\mathscr{O}$-linear map induced by $\nabla$. By 2.2.1(3), the bundle

$\operatorname{gr}^{\bullet}_{F^{\bullet}}\mathscr{E}=F\oplus\mathscr{E}/F$

with the Higgs field

$\begin{pmatrix}0&\theta\\\ 0&0\end{pmatrix}$

is a parabolically stable Higgs bundle of slope zero.

As $F$ is a quotient of this Higgs bundle, it must have positive parabolic
degree. Hence in particular, as it is a saturated subbundle of $\mathscr{E}$,
it must be the maximal destabilizing subbundle of $\mathscr{E}$ (as a
parabolic bundle). As $X$ is a general fiber of $\pi^{\circ}$, we have

$\underline{\operatorname{Hom}}(F_{\star},(\mathscr{E}/F)_{\star})_{0}=\mathscr{O}(-1)$

by 2.3.1.

Now let $0\leq\alpha_{i}\leq\beta_{i}<1$ be the parabolic weights of
$\mathscr{E}$ at $x_{i}$. By assumption $\alpha_{i}=0$ for $i=1,\cdots,n$. Let

$m=\sum_{i}(\alpha_{i}+\beta_{i})=\alpha_{n+1}+\sum_{i}\beta_{i}.$

We have $\deg\mathscr{E}=-m$; thus it suffices to show $\deg F=-m+1.$ Note
that $\deg F\geq-m+1$, as

$-\deg
F-m=\deg\mathscr{E}/F\leq\operatorname{par.deg.}(\mathscr{E}/F)_{\star}<0,$

again by parabolic stability of
$\operatorname{gr}^{\bullet}_{F^{\bullet}}\mathscr{E}$.

So set $f=-\deg F$; we know $f\leq m-1$ and wish to show $f\geq m-1$. Let
$S\subset\\{1,\cdots,n\\}$ be the set of $i$ such that $F$ has nonzero
parabolic weight at $x_{i}$. As the parabolic degree of $F$ is positive, we
must have that $|S|\geq f$. Hence

$\mathscr{O}(-1)=\underline{\operatorname{Hom}}(F_{\star},(\mathscr{E}/F)_{\star})_{0}\subset\underline{\operatorname{Hom}}(F,\mathscr{E}/F)(-D_{S}),$

where $D_{S}=\cup_{i\in S}\\{x_{i}\\}$ has degree at least $f$. Hence we have

$\displaystyle-1$
$\displaystyle\leq\deg\underline{\operatorname{Hom}}(F,\mathscr{E}/F)(-D_{S})$
$\displaystyle=\deg\mathscr{E}/F-\deg F-\deg D_{S}$ $\displaystyle=2f-m-\deg
D_{S}$ $\displaystyle\leq f-m$

Thus $f\geq m-1$ as desired. ∎

###  4.3. The main technical result

We may now prove the main technical results of the paper.

Let $\pi^{\circ}:\mathscr{C}^{\circ}\to\mathscr{M}$ be a versal family of
$n$-punctured projective lines. Let $m\in\mathscr{M}$ be a general point and
let $X$ be the fiber of $\pi^{\circ}$ over $m$.

######  Proposition 4.3.1.

Let $\mathbb{V}$ be a rank $2$ $\mathbb{C}$-local system on
$\mathscr{C}^{\circ}$ such that $\mathbb{V}|_{X}$ is irreducible. Suppose that
$\mathbb{V}$ underlies a polarizable complex variation of Hodge structure with
$h^{1,0}=h^{0,1}=1$, and $h^{p,q}=0$ for $(p,q)\neq(1,0),(0,1)$. Then applying
Katz’s algorithm (§4.1) to $\mathbb{V}$ yields a local system with unitary
monodromy.

###### Proof.

Without loss of generality, we may assume $n\geq 3$. Indeed, if $n<3$, then
$\pi_{1}(\mathbb{P}^{1}\setminus D)$ is abelian, so there are no irreducible
local systems of rank $2$ over $\mathbb{P}^{1}\setminus D$.

We first apply steps (1) and (2) of Katz’s algorithm, obtaining a local system
$\mathbb{V}^{\prime\prime}$ with no non-trivial scalar monodromies, and with
$1$ as an eigenvalue at each point of $D\setminus\\{\infty\\}$. Let $\chi$ be
as in step (3) of Katz’s algorithm. The main idea of the proof is to perform
middle convolution _in families_.

We consider
$\mathscr{U}=\mathscr{C}^{\circ}\times_{\mathscr{M}}\mathscr{C}^{\circ}\setminus\Delta$,
where $\Delta$ is the diagonal, and let
$j:\mathscr{U}\hookrightarrow\mathbb{P}^{1}\times\mathscr{C}^{\circ}$ be the
natural inclusion. Let $\pi_{1}:\mathscr{U}\to\mathscr{C}^{\circ}$ be the
projection onto the first coordinate, and
$\pi_{2}:\mathbb{P}^{1}\times\mathscr{C}^{\circ}\to\mathscr{C}^{\circ}$ the
projection onto the second coordinate. Note that $\pi_{2}\circ j$ is a versal
family of $(n+1)$-punctured projective lines. As in § 3, there is a map
$\alpha:\mathscr{U}\to\mathbb{G}_{m}$ given fiberwise by $(x,y)\mapsto x-y$.

We claim the versal family $\pi_{2}\circ j$, with the local system
$\pi_{1}^{*}\widetilde{\mathbb{V}}\otimes\alpha^{*}\chi$, satisfies the
hypotheses of 4.2.1. This suffices because the restriction of
$R^{1}\pi_{2*}j_{*}(\pi_{1}^{*}\widetilde{\mathbb{V}}\otimes\alpha^{*}\chi)$
to $X$ is precisely the output of Katz’s algorithm. We need to check that the
restriction of $\pi_{1}^{*}\widetilde{\mathbb{V}}\otimes\alpha^{*}\chi$ to a
fiber of $\pi_{2}\circ j$ satisfies:

  1. (1)

$h^{1,0}=h^{0,1}=1$, and

  2. (2)

has $1$ as an eigenvalue of at least $n$ of the $n+1$ local monodromy
matrices.

\begin{overpic}[trim=56.9055pt 426.79134pt 256.0748pt 0.0pt,clip]{zoomed-
middle-convolution} \put(30.0,8.0){$y$} \put(38.0,31.0){$(1,\lambda_{1})$}
\put(38.0,41.0){$(1,\lambda_{2})$} \put(38.0,53.0){$(\nu,\nu)$}
\put(38.0,63.0){$(1,\lambda_{3})$} \put(38.0,72.0){$(1,\lambda_{\infty})$}
\put(21.0,31.0){$x_{1}$} \put(21.0,41.0){$x_{2}$} \put(21.0,46.0){$\Delta$}
\put(21.0,63.0){$x_{3}$} \put(21.0,72.0){$\infty$}
\put(32.0,17.0){$\pi_{2}\circ j$} \end{overpic}

Figure 2. A neighborhood of the fiber of $\pi_{2}\circ j:X\times
X\setminus\Delta\to X$ over $y$, with eigenvalues of the local monodromy
matrices of $\pi_{1}^{*}\widetilde{\mathbb{V}}\otimes\alpha^{*}\chi$ indicated
on the right of the diagram.

