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\title{Two-step monodromy of special quasi-projective varieties}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
Every complex linear representation of the ordinary fundamental group of a
connected smooth special complex quasi-projective variety has virtually
nilpotent image of nilpotency class at most two. We give a complete
independent proof. The engine is a structural theorem: whenever the
quasi-Albanese map has exact fundamental-group sequence, the entire
rational unipotent completion of the fundamental group is nilpotent of
class at most two; a mixed Hodge weight argument controls the fibre image
at each finite nilpotent stage. We also construct a smooth special open
surface whose general quasi-Albanese fibre is not special, contradicting a
fibre-specialness assertion of Cao-Deng-Hacon-Paun.
\end{abstract}

\section{Introduction}\label{sec:intro}

A \emph{special} variety, in Campana's sense, is one that admits no
fibration of general type: no dominant map onto a positive-dimensional base
whose orbifold base is of general type. Specialness is a weak form of
non-hyperbolicity with sharp consequences for the topology of the total
space. In the compact K\"ahler category, Campana's abelianity conjecture
asks that the ordinary fundamental group of a special variety be virtually
abelian. In the open (quasi-projective) category the result cannot hold
verbatim, because nilpotent non-abelian groups do occur; the sharp
statement instead bounds the nilpotency class, and it bounds it at two.

For a group $G$, put $\gamma_1 G = G$ and $\gamma_{r+1} G = [G,\gamma_r G]$.
$G$ is in \emph{nilpotency class at most two} if $\gamma_3 G = 1$,
equivalently if every commutator $[a,b]=aba^{-1}b^{-1}$ is central. A
property holds \emph{virtually} for $G$ if it holds in a subgroup of finite
index.

\subsection{The linear monodromy theorem}

\begin{theorem}[Two-step linear monodromy]\label{thm:linear}
Let $V$ be a connected smooth special complex quasi-projective variety.
For every positive integer $d$ and every representation
$\rho:\pi_1(V)\to\GL_d(\C)$, the image $\rho(\pi_1(V))$ has a subgroup of
finite index and nilpotency class at most two.
\end{theorem}

Cadorel-Deng-Yamanoi \cite[Theorem A/4.1]{CDY2025} proved virtual nilpotence
of such linear images (the identity component of their Zariski closure is
a product of a unipotent group and a torus). The sharp class-two conclusion
of Theorem~\ref{thm:linear} was announced by Cao-Deng-Hacon-P\u{a}un
\cite[Theorem A]{CDHP2026}. Our proof is independent of theirs. It uses the
CDY inputs but assembles them differently: the mixed Hodge-theoretic weight
analysis is carried out not on an arbitrary representation quotient, but
directly on the canonical quotients of the rational unipotent completion of
$\pi_1(V)$.

The bound is optimal. Cadorel-Deng-Yamanoi
\cite[Example 4.25 and Remarks 4.28--4.29]{CDY2025} construct a smooth special
quasi-projective surface whose ordinary fundamental group is a Heisenberg
central extension, a two-step nilpotent non-abelian group. The surface is
the complement of the zero section in a degree-one line bundle over an
elliptic curve; its ruled compactification has log-canonical divisor zero.

\subsection{A structural theorem on unipotent completion}\label{sec:intro-structural}

A \emph{semiabelian variety} is a connected algebraic group $A$ fitting
into an exact sequence
\[
1\longrightarrow\Gm^r\longrightarrow A\longrightarrow B\longrightarrow 1
\]
with $B$ abelian. For a connected smooth quasi-projective $V$ and a base
point $v$, the algebraic \emph{quasi-Albanese map} $a_V:V\to\Alb(V)$ is the
universal based algebraic morphism from $V$ to a semiabelian variety. We use
the construction and the universal property recalled by Fujino
\cite[Definition 2.18 and Theorem 3.16]{Fujino2024}.