Point (1) above is immediate from the fact that the same is true for
$\mathbb{V}$, by hypothesis; tensoring with unitary rank one local systems
cannot change this property. (As $\mathbb{V}$ underlies a polarizable complex
variation and hence the eigenvalues of its local monodromy matrices have
absolute value $1$, the local systems $\mathbb{L}_{1},\mathbb{L}_{2},\chi$ are
necessarily unitary.)

Point (2) follows from the choices in Katz’s algorithm, as we now explain; the
reader may find it useful to refer to Figure 2. It suffices to check point (2)
on any fiber of $\pi_{2}\circ j$. Consider $y\in X\subset\mathscr{C}^{\circ}$;
by construction $X^{\prime}=(\pi_{2}\circ j)^{-1}(y)$ is isomorphic to
$X\setminus\\{y\\}$; for notational convenience, we choose this fiber. The
local monodromy of
$\pi_{1}^{*}\widetilde{\mathbb{V}}\otimes\alpha^{*}\chi|_{X^{\prime}}$ at all
points of $D$ except $\infty$ has $1$ as an eigenvalue by step (2) of Katz’s
algorithm. $X^{\prime}$ has two remaining punctures: $y$ and $\infty$. The
local monodromy at $y$ is scalar, so we must show the local monodromy at
$\infty$ has $1$ as an eigenvalue. But this is immediate from our choice of
$\chi$ in step (3) of Katz’s algorithm. ∎

######  Theorem 4.3.2.

Let $D\subset\mathbb{P}^{1}$ be a generic effective divisor of degree $n$
containing $\infty$, and let $\mathbb{V}$ be a rank $2$
$\mathscr{O}_{K}$-local system on $X=\mathbb{P}^{1}\setminus D$ with non-
scalar monodromy at $\infty$, such that for each embedding
$\iota:\mathscr{O}_{K}\hookrightarrow\mathbb{C}$,
$\mathbb{V}\otimes_{\iota}\mathbb{C}$ is irreducible and underlies a non-
unitary polarizable complex variation of Hodge structure. Then applying Katz’s
algorithm (§ 4.1) to $\mathbb{V}$ yields a local system with finite monodromy.

###### Proof.

Without loss of generality, we may assume $n\geq 3$. Indeed, if $n<3$, then
$\pi_{1}(\mathbb{P}^{1}\setminus D)$ is abelian, so there are no irreducible
local systems of rank $2$ over $\mathbb{P}^{1}\setminus D$.

By Corlette-Simpson [CS08, Theorem 2 and §9], $\mathbb{V}$ is of geometric
origin, and hence, spreading out, there exists a versal family of
$n$-punctured projective lines $\mathscr{C}^{\circ}\to\mathscr{M}$, containing
$X$ as a fiber, with
$\mathscr{C}^{\circ}\subset\mathbb{P}^{1}\times\mathscr{M}$, and an
$\mathscr{O}_{K}$-local system $\widetilde{\mathbb{V}}$ on
$\mathscr{C}^{\circ}$ restricting to $\mathbb{V}$ on $X$.

Choosing $\mathbb{L}_{1},\mathbb{L}_{2},\chi$ as in Katz’s algorithm (§ 4.1),
we have by construction that $\mathbb{L}_{1},\mathbb{L}_{2},\chi$ are valued
in $\mathscr{O}_{K^{\prime}}^{\times}$, for $K^{\prime}$ some finite extension
of $K$. Now by hypothesis, for each embedding
$\iota:\mathscr{O}_{K^{\prime}}\hookrightarrow\mathbb{C}$,
$\widetilde{\mathbb{V}}\otimes_{\iota}\mathbb{C}$ satisfies the hypotheses of
4.3.1 (as it is non-unitary). Hence for each $\iota$, the output of Katz’s
algorithm (§ 4.1) is unitary and defined over $\mathscr{O}_{K^{\prime}}$,
whence it has finite monodromy by e.g. [LL22b, Lemma 7.2.1]. ∎

######  Proposition 4.3.3 (Output of Katz’s algorithm).

With hypotheses as in Theorem 4.3.2, let $D^{\prime}\subset D$ be the set of
points at which $\mathbb{V}$ has non-scalar monodromy. Then the local system
$\mathbb{U}$ produced by Katz’s algorithm (§ 4.1) has the following
properties:

  1. (1)

$\operatorname{rk}\mathbb{U}=|D^{\prime}|-2$

  2. (2)

The monodromy group of $\mathbb{U}$ is a finite complex reflection group.

  3. (3)

The local monodromy of $\mathbb{U}$ at each $x_{i}\in
D^{\prime}\setminus\\{\infty\\}$ is a pseudoreflection. At each point of
$D\setminus D^{\prime}$, the local monodromy is trivial.

  4. (4)

The local monodromy of $\mathbb{U}$ at $\infty$ is $\nu^{-1}R$, where
$\nu^{-1}$ is the local monodromy at $\infty$ of the local system $\chi$ on
$\mathbb{G}_{m}$ chosen in Katz’s algorithm, and $R$ is a pseudoreflection.

###### Proof.

Point (1) is immediate from the computation [Kat96, Corollary 3.3.6] of Katz
of the rank of the middle convolution (see also [DR07, Theorem 2.2]). Point
(2) follows from (3) and Theorem 4.3.2. Point (3) follows directly from
3.2.2(1). Finally, (4) follows from 3.2.2(2), noting that by Theorem 4.3.2,
the local system $\mathbb{U}$ has finite monodromy and hence all Jordan blocks
have size $1$. ∎

Note that middle convolution is invertible by 3.2.1, so Theorem 4.3.2 tells us
that $\mathbb{V}=MC_{\chi}(\mathbb{U})\otimes\mathbb{L}$, for appropriate rank
one local systems $\chi$ on $\mathbb{G}_{m}$ and $\mathbb{L}$ on $X$, and
$\mathbb{U}$ a local system on $X$ satisfying the conclusions of 4.3.3.

######  Remark 4.3.4.

The proof of 4.3.1 in fact shows that applying Katz’s algorithm (§ 4.1) to
irreducible _supra-maximal_ local systems [DT19] yield a local systems with
unitary monodromy. In our language, irreducible supra-maximal local systems
have the following properties: they are rank $2$ local systems with trivial
determinant, underlying a polarizable $\mathbb{C}$-VHS
$(F,\mathscr{E},\nabla)$ such that
$\underline{\operatorname{Hom}}(F_{\star},(\mathscr{E}/F)_{\star})_{0}=\mathscr{O}(-1)$
(combining [DT19, Theorem 5] and 2.3.1). This answers a question of Deroin and
Tholozan; we are grateful to them for pointing out this consequence of our
work. This gives a different proof of the compactness of the space of supra-
maximal representations, which is [DT19, Corollary 3].