The \emph{rational unipotent completion} of a group $\Gamma$ is the universal
pro-unipotent algebraic group receiving $\Gamma$; every finite-dimensional
unipotent representation of $\Gamma$ over $\Q$ factors uniquely through it.
For a finitely generated $\Gamma$ and $c\geq 1$, the \emph{canonical
$c$-step quotient} is the universal unipotent quotient of nilpotency class
at most $c$; Section~\ref{sec:mhs} gives its Lie-algebraic construction and
the mixed Hodge structure that sits on it.

\begin{theorem}[Completion under quasi-Albanese exactness]\label{thm:completion}
Let $V$ be a connected smooth complex quasi-projective variety and let
$a:V\to A=\Alb(V)$ be its algebraic quasi-Albanese map. Suppose $a$ is
dominant, choose a smooth connected general fibre $F$ and a base point
$v\in F$, and assume the sequence of ordinary fundamental groups
\[
\pi_1(F,v)\longrightarrow\Gamma:=\pi_1(V,v)
\xrightarrow{\ a_* \ }\Lambda:=\pi_1(A,a(v))\longrightarrow 1
\tag{1.1}\label{eq:exact}
\]
is exact. Then every canonical rational unipotent quotient of $\Gamma$ has
nilpotency class at most two. The full rational unipotent completion is a
finite-dimensional unipotent algebraic group of class at most two. Every
finite-dimensional complex unipotent representation of $\Gamma$ has image of
class at most two.
\end{theorem}

Exactness of \eqref{eq:exact} identifies the image of the fibre group with
the whole kernel of $a_*$; the first arrow need not be injective.
Theorem~\ref{thm:completion} assumes neither specialness of $V$ nor
specialness of $F$ nor any nilpotence of the discrete group $\Gamma$. Its
two geometric hypotheses are exactly what the proof uses: the quasi-Albanese
map makes first homology an isomorphism modulo torsion, and the algebraic
inclusion of the smooth fibre $F\hookrightarrow V$ gives a morphism of
pointed mixed Hodge structures on the fibre group. Together these force
the commutator of each canonical unipotent quotient to be pure of weight
$-2$, which in turn forces the canonical quotient to have nilpotency class
at most two.

\subsection{A special surface whose quasi-Albanese fibre is nonspecial}

Campana \cite[Question 5.4, p.~577]{Campana2004} asked whether the general
Albanese fibre of a compact special variety is special. Campana-Claudon
\cite[Theorem 2.4]{CampanaClaudon2016} answered this positively for smooth
projective varieties. We show that the analogous assertion with reduced
open boundary is false.

\begin{theorem}\label{thm:example}
Let $E$ be a complex elliptic curve, let $q_1,\ldots,q_n$ be distinct points
of $\PP^1$ with $n\geq 3$, and choose arbitrary points $e_1,\ldots,e_n$ on $E$.
Let
\[
b:Y=\Bl_{\{(e_i,q_i)\}_{i=1}^{n}}(E\times\PP^1)\longrightarrow E\times\PP^1
\]
be the blowup at the $n$ distinct points. Write $D_i$ for the strict
transform of $E\times\{q_i\}$, put $D=\sum_{i=1}^n D_i$ and $X=Y\setminus D$.
Then:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2em]
\item $(Y,D)$ is a smooth projective special pair in the orbifold sense of
      Definition~\ref{def:pair-special}, including every elementary
      birational orbifold upper model.
\item After choosing origins, $a=(\operatorname{pr}_E\circ b)|_X:X\to E$ is
      the algebraic quasi-Albanese map $a_X$. Every fibre of $a$ is
      connected; the general fibre is $F\simeq\PP^1\setminus\{q_1,\ldots,q_n\}$.
\item The induced map $a_*:\pi_1(X)\to\pi_1(E)$ is an isomorphism. Hence
      $\pi_1(X)\simeq\Z^2$, and the map $\pi_1(F)\to\pi_1(X)$ is the zero map.
\end{enumerate}
\end{theorem}

The general fibre $F$ has logarithmic canonical degree $n-2>0$, so $F$ is
not special (Section~\ref{sec:definitions} records the curve criterion).
The total space $X$ is special by (i) and the comparison of conventions
after Definition~\ref{def:pair-special}. So a special smooth quasi-projective
surface has a general quasi-Albanese fibre that is not special, and the
fibre-specialness assertion of \cite[Theorem E, equivalently Theorem
7.15]{CDHP2026} fails on this example. The fibre's ordinary fundamental
group is free of rank $n-1$, but its image inside $\pi_1(X)=\Z^2$ is trivial
by (iii), so the total space still satisfies the monodromy bound.

Geometrically, at each $q_i$ the complete scheme fibre of
$j=\operatorname{pr}_{\PP^1}\circ b:Y\to\PP^1$ is $D_i+A_i$ with both
components of multiplicity one. The retained component $A_i$ (an
exceptional $\PP^1$) makes the direct orbifold multiplicity at $q_i$ equal
to one; contracting $A_i$ first discards its contribution, so the iterated
base computed by Cao-Deng-Hacon-P\u{a}un differs. Campana's composition rule
records exactly the exceptional-divisor hypothesis under which the two
bases agree. Section~\ref{sec:composition} identifies the failure in
\cite[Claim 7.17]{CDHP2026} on these models: the proof treats flatness over
the Iitaka base $J$ as flatness over a different, contracted intermediate
space, and it is the latter that fails.

\subsection{Structure of the paper}\label{sec:structure}

Sections~\ref{sec:mhs}--\ref{sec:linear} prove Theorems~\ref{thm:completion}
and~\ref{thm:linear}. Sections~\ref{sec:definitions}--\ref{sec:albanese} prove
Theorem~\ref{thm:example}. Section~\ref{sec:composition} identifies the
composition failure. Section~\ref{sec:ordinary} records the remaining open
question on the ordinary discrete fundamental group, which our method does
not decide.

\section{The canonical quotients of the rational unipotent completion}\label{sec:mhs}

This section prepares the proof of Theorem~\ref{thm:completion}. Fix a
connected smooth complex quasi-projective variety $V$ and a base point
$v\in V$. Put $\Gamma=\pi_1(V,v)$ and let $\m_V$ be the complete
pronilpotent Lie algebra of the rational unipotent completion of $\Gamma$.
The section recalls three things: the canonical finite-dimensional
quotients of $\m_V$ and their universal properties, the mixed Hodge
structures on them, and the first-homology computation from the
quasi-Albanese map. No specialness hypothesis enters.

\subsection{Canonical quotients and their weight filtration}

Let $\gamma_1\m_V=\m_V$ and $\gamma_{r+1}\m_V=[\m_V,\gamma_r\m_V]$, and
let $\cl{\gamma_{r}\m_V}$ denote the closure in the inverse-limit topology.
For $c\geq 1$, the \emph{canonical $c$-step quotient} is
\[
\g_c=\m_V/\cl{\gamma_{c+1}\m_V},\qquad U_c=\exp(\g_c).
\tag{2.1}\label{eq:canonical-quotient}
\]
The exponential and logarithm of the Baker-Campbell-Hausdorff multiplication
identify the nilpotent Lie algebra $\g_c$ with a unipotent algebraic group
$U_c$. Because $\Gamma$ is finitely generated, each $\g_c$ is
finite-dimensional. The canonical image of $\Gamma$ in $U_c$ is Zariski
dense. The universal property is: every homomorphism from $\Gamma$ to a
unipotent algebraic group of nilpotency class at most $c$ factors uniquely
through $U_c$. These facts follow from the completed group algebra; see
Hain \cite[\S2.5, Theorem 2.5.3 and Proposition 2.5.5]{Hain1987}.