###  4.4. Proofs of Theorem 1.1.2 and 1.1.7

###### Proof of Theorem 1.1.2.

Let $\mathbb{V}$ be a local system as in the statement of Theorem 1.1.2. It
suffices by Theorem 4.3.2 and 4.3.3 to show that $\mathbb{V}$ satisfies the
hypotheses of Theorem 4.3.2. As $\mathbb{V}$ is of geometric origin, its local
monodromy matrices are quasi-unipotent, so the assumption that one of them has
infinite order implies it is not diagonalizable. Hence no Galois conjugate of
$\mathbb{V}$ can be unitary, as desired.

Now Theorem 4.3.2 and 4.3.3 tell us that applying Katz’s algorithm to
$\mathbb{V}$ yields a rank one finite order local system $\mathbb{L}$ on
$\mathbb{P}^{1}\setminus D$ and a non-trivial finite order rank one local
system $\chi^{\prime}$ on $\mathbb{G}_{m}$ such that
$\mathbb{U}:=MC_{\chi^{\prime}}(\mathbb{V}\otimes\mathbb{L})$ has monodromy a
finite complex reflection group. Hence we have
$\mathbb{V}=MC_{\chi}(\mathbb{U})\otimes\mathbb{M}$, where
$\mathbb{M}=\mathbb{L}^{-1}$ and $\chi=(\chi^{\prime})^{-1}$. ∎

###### Proof of 1.1.7.

We recall the setup from the introduction. Let

$\rho:\pi_{1}(\Sigma_{0,n})\to\operatorname{SL}_{2}(\mathbb{C})$

be a Zariski-dense, MCG-finite representation, with at least one local
monodromy matrix of infinite order. Without loss of generality we may assume
$n\geq 3$, as if $n<3$, no Zariski-dense $\rho$ exist, since
$\pi_{1}(\Sigma_{0,n})$ is abelian.

By [LL22b, Corollary 2.3.5], there exists a versal family of $n$-punctured
projective lines

$\pi^{\circ}:\mathscr{C}^{\circ}\to\mathscr{M},$

and a local system $\mathbb{V}$ on $\mathscr{C}^{\circ}$, such that the
restriction of $\mathbb{V}$ to a fiber of $\pi^{\circ}$ has monodromy given by
$\rho$. Suppose $\mathbb{V}$ does not have quasi-unipotent local monodromy at
infinity. Then by [LPT16, Theorem A], $\mathbb{V}$ is pulled back through some
map $\mathscr{C}^{\circ}\to Y$, with $Y$ a Deligne-Mumford curve, that is, we
are in case (1) of the corollary, classified by Diarra [Dia13, §3-§5].

Thus we may suppose $\mathbb{V}$ has quasi-unipotent monodromy at infinity. In
this case, if $\mathbb{V}$ is not pulled-back through a map to a Deligne-
Mumford curve, it is, by [CS08, Theorem 2], of geometric origin. Hence its
restriction to a general fiber of $\pi^{\circ}$ satisfies the hypotheses of
Theorem 1.1.2, and we may conclude. ∎

##  5\. Finite complex reflection groups

In this section we analyze the finite complex reflection groups that may arise
from Katz’s algorithm § 4.1, using the Shephard-Todd classification.

###  5.1. Shephard-Todd classification

When we refer to a complex reflection group, we will always view it as
realized on a complex vector space $V$; we generally denote this
representation by $\rho$, and by the trace field of $G$ we mean the field
generated by the character values of $\rho$. We recall the following examples:

######  Definition 5.1.1.

Let $m,N$ be positive integers. An $N\times N$ matrix is _monomial_ if it has
precisely one non-zero entry in each row and column. The group
$G(m,1,N)\subset\operatorname{GL}_{N}(\mathbb{C})$ is the group of monomial
matrices whose non-zero entries are all $m$-th roots of unity.

For a positive integer $p$ dividing $m$, $G(m,p,N)\subset G(m,1,N)$ is the
subgroup consisting of the matrices such that the product of all non-zero
entries is an $m/p$-th root of unity. The representation
$G(m,p,N)\subset\operatorname{GL}_{N}(\mathbb{C})$ is irreducible except for
$G(1,1,N)$ and $G(2,2,2)$ (see [LT09, Proposition 2.10]).

######  Remark 5.1.2.

Note that $G(1,1,n)=W(A_{n}),G(2,1,n)=W(B_{n})=W(C_{n}),G(2,2,n)=W(D_{n})$,
the Weyl groups of the infinite series of Dynkin diagrams, and $G(m,m,2)$ is
the dihedral group of order $2m$.

We recall the Shephard-Todd classification of complex reflection groups.

######  Theorem 5.1.3 ([ST54], [LT09, Theorem 8.29]).

Suppose $G\subset\operatorname{GL}_{N}(\mathbb{C})$ is an irreducible complex
reflection group. Then it belongs to one of the following classes (some of
which have overlaps):

  1. (1)

$G=G(m,p,N)$ for some $m,p$;

  2. (2)

$N=2$ and $G$ is one of 19 examples;

  3. (3)

$G$ is the Weyl group of a semisimple Lie algebra (which we refer to as
_classical Weyl groups_), and $G\subset\operatorname{GL}_{N}(\mathbb{C})$ is
the standard representation, i.e. the representation on the corresponding
complexified root lattice;

  4. (4)

$G$ is one of the following:

$W(H_{3}),W(J_{3}^{(4)}),W(J_{3}^{(5)}),W(L_{3}),W(M_{3}),W(H_{4}),W(L_{4}),W(N_{4}),W(O_{4}),W(K_{5}),W(K_{6}).$

For the last class, $W(X)$ denotes the Weyl group of the complex root system
$X$, and the subscript refers to the rank of the complex reflection group,
i.e. the dimension of the vector space on which it is realized.

Our main goal in § 5 is to prove:

######  Corollary 5.1.4.

For $N\geq 5$, the complex reflection group
$G(m,p,N)\subset\operatorname{GL}_{N}(\mathbb{C})$ does not arise as the
output of the Katz algorithm of Theorem 4.3.2; the same is true for
$W(J_{3}^{(4)}),$ $W(H_{4}),$ $W(K_{5}),$ $W(K_{6}),$ $W(E_{6}),$ $W(E_{7}),$
$W(E_{8})$. In particular, by the Shephard-Todd classification, there are no
irreducible rank two motivic local systems $\mathbb{V}$ on
$\mathbb{P}^{1}\setminus D$ with

  1. (1)

no unitary Galois conjugates, and

  2. (2)

with $D$ generic, $|D^{\prime}|\geq 7$, where $D^{\prime}\subset D$ denotes
the subset with non-scalar local monodromies.