For the abelianization, the Hurewicz map gives
\[
\g_c/[\g_c,\g_c]\simeq H_1(V,\Q).
\tag{2.2}\label{eq:hurewicz}
\]
Indeed, the abelianization of $\m_V$ is the commutator quotient of the
completed group algebra, which is exactly the rational completion of
$H_1(V,\Z)$; the identification \eqref{eq:hurewicz} is canonical.

\subsection{Mixed Hodge structures and weight bounds}\label{sec:mhs-recall}

A rational \emph{mixed Hodge structure} is a finite-dimensional $\Q$-vector
space $W$ with an increasing weight filtration $W_\bullet$ and a decreasing
Hodge filtration $F^\bullet$ on the complexification $W_\C$, such that each
graded piece $\mathrm{gr}^W_r$ with its induced Hodge filtration is a pure
Hodge structure of weight $r$. We say $W$ has \emph{weights at most $r$} if
$W_r=W$, and is \emph{pure of weight $r$} if also $W_{r-1}=0$.
Morphisms of mixed Hodge structures preserve both filtrations. The
category is abelian, and morphisms are strict: the image of a morphism in
the target inherits the same filtration it has as a quotient of the
source \cite[Theorem 2.3.5 and \S2.3.8]{Deligne1971}. Strictness is the
key property that will produce both weight bounds in
Section~\ref{sec:mhs-proof}: a surjection
$H_1(F,\Q)\to B$ forces $B$ to have no weights below the lowest weight of
$H_1(F,\Q)$, and an inclusion $\kk\hookrightarrow\g_c$ forces $\kk$ to
have no weights above the highest weight of $\g_c$. Tensor product adds
weights.

For a smooth quasi-projective $V$, the first cohomology $H^1(V,\Q)$ carries
a mixed Hodge structure of weights $1$ and $2$, hence the dual first
homology $H_1(V,\Q)$ has weights $-1$ and $-2$
\cite[Corollary 3.2.15(ii)]{Deligne1971}. This is the only place the
quasi-projective hypothesis is used below in the abelian part of the
argument.

\subsection{Mixed Hodge structures on the canonical quotients}

Hain \cite[Theorem 6.3.1, p.~321]{Hain1987} equips the completed group
algebra of $\Gamma$ and hence the complete Lie algebra $\m_V$ with a
pro-mixed Hodge structure compatible with the bracket. The closed
lower-central ideals $\cl{\gamma_r\m_V}$ are topologically generated by
iterated brackets of elements that lift from $H_1(V,\Q)$, so they are
pro-mixed-Hodge subobjects. The finite-dimensional quotients $\g_c$ then
carry ordinary mixed Hodge structures, and the transition maps between
different $c$'s, the Lie bracket $[\g_c,\g_c]\to\g_c$, and the Hurewicz map
\eqref{eq:hurewicz} are all morphisms of mixed Hodge structures.

The functoriality we need is pointed: a based algebraic morphism
$(V,v)\to(V',v')$ of smooth complex quasi-projective varieties induces a
homomorphism of completed group algebras and a morphism
$\g_c(V)\to\g_c(V')$ compatible with mixed Hodge structures. In
Section~\ref{sec:mhs-proof} we apply this to the inclusion of the smooth
general fibre $F\hookrightarrow V$ at a base point $v\in F$. This is a
pointed algebraic map, and its induced map $\g_c(F)\to\g_c(V)$ is a
morphism of mixed Hodge structures.

A historical caveat: Morgan's foundational construction \cite{Morgan1978}
of nilpotent homotopy data has a functoriality assertion that was corrected
in \cite{Morgan1986}. The stronger pointed statement we use is due to
Hain \cite{Hain1987}. We do not attempt to prove the mixed Hodge statement
here; it is a foundational input, cited by number.

For the remainder of the proof, all images, kernels and abelianizations of
mixed Hodge subquotients are taken in the abelian category of mixed Hodge
structures. Arbitrary unipotent representation quotients of $\Gamma$ need
not themselves carry the mixed Hodge structure; the strategy is to prove
the canonical quotients first and then pass to arbitrary representations
by the universal property of the completion in Section~\ref{sec:mhs-complex}.

\section{Proof of Theorem~\ref{thm:completion}}\label{sec:mhs-proof}

We now prove Theorem~\ref{thm:completion}. Fix a connected smooth complex
quasi-projective $V$, its algebraic quasi-Albanese map $a:V\to A=\Alb(V)$,
a smooth connected general fibre $F$ and a base point $v\in F$. Assume $a$
is dominant and that the ordinary sequence
\[
\pi_1(F,v)\longrightarrow\Gamma:=\pi_1(V,v)\xrightarrow{\ a_*\ }\Lambda
:=\pi_1(A,a(v))\longrightarrow 1
\tag{3.0}\label{eq:exact-ref}
\]
is exact. Write $N=\im(\pi_1(F,v)\to\Gamma)$. By exactness $N=\ker(a_*)$.

The proof uses only five inputs: (I) the quasi-Albanese first homology
computation \eqref{eq:alb-h1} below, (II) the finite abelian quotient
\eqref{eq:almost-derived}, (III) strictness of morphisms of mixed Hodge
structures \eqref{eq:mhs-strict}, (IV) the weight decomposition of
$H_1(V,\Q)$ and of $H_1(F,\Q)$ (both weights $-1$ and $-2$), and (\V) the
algebraicity of the commutator morphism of a unipotent algebraic group.
Everything else is elementary Lie algebra.

\subsection{The quasi-Albanese first-homology computation}

Fujino \cite[Lemmas 3.11--3.12, arXiv v2 PDF, pp.~20--21]{Fujino2024}
shows that the quasi-Albanese map $a:V\to A$ induces an isomorphism
\[
H_1(V,\Z)\big/\Tor H_1(V,\Z)\xrightarrow{\ \sim\ }H_1(A,\Z).
\tag{3.1}\label{eq:alb-h1}
\]
The first cited lemma states integral surjectivity with torsion kernel;
the proof of the second gives the rational mixed-Hodge identification.
The group $A$ is a semiabelian variety, so $\Lambda=\pi_1(A,a(v))$ is free
abelian, and $\Lambda=H_1(A,\Z)$.

Two consequences. First, since $\Lambda$ is abelian, every commutator of
$\Gamma$ dies in $\Lambda$, so
\[
[\Gamma,\Gamma]\subseteq N.
\tag{3.2}\label{eq:derived-in-N}
\]
Second, by \eqref{eq:alb-h1} the kernel of $H_1(V,\Z)\to H_1(A,\Z)$ is the
torsion subgroup. Combining with the definition of $N$ as preimage of the
identity,
\[
N/[\Gamma,\Gamma]\simeq\Tor H_1(V,\Z),
\tag{3.3}\label{eq:almost-derived}
\]
a finite abelian group. In particular every element of $N$ has a positive
power in $[\Gamma,\Gamma]$. When the torsion vanishes, $N=[\Gamma,\Gamma]$
identically; we keep the possible finite discrepancy because the
completion arguments below handle both cases uniformly.

\subsection{Fibre image equals commutator at each finite stage}

Fix $c\geq 1$. Let $K_c\subseteq U_c$ be the Zariski closure of the image
of $N$ in $U_c$, and $\kk_c=\Lie(K_c)$. The inclusion $F\hookrightarrow V$
is a pointed algebraic morphism (base point $v\in F$), so by
Section~\ref{sec:mhs} it induces a morphism of canonical quotients
$\g_c(F)\to\g_c(V)$ respecting mixed Hodge structures. Its image is exactly
$\kk_c$: the image of a morphism of unipotent algebraic groups is closed,
and $\pi_1(F,v)$ is Zariski dense in $U_c(F)$, so the image in $U_c(V)$ is
closed and equals $\kk_c$. Thus $\kk_c$ is the image of $\g_c(F)$ and
inherits a mixed Hodge structure.