We give the proof in § 5.7, along with the proof of 1.1.8.

###  5.2. Weyl groups

######  Corollary 5.2.1.

With notation as in Theorem 4.3.2, suppose $\mathbb{U}$, with monodromy a
classical Weyl group $G\subset\operatorname{GL}_{N}(\mathbb{C})$ for $N\geq
2$, is the output of Katz’s algorithm applied to the rank two local system
$\mathbb{V}$ satisfying the hypotheses of Theorem 4.3.2. Then, after possibly
twisting by a rank one local system, $\mathbb{V}$ has coefficients in
$\mathbb{Q}$, and is pulled back from the modular curve $X(1)$.

###### Proof.

By 4.3.3 and 3.2.2, the local monodromies of $\mathbb{U}$ have eigenvalues

  1. (1)

$\\{\lambda\beta_{i},1,\cdots,1\\}$ at a finite point $x_{i}\in D^{\prime}$,
where $1,\beta_{i}$ are the generalized eigenvalues of
$MC_{\chi^{-1}}(\mathbb{U})$ at $x_{i}$,

  2. (2)

$\\{\lambda^{-1}\gamma,\lambda^{-1},\cdots,\lambda^{-1}\\}$ at $\infty$, where
$\lambda,\gamma$ are the generalized eigenvalues of
$MC_{\chi^{-1}}(\mathbb{U})$ at $\infty$.

We first show that $\lambda=-1$. Since we are assuming
$G\subset\operatorname{GL}_{N}(\mathbb{C})$ is the representation of a Weyl
group on the corresponding root lattice, each local monodromy matrix of
$\mathbb{U}$ is defined over $\mathbb{Z}$; by considering the trace of
monodromy at $\infty$, we obtain

$\lambda^{-1}(\gamma+N-1)\in\mathbb{Z}.$

Since $|\lambda|=1$ and $|\gamma|=1$, by taking the norm, we have

(5.1) $|\gamma+N-1|^{2}=1+(N-1)(\gamma+\bar{\gamma})+(N-1)^{2}=m^{2}$

for some $m\in\mathbb{Z}$. This implies that $\gamma$ is quadratic over
$\mathbb{Q}$, i.e. it is a $2,3,4,$ or $6$-th root of unity. Hence
$\gamma+\bar{\gamma}=0,\pm 1$ or $\pm 2$, and if moreover $N\geq 3$ we must
have $\gamma+\bar{\gamma}=\pm 2$, i.e. $\gamma=\pm 1$; the latter implies,
when $N\geq 3$, that $\lambda\in\mathbb{Q}$, and since it is a non-trivial
root of unity, we have $\lambda=-1$.

It remains to treat the case $N=2$, in which case $G$ is one of
$W(A_{3}),W(B_{2}),W(G_{2})$, and it is a straightforward check that
$\lambda=-1$ in these cases also.

Therefore we have shown that $\chi$ corresponds to the local system on
$\mathbb{G}_{m}$ with monodromy $-1$ around zero. Since $\mathbb{U}$ is in
fact a $\mathbb{Z}$-VHS, the same is true of $MC_{\chi^{-1}}(\mathbb{U})$; in
other words, the latter is a rank two polarizable $\mathbb{Z}$-VHS of type
$(1,1)$, and so comes from a family of elliptic curves, as required. ∎

###  5.3. $G(m,p,N)$ for $N\geq 5$

Let $S_{N}$ denote the $N$-th symmetric group. We first recall that there is a
natural projection map $\pi:G(m,p,N)\rightarrow S_{N}$: we view
$S_{N}\subset\operatorname{GL}_{N}(\mathbb{C})$ as the monomial matrices all
of whose non-zero entries are $1$, and $\pi$ sends a matrix in $G(m,p,N)$ to
$S_{N}$ by replacing each non-zero entry by $1$.

######  Proposition 5.3.1 ([LT09, Lemma 2.8]).

Suppose $\gamma\in G(m,p,N)$ is a pseudoreflection. Then $\pi(\gamma)$ is
either trivial or a transposition.

######  Lemma 5.3.2.

For $N\geq 5$, there are no collections of $N+2$ elements
$s_{1},\cdots,s_{N+2}\in G(m,p,N)$ such that

  * •

$s_{1}\cdots s_{N+2}=e$,

  * •

$s_{i}$ is a pseudoreflection for $i=1,\cdots,N+1$,

  * •

$s_{N+2}$ is a scalar multiple of a pseudoreflection (the pseudoreflection is
not assumed to be in $G(m,p,N)$), and

  * •

$s_{1},\cdots,s_{N+2}$ generate $G(m,p,N)$.

###### Proof.

We proceed by contradiction, so let us assume such a collection $s_{i}\in
G(m,p,N)$ exists. For $i=1,\cdots,N+2$, denote by $t_{i}$ be the elements
$\pi(s_{i})\in S_{N}$. Note that each $t_{i}$ is either the trivial element or
a transposition. This is clear for $i=1,\cdots,N+1$ so it remains to check it
for $t_{N+2}$. By our assumption $s_{N+2}=\mu R$ for $\mu$ a root of unity and
$R\in\operatorname{GL}_{N}(\mathbb{C})$ a pseudoreflection; the group
generated by $s_{1},\cdots,s_{N+1},R$ is certainly contained in
$G(m^{\prime},1,N)$ for some $m^{\prime}$, and we may apply 5.3.1 to conclude.

The product of the $t_{i}$’s is then trivial. Note that these transpositions
must act transitively on $\\{1,2,...,N\\}$, since otherwise the group
generated by the $s_{i}$’s is reducible.

We discard the $t_{i}$ which are trivial; let $k\leq N+2$ be the number of
transpositions remaining, and relabel the $t_{i}$ so that $t_{1},\ldots,t_{k}$
are transpositions with $t_{1}\cdots t_{k}=e$, and generating a transitive
subgroup of $S_{N}$. We refer to the sequence
$(j,t_{k}(j),t_{k-1}t_{k}(j),\ldots,t_{1}\cdots t_{k}(j)=j)$ with consecutive
repetitions removed as the cycle of $j$. For each $j\in\\{1,\ldots,N\\}$ let
$c_{j}$ denote one fewer than the number of elements in the cycle of $j$.
Informally, $c_{j}$ is the “cycle length” of $j$. For $i\in\\{1,\ldots,N\\}$
let $a_{i}$ denote the number of $t_{1},\ldots,t_{k}$ among which $i$ appears.
Note that every $a_{i}\geq 2$, as otherwise $i$ would not be not be fixed by
$t_{1}\cdots t_{k}$, and every $c_{j}\geq 2$, since the action is irreducible.