\begin{lemma}\label{lem:fibre-eq-comm}
$K_c=[U_c,U_c]$ and $\kk_c=[\g_c,\g_c]$.
\end{lemma}
\begin{proof}
First, $K_c\subseteq[U_c,U_c]$. Let $n\in N$. By \eqref{eq:almost-derived}
some positive power $n^k$ lies in $[\Gamma,\Gamma]$. The abelianization
$U_c/[U_c,U_c]=\g_c/[\g_c,\g_c]$ is an additive vector group over $\Q$; it
has no nontrivial torsion. The composition $N\to U_c\to U_c/[U_c,U_c]$ thus
sends every $n$ to zero, because $n^k$ lies in $[\Gamma,\Gamma]$ which maps
to the trivial subgroup. Hence $K_c\subseteq[U_c,U_c]$.

Conversely, $[U_c,U_c]\subseteq K_c$. Normality of $N$ in $\Gamma$ together
with density of $\Gamma$ in $U_c$ implies $K_c$ is normal in $U_c$.
Consider the quotient unipotent group $U_c/K_c$. Its image of $\Gamma$ is
dense, and is abelian because $[\Gamma,\Gamma]\subseteq N\subseteq K_c$.
The commutator morphism
$(U_c/K_c)\times(U_c/K_c)\to U_c/K_c$, $(x,y)\mapsto xyx^{-1}y^{-1}$
vanishes on the dense subset (image of $\Gamma$)$\times$(image of $\Gamma$).
An algebraic function that vanishes on a Zariski-dense subset vanishes
identically, so $U_c/K_c$ is abelian. Hence $[U_c,U_c]\subseteq K_c$. This
gives $K_c=[U_c,U_c]$, and passing to Lie algebras yields
$\kk_c=[\g_c,\g_c]$.
\end{proof}

Note what the lemma does NOT use: it does not assume that unipotent
completion is left exact, and it does not assume that $\pi_1(F,v)\to\Gamma$
is injective. The single point of intersection with exactness is the
inclusion $[\Gamma,\Gamma]\subseteq N$, which came from the equality
$\Lambda=H_1(A,\Z)$ and was refined into the finite quotient
\eqref{eq:almost-derived}.

\subsection{Two weight bounds}\label{sec:mhs-weights}

The lower central series of $\g_c$ now yields our first weight bound. For
$r\geq 1$, the $r$-fold iterated bracket induces a surjective morphism of
mixed Hodge structures
\[
H_1(V,\Q)^{\otimes r}\longrightarrow\gamma_r\g_c/\gamma_{r+1}\g_c.
\tag{3.4}\label{eq:lower-central-weight}
\]
The tensor product on the left has weights at most $-r$ (weights add, and
$H_1(V,\Q)$ has top weight $-1$). Hence the target has weights at most
$-r$. By downward induction on $r$ using extensions in the abelian
category of mixed Hodge structures, each $\gamma_r\g_c$ itself has weights
at most $-r$:
\[
W_{-r}\bigl(\gamma_r\g_c\bigr)=\gamma_r\g_c.
\tag{3.5}\label{eq:gamma-r-weights}
\]
For $r=1$ this says $W_{-1}\g_c=\g_c$. For $r=2$, since
$\kk_c=[\g_c,\g_c]=\gamma_2\g_c$ by Lemma~\ref{lem:fibre-eq-comm},
\[
\kk_c\subseteq W_{-2}\g_c.
\tag{3.6}\label{eq:kk-weight-bound}
\]