We first consider the case $N$ is odd. Since a product of an odd number of
transpositions is never the identity, we must have $k\leq N+1$. Since there
are at most $N+1$ transpositions,

$\sum_{j=1}^{N}c_{j}=\sum_{i=1}^{N}a_{i}\leq 2(N+1),$

as each transposition contributes $2$ to both $\sum_{j}c_{j}$ and
$\sum_{i}a_{i}$.

Since $N\geq 5$, the pigeonhole principle implies there are at least $2$
values of $j$ for which $c_{j}=2$. Without loss of generality, we may assume
$c_{1}=2$ and $c_{2}=2$. The cycle of $1$ cannot be $1,2,1$, as then the cycle
of $2$ would be $2,1,2$, and the action would be reducible. We can therefore
assume the cycle of $1$ is $1,3,1$ and the cycle of $2$ is $2,4,2$, meaning
that the transpositions $(13),(31),(24),(42)$ appear among the $t_{k}$. In
order for $c_{1}=2$, we must either have (considering the cycle of $3$)
$a_{3}\geq 4$ or both $a_{1}\geq 3$ and $a_{3}\geq 3$. Similarly, we must
either have $a_{4}\geq 4$ or both $a_{2}\geq 3$ and $a_{4}\geq 3$. This
implies $a_{1}+a_{2}+a_{3}+a_{4}\geq 12$. Since each $a_{i}\geq 2$ we find
$12+2(N-4)\leq(a_{1}+a_{2}+a_{3}+a_{4})+\sum_{i=5}^{N}a_{i}\leq 2N+2$, a
contradiction.

The case $N$ is even and $N\geq 8$ is similar to the case $N$ is odd, where
one argues by showing there are at least $3$ values of $j$ with $c_{j}=2$, and
one obtains that $18\leq\sum_{i=1}^{6}a_{i}$, leading to a similar
contradiction. The case $N=6$ follows from a brute-force verification. ∎

######  Lemma 5.3.3.

Suppose $\mathbb{V}$ satisfies the hypotheses of Theorem 4.3.2. Then
$G(m,p,N)$ does not arise as the output of applying Katz’s algorithm to
$\mathbb{V}$ if $N\geq 5$.

###### Proof.

Applying the Katz algorithm to $\mathbb{V}$ as in Theorem 4.3.2, the result is
irreducible, and the local monodromies satisfy the conditions of 5.3.2 by
4.3.3, which in turn implies that the resulting complex reflection group
$G\subset\operatorname{GL}_{N}(\mathbb{C})$ cannot be $G(m,p,N)$ for $N\geq
5$. ∎

###  5.4. The groups $W(J_{3}^{(4)})$, $W(H_{4})$, $W(K_{5})$

######  Lemma 5.4.1.

Suppose $\mathbb{V}$ satisfies the hypotheses of Theorem 4.3.2. The groups
$W(J_{3}^{(4)})$, $W(H_{4})$, $W(K_{5})$ do not arise as the output of
applying Katz’s algorithm to $\mathbb{V}$.

###### Proof.

We follow the notation of Theorem 4.3.2 and 4.3.3. Let $G$ denote any of
$W(J_{3}^{(4)}),W(H_{4}),W(K_{5})$, and $\rho$ its natural representation
coming from the description as a complex reflection group. We use the
following properties of $G$:

  1. (1)

The trace field of $\rho$ is not totally real: it is $\mathbb{Q}(\sqrt{-7})$
for $W(J_{3}^{(4)})$, $\mathbb{Q}(\omega,\sqrt{5})$ for $W(H_{4})$, and
$\mathbb{Q}(\omega)$ for $W(K_{5})$, where $\omega$ is a primitive third root
of unity (see e.g. [Fei03, Table 1]),

  2. (2)

all pseudoreflections are of order 2, and

  3. (3)

its center is $\mathbb{Z}/2$.

Suppose $(G,\rho)$ arises from the Katz’ algorithm, i.e. there is a local
system $\mathbb{U}$ on a punctured $\mathbb{P}^{1}$ with monodromy $G$ and
satisfying the properties listed in 4.3.3.

The monodromy $s_{\infty}$ at $\infty$ is given by $\lambda^{-1}R$ for some
root of unity $\lambda\neq 1$ and pseudoreflection $R$. Note that $R$ is not a
priori in $G$, but this is true in our case: adding in the scalar element
$\lambda$, we obtain a possibly larger irreducible finite complex reflection
group $G^{\prime}$ of the same rank, but the Shephard-Todd classification
shows that this must be $G$ itself. Indeed, we may rule out the case
$G^{\prime}=G(m,p,n)$ since this would imply $G$ is _imprimitive_ , and the
only imprimitive finite complex reflection groups are the $G(m,p,n)$’s
themselves—see [LT09, Chapter 2]. For the exceptional ones, we simply rule
them out by inspection—for example, their orders are not divisible by that of
$G$. Therefore $R\in G$, and so $\lambda$ is in the center of $G$, which in
turn implies $\lambda=-1$.

Hence $\chi$ (as defined in § 4.1) is the rank one local system on
$\mathbb{G}_{m}$ with monodromy $-1$ around $0$. Applying
$MC_{\chi^{-1}}=MC_{\chi}$ to $\rho$, the result is a rank two local system
$\mathbb{V}$ with the same trace field as $\rho$. On the other hand, by 3.2.2
and point (2) above, the monodromy at each finite point of $D$ is conjugate to
$J(1,2)$, and therefore $\mathbb{V}$ has trivial determinant; $\mathbb{V}$ and
all of its Galois conjugates underly polarizable variations of Hodge
structure, and the unipotent local monodromies force all of these to have type
$(1,1)$. In other words, the monodromy of $\mathbb{V}$ and each of its Galois
conjugates takes values in $\operatorname{SL}_{2}(\mathbb{R})$, and therefore
$\mathbb{U}$ has totally real trace field, contradicting point (1) above. ∎

###  5.5. The groups $W(J_{3}^{(5)}),W(N_{4}),W(O_{4}),W(K_{6})$

######  Lemma 5.5.1.

Suppose $\mathbb{V}$ satisfies the hypotheses of Theorem 4.3.2, and in
addition $\mathbb{V}$ has local monodromy of infinite order at some point.
Then $W(J_{3}^{(5)}),W(N_{4}),W(O_{4}),W(K_{6})$ do not arise as the output of
applying Katz’s algorithm to $\mathbb{V}$.

###### Proof.

Set $G$ to be one of $W(J_{3}^{(5)}),W(N_{4}),W(O_{4}),W(K_{6})$. We recall
the following facts about $G$:

  1. (1)

The trace field of the tautological representation of $G$ is not totally real.
For $W(J_{3}^{(5)})$ it is $\mathbb{Q}(\omega,\sqrt{5})$, where $\omega$ is a
primitive cube root of unity; for $W(N_{4}),W(O_{4})$ it is $\mathbb{Q}(i)$;
and for $W(K_{6})$, it is $\mathbb{Q}(\omega)$ [Fei03, Table 1].

  2. (2)

All pseudoreflections are of order $2$.

The hypotheses of Theorem 4.3.2 imply that $\mathbb{V}$ has quasi-unipotent
local monodromy; hence the hypothesis that $\mathbb{V}$ has local monodromy of
infinite order at some point implies there exists a point where the local
monodromy of $\mathbb{V}$ is not semisimple. But as all pseudoreflections in
$G$ have order $2$, this implies by 3.2.2 that $\chi$ (as defined in § 4.1) is
the rank one local system on $\mathbb{G}_{m}$ with monodromy $-1$ around $0$.
Now we may proceed as in the proof of 5.4.1—$\mathbb{V}$, and hence
$\mathbb{U}$, has totally real trace field, contradicting (1) above. ∎

###  5.6. Further analysis of Mitchell’s group

We recall the following facts about Mitchell’s group $W(K_{6})$:

  1. (1)

the trace field of $W(K_{6})$ is $\mathbb{Q}(\omega)$, where $\omega$ is a
primitive third root of unity [Fei03, Table 1], and

  2. (2)

all of its 126 pseudoreflections are of order 2, and its center is cyclic of
order 6.

Picking a basepoint $x\in\mathbb{P}^{1}\setminus\\{x_{1},\cdots,x_{n}\\}$,
suppose that $(A_{1},\cdots,A_{n})$ represents a point of $X_{n}$. Then for
any $i=1,\cdots,n-1$, the tuple $(A_{1},\cdots,A_{i}A_{i+1},\cdots,A_{n})$
also has product equal to the identity, and we call it a _coalescence_ of
$(A_{1},\cdots,A_{n})$. We also call $(A_{n}A_{1},\cdots,A_{n-1})$ a
coalescence of $(A_{1},\cdots,A_{n})$; it is straightforward to see that any
coalescence of a MCG-finite tuple is again MCG-finite.

######  Lemma 5.6.1.

Suppose $\mathbb{V}$ is an irreducible MCG-finite
$\operatorname{SL}_{2}$-local system on an $n$-punctured $\mathbb{P}^{1}$ with
$n\geq 8$, whose local monodromies around the punctures are non-scalar, and
which moreover underlies a polarizable $\mathbb{C}$-VHS of type $(1,1)$.
Suppose $\mathbb{V}^{\prime}$ is a coalescence of $\mathbb{V}$ which stays
irreducible. Then either

  * •

$\mathbb{V}^{\prime}$, viewed as a local system on
$\mathbb{P}^{1}\setminus\\{y_{1},\cdots,y_{n-1}\\}$, also underlies a
polarizable $\mathbb{C}$-VHS of type $(1,1)$, or

  * •

the Zariski closure of the monodromy of $\mathbb{V}^{\prime}$ is contained in,
up to conjugation, the infinite dihedral group

$D_{\infty}=\bigg{\\{}\begin{pmatrix}c&0\\\
0&c^{-1}\end{pmatrix}\bigg{|}c\in\mathbb{C}^{\times}\bigg{\\}}\cup\bigg{\\{}\begin{pmatrix}0&c\\\
-c^{-1}&0\end{pmatrix}\bigg{|}c\in\mathbb{C}^{\times}\bigg{\\}}.$

###### Proof.

Note that $\mathbb{V}^{\prime}$ cannot come from the pullback construction by
[Dia13, Théorème 5.1]. If it has Zariski-dense monodromy, Corlette-Simpson
implies that $\mathbb{V}^{\prime}$ underlies a polarizable $\mathbb{C}$-VHS.
On the other hand, the fiber $\mathbb{V}_{x}$ carries a Hermitian form of
signature $(1,1)$ invariant under the group generated by
$A_{1},\cdots,A_{i}A_{i+1},\cdots,A_{n}$, inherited from the Hermitian form on
$\mathbb{V}$, and since $\mathbb{V}^{\prime}$ is assumed irreducible, there is
a unique invariant form up to scaling. Therefore $\mathbb{V}^{\prime}$ must
also underly a polarizable $\mathbb{C}$-VHS of type $(1,1)$, as claimed.

On the other hand if $\mathbb{V}^{\prime}$ does not have Zariski-dense
monodromy, by the classification of algebraic subgroups (up to conjugation) of
$\operatorname{SL}_{2}$ (see e.g. [VdPS12, Theorem 4.29]), since we are
assuming $\mathbb{V}^{\prime}$ is irreducible, the image of monodromy is
either contained in $D_{\infty}$, or a finite irreducible subgroup
$G\subset\operatorname{SL}_{2}(\mathbb{C})$. The latter cannot happen since it
would imply that $G$ has an invariant hermitian form of signature $(1,1)$,
contradiction. Therefore the monodromy of $\mathbb{V}^{\prime}$ is contained
in $D_{\infty}$, as required. ∎

######  Lemma 5.6.2.

The group $G=W(K_{6})$ does not arise as the output of Katz’s algorithm,
applied to $\mathbb{V}$ satisfying the hypotheses of Theorem 4.3.2.

###### Proof.

Suppose the local system $\mathbb{U}$, with monodromy $G$, arises from the
Katz algorithm from a MCG-finite, irreducible, $\operatorname{SL}_{2}$-local
system $\mathbb{V}$, which we may take to be on
$\mathbb{P}^{1}\setminus\\{x_{1},\cdots,x_{7},\infty\\}$, whose local
monodromies around the eight punctures are non-scalar; recall that
$\mathbb{V}$ and each of its Galois conjugates are assumed to underly a
polarizable $\mathbb{C}$-VHS of type $(1,1)$, so that Theorem 4.3.2 applies.
Note that all local monodromies of $\mathbb{V}$ must be semisimple, as
otherwise the same argument as in 5.4.1 applies: indeed, since the
pseudoreflections in $G$ are all of order 2, the only way to get non-
semisimple monodromies is if $\chi$ has monodromy $-1$, where $\chi$ is the
character for the middle convolution in the Katz algorithm.

We will say that a $2\times 2$ matrix is _neat_ if it is regular semisimple,
i.e. semisimple with distinct eigenvalues. Picking a basepoint and loops
around $x_{1},\cdots,x_{7},\infty$, we write $\mathbb{V}$ equivalently as a
tuple of matrices $(A_{1},\cdots,A_{7},A_{8})$, each of which is neat.

######  Claim.

Coalescing the punctures $x_{j},x_{j+1}$, the obtained local system
$\mathbb{V}^{\prime}$ cannot stay irreducible.

###### Proof of claim.

Suppose $\mathbb{V}^{\prime}$ is irreducible. By 5.6.1, either
$\mathbb{V}^{\prime}$ has image in $D_{\infty}$, or it underlies a VHS of type
$(1,1)$. We now rule out the $D_{\infty}$ case. Note that there is the sign
homomorphism $\operatorname{sgn}:D_{\infty}\rightarrow\\{\pm 1\\}$, given by
quotienting out the diagonal matrices. We must have
$\operatorname{sgn}(A_{i})=1$ for $i\neq j,j+1$: indeed, if
$\operatorname{sgn}(A_{i})=-1$, then $A_{i}$ has trace zero, and hence has
eigenvalues $\pm\sqrt{-1}$. This may be ruled out by analyzing the local
monodromies of $MC_{\chi^{-1}}(\mathbb{U})$, combined with the fact that
$\chi$ must have monodromy a sixth root of unity. Since $A_{1}A_{2}\cdots
A_{8}=\operatorname{id}$, the above implies
$\operatorname{sgn}(A_{j}A_{j+1})=1$, which in turn implies that
$\mathbb{V}^{\prime}$ is reducible, a contradiction.

Therefore $\mathbb{V}^{\prime}$ and all of its Galois conjugates underlie VHS
of type $(1,1)$. We may then apply the Katz algorithm to get as output an
irreducible rank 5 complex reflection group; it must then be $G(m,p,5)$ for
some $m,p$, or $W(K_{5})$. The former is ruled out by 5.3.3, and the latter by
5.4.1. ∎

Hence we may assume that any coalescence of two points gives a reducible local
system.

Coalescing the points $x_{i},x_{i+1}$, where $i=1,\cdots,7,$ we obtain the
tuple $(A_{1},\cdots,A_{i}A_{i+1},\cdots,A_{8})$. By the above, this tuple
corresponds to a reducible representation, i.e. a representation into the
group of upper triangular matrices
$B\subset\operatorname{SL}_{2}(\mathbb{C})$. Composing with the natural map
$B\rightarrow\mathrm{Aff}(\mathbb{C})$, the latter being the group of affine
transformations of $\mathbb{C}$, we obtain a representation into
$\mathrm{Aff}(\mathbb{C})$. We now apply the results of [CM18b] which classify
MCG-finite representations into $\rm{Aff}(\mathbb{C})$.

Case 1: Suppose for some $j=1,\cdots,7$, $A_{j}A_{j+1}$ is neat. Now we apply
[CM18b, Theorem 2.3.4]. As all the matrices in the tuple $(A_{1},\cdots
A_{j}A_{j+1},\cdots,A_{\infty})$ are neat, loc.cit. implies that the only such
local systems are on an $n$-punctured $\mathbb{P}^{1}$ for $n\leq 6$, or a sum
of two characters; since we now have seven punctures, we must be in the latter
case. Therefore, in this case, we may assume that $A_{i}$ is diagonal for any
$i=1,\cdots,j-1,j+2,\cdots,\infty$, along with $A_{j}A_{j+1}$; in particular
these matrices commute with one another.

Now we instead coalesce $x_{j+1},x_{j+2}$ (where $A_{9}=A_{1}$) for the
original tuple $(A_{1},\cdots,A_{8})$. If $A_{j+1}A_{j+2}$ is also neat, then
the same argument as above shows that $A_{j}$ commutes with $A_{j-1}$ (with
$A_{-1}=A_{8}$), and hence $A_{j}$ is also diagonal, and hence so is
$A_{j+1}$, implying $\mathbb{V}$ is reducible, contradiction. If
$A_{j+1}A_{j+2}$ is not neat, then [CM18b, Theorem 2.3.3] implies it is
scalar, and hence $A_{j+1}$ is diagonal, again implying $\mathbb{V}$
reducible, which is a contradiction.

Case 2: therefore we may assume that $A_{i}A_{i+1}$ is not neat, for any
choice of $i=1,\cdots,7$. On the other hand, [CM18b, Theorem 2.3.3] then
implies that $A_{i}A_{i+1}=\pm 1$ for each such $i$, which in turn implies
that $\mathbb{V}$ is reducible, which is again a contradiction. ∎

###  5.7. Ruling out local systems on $\mathbb{P}^{1}\setminus D$, with $|D|$
large

We may now prove 5.1.4 and 1.1.8.

###### Proof of 5.1.4.

Recall from the statement of 5.1.4 that $D^{\prime}$ is the number of points
at which $\mathbb{V}$ has non-scalar monodromy. By 4.3.3, the result of
applying our variant Katz’s algorithm of Theorem 4.3.2 to $\mathbb{V}$ is a
local system $\mathbb{U}$ of rank $N=|D^{\prime}|-2$, with monodromy group a
finite complex reflection group. We now study the possible monodromy groups of
$\mathbb{U}$.

5.3.3 rules out the group $G(m,p,N)$ for $N\geq 5$. All classical Weyl groups
of rank at least $5$ are handled by the previous sentence and 5.1.2, except
$W(E_{6}),W(E_{7}),W(E_{8})$. That $W(E_{6}),W(E_{7}),W(E_{8})$ do not appear
follows from 5.2.1, as the corollary shows that classical Weyl groups give
rise to “pullback representations” with non-scalar monodromy at
$|D^{\prime}|=N+2$ points by 5.2.1. But these do not exist for
$|D^{\prime}|\geq 7$ by [Dia13, Théorème 5.1]. Similarly, we have ruled out
$W(J_{3}^{(4)}),W(H_{4}),W(K_{5})$ in 5.4.1 and $W(K_{6})$ in 5.6.2.

The second claim now follows from the Shephard-Todd classification; we have
ruled out all finite complex reflection groups of rank $N\geq 5$, and hence
all motivic local systems as in the statement with non-scalar local monodromy
at $|D^{\prime}|=N+2\geq 7$ points. ∎

###### Proof of 1.1.8.

This is immediate by combining 5.1.4 with [Dia13, Théorème 5.1]. Diarra’s
result rules out finite orbits of “pullback type,” and ours rules out local
systems of geometric origin. As discussed in the introduction and in the proof
of 1.1.7, the results of [CS08, Theorem 2] and [LPT16, Theorem A] show that
these are the only two possibilities, completing the proof. ∎

###  5.8. Examples

####  5.8.1. Dihedral groups

As explained in [LL23], there is an infinite collection of finite orbits of
the $\operatorname{Mod}_{0,4}$-action on $X_{4}$ arising via middle
convolution from rank $2$ local systems with dihedral monodromy. We now give a
brief explanation of this.

For any $\lambda\in\mathbb{C}$ different from $0,1$, let $X$ denote
$\mathbb{P}^{1}\setminus\\{0,1,\lambda,\infty\\}$, and let
$\pi:E\rightarrow\mathbb{P}^{1}$ denote the double cover branched at
$0,1,\lambda,\infty$. Denote by $E^{\circ}$ the curve $\pi^{-1}(X)$, so that
we have the étale map $\pi^{\circ}:E^{\circ}\rightarrow X$.

In loc.cit it was shown (see [LL23, Proposition 4.2.2]) that each irreducible
rank two local system $\mathbb{V}$ on $X$ of geometric origin and satisfying a
local monodromy condition $(\star)$ (see [LL23, p.2]) is of the form
$MC_{\chi}(\pi^{\circ}_{*}\mathbb{L}|_{E^{\circ}})$; here $\mathbb{L}$ is a
torsion rank one local system on $E$, and $\chi$ has monodromy $-1$. Note that
$\pi^{\circ}_{*}\mathbb{L}|_{E^{\circ}}$ has monodromy a dihedral group; in
other words $\mathbb{V}$ arises as the middle convolution of a local system
with monodromy a complex reflection group. An explicit description of the
monodromy representation on these $\mathbb{V}$ is given in [LL23, Example
1.1.7].

####  5.8.2. $W(H_{3})$

The group $W(H_{3})$ is the group of symmetries of the icosahedron, so that
$W(H_{3})\simeq\mathbb{Z}/2\times A_{5}$, with the representation realizing it
as a reflection group being either of the two (Galois-conjugate) rank 3
irreducible representations with non-trivial central character. The trace
field of $W(H_{3})$ is $\mathbb{Q}(\sqrt{5})$.

  1. (1)

The center of $W(H_{3})$ is given by the $\mathbb{Z}/2$ factor, which acts by
$-1$ on the corresponding three dimensional representation; we denote a
generator of the $\mathbb{Z}/2$ by $-1$.

  2. (2)

The pseudoreflections are precisely the products of $-1$ with the elements of
$A_{5}$ which are products of two disjoint transpositions (i.e. the conjugacy
class of $(12)(34)$), so that there are 15 of such.

Combining these, we see that each appearance of $W(H_{3})$ as an output of
Katz’s algorithm in Theorem 4.3.2 corresponds to five elements
$\sigma_{1},\cdots,\sigma_{5}\in A_{5}$ as in point (2) above such that
$\sigma_{1}\cdots\sigma_{5}=e$. For example, we can take the tuple
$(\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4},\sigma_{5})$ to be

$((12)(34),(34)(51),(51)(23),(23)(45),(45)(12)).$

This produces, via middle convolution, a rank two local system $\mathbb{V}$
with non-semisimple monodromy at all punctures. Since the trace field
$\mathbb{Q}(\sqrt{5})$ is quadratic over $\mathbb{Q}$, $\mathbb{V}$ arises in
the cohomology of a family of abelian surfaces with real multiplication by
$\mathbb{Q}(\sqrt{5})$ over a generic 5-punctured $\mathbb{P}^{1}$; moreover,
this family has (potentially) totally degenerate reduction at each puncture by
Grothendieck’s semistable reduction criterion.

####  5.8.3. $G(m,p,3)$

Suppose the elements

$s_{1}=\begin{pmatrix}&a&\\\ a^{-1}&&\\\
&&1\end{pmatrix},s_{2}=\begin{pmatrix}&b&\\\ b^{-1}&&\\\
&&1\end{pmatrix},s_{3}=\begin{pmatrix}1&&\\\ &&c\\\
&c^{-1}&\end{pmatrix},s_{4}=\begin{pmatrix}1&&\\\ &&d\\\
&d^{-1}&\end{pmatrix},$

of $G(m,p,3)$ generate the group and satisfy

$s_{1}s_{2}s_{3}s_{4}=\lambda^{-1}R$

with $R$ a pseudoreflection and $\lambda\neq 1$. Then the tuple
$(s_{1},s_{2},s_{3},s_{4},\lambda R)$ corresponds to a local system on a
5-punctured $\mathbb{P}^{1}$, with monodromy $G(m,p,3)$; applying $MC_{\chi}$,
with $\chi$ having monodromy $\lambda$, we obtain a rank two, irreducible,
MCG-finite local system. It is straightforward to find examples of such
elements $s_{1},\cdots,s_{4}$; for example, any choice of roots of unity
$a,b,c,d$ such that $a/b=d/c\neq bc/ad$ suffices.

####  5.8.4. $W(D_{4})$

The elements

$s_{1}=\begin{pmatrix}&-1&&\\\ -1&&&\\\ &&1&\\\
&&&1\end{pmatrix},s_{2}=\begin{pmatrix}&1&&\\\ 1&&&\\\ &&1&\\\
&&&1\end{pmatrix},s_{3}=\begin{pmatrix}1&&&\\\ &1&&\\\ &&&-1\\\
&&-1&\end{pmatrix},$ $s_{4}=\begin{pmatrix}1&&&\\\ &1&&\\\ &&&1\\\
&&1&\end{pmatrix},s_{5}=\begin{pmatrix}&&-1&\\\ &1&&\\\ -1&&&\\\
&&&1\end{pmatrix},s_{6}=\begin{pmatrix}&&1&\\\ &-1&&\\\ 1&&&\\\
&&&-1\end{pmatrix},$

of $W(D_{4})=G(2,2,4)$ satisfy $s_{1}\cdots s_{6}=1$, and $-s_{6}$ is a
pseudoreflection. The tuple $(s_{1},\cdots,s_{6})$ corresponds to a rank 4
local system on a 6-punctured $\mathbb{P}^{1}$ with monodromy $W(D_{4})$;
applying $MC_{\chi}$ where $\chi$ has monodromy $-1$, we obtain a rank two,
irreducible, MCG-finite local system on the projective line minus six points,
with non-semisimple monodromy at all $6$ points. This shows that 5.1.4 and
1.1.8 are sharp.

###  5.9. Table

We summarize some data regarding the finite complex reflection groups which
may arise from middle convolution as above:

Complex reflection group | Arising from Katz algorithm? | Geometric origin of $\operatorname{SL}_{2}$-local system
---|---|---
Dihedral groups | [LL23] | AVs with RM by $\mathbb{Q}(\zeta+\zeta^{-1})$
other rank 2 | see [LT14] | many cases
$G(m,p,3)$ | § 5.8.3 | many cases
$G(m,p,N)$, $N\geq 5$ | None exists: 5.1.4 | –
$W(H_{3})$ | § 5.8.2 | AVs with RM by $\mathbb{Q}(\sqrt{5})$
$W(J_{3}^{(4)}),W(H_{4}),W(K_{5})$ | None exists: 5.4.1 | –
$W(K_{6})$ | None exists: 5.6.2 | –
$W(D_{4})$ | § 5.8.4 | Elliptic curves
Classical Weyl groups | Pullback families | Elliptic curves (by 5.2.1)

######  Remark 5.9.1.

We leave open the analysis for the cases
$G(m,p,4),W(L_{3}),W(M_{3}),W(J_{3}^{(5)}),W(L_{4})$,
$W(N_{4}),W(O_{4}),W(F_{4})$, except for the special cases considered in
5.5.1, where $\mathbb{V}$ has a point with local monodromy of infinite order.

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