The second bound comes from the fibre's first homology. Consider
$B_c=\kk_c/[\kk_c,\kk_c]$ with the quotient mixed Hodge structure (a
quotient of $\kk_c$). The surjection $\g_c(F)\to\kk_c$ proved above,
combined with the natural projection $\kk_c\to B_c$, factors through the
abelianization:
\[
H_1(F,\Q)\longrightarrow B_c
\tag{3.7}\label{eq:fibre-h1-onto}
\]
is surjective as a morphism of mixed Hodge structures. Since $F$ is smooth
quasi-projective, $H_1(F,\Q)$ has weights $-1$ and $-2$ only; strictness
of \eqref{eq:fibre-h1-onto} forces
\[
W_{-3}B_c=0.
\tag{3.8}\label{eq:Bc-below-zero}
\]
On the other hand $B_c$ is a quotient of $\kk_c\subseteq W_{-2}\g_c$, so
\[
W_{-2}B_c=B_c.
\tag{3.9}\label{eq:Bc-above-minus-2}
\]
Combining \eqref{eq:Bc-below-zero} and \eqref{eq:Bc-above-minus-2}, $B_c$
is pure of weight $-2$. This is precisely where the geometry of the fibre
enters: its first homology forbids weights below $-2$, even though $B_c$
sits inside (the commutator of) $U_c$.

\subsection{Centrality and collapse to class two}\label{sec:mhs-centrality}

Now bracket $\g_c$ with $\kk_c$ and project to $B_c$. Since
$\kk_c=[\g_c,\g_c]$ is an ideal of $\g_c$, the composite
\[
[\cdot,\cdot]:\g_c\otimes\kk_c\longrightarrow\kk_c\longrightarrow B_c
\tag{3.10}\label{eq:bracket-to-Bc}
\]
is a morphism of mixed Hodge structures. The source has weights at most
$-1+(-2)=-3$ by \eqref{eq:gamma-r-weights}, while the target has
$W_{-3}B_c=0$ by \eqref{eq:Bc-below-zero}. A mixed-Hodge morphism from
something of weights at most $-3$ into a structure with $W_{-3}=0$ must
vanish. Hence
\[
[\g_c,\kk_c]\subseteq[\kk_c,\kk_c].
\tag{3.11}\label{eq:action-zero}
\]

Now read off the lower-central consequence. By Lemma~\ref{lem:fibre-eq-comm}
we have $\kk_c=\gamma_2\g_c$; substituting into \eqref{eq:action-zero},
\[
\gamma_3\g_c=[\g_c,\gamma_2\g_c]=[\g_c,\kk_c]
\subseteq[\kk_c,\kk_c]=[\gamma_2\g_c,\gamma_2\g_c]\subseteq\gamma_4\g_c.
\tag{3.12}\label{eq:gamma3-in-gamma4}
\]
The final inclusion is the standard lower-central estimate
$[\gamma_p\g_c,\gamma_q\g_c]\subseteq\gamma_{p+q}\g_c$ with $p=q=2$. The
reverse inclusion $\gamma_4\g_c\subseteq\gamma_3\g_c$ is by definition, so
\[
\gamma_3\g_c=\gamma_4\g_c.
\tag{3.13}\label{eq:gamma3-eq-gamma4}
\]
Repeated bracketing with $\g_c$ propagates the equality down the lower
central series: $\gamma_4\g_c=[\g_c,\gamma_3\g_c]=[\g_c,\gamma_4\g_c]
=\gamma_5\g_c$, and so on. So every $\gamma_k\g_c$ with $k\geq 3$ equals
$\gamma_3\g_c$.

But $\g_c$ is nilpotent: there exists $N\geq 1$ with $\gamma_N\g_c=0$. If
$N\geq 3$, combine the equalities with this vanishing to get
\[
\gamma_3\g_c=\gamma_N\g_c=0.
\]
If $N<3$, then $\gamma_3\g_c=0$ trivially. In every case
\[
\gamma_3\g_c=0\qquad\text{for each }c\geq 1.
\tag{3.14}\label{eq:gamma3-zero}
\]
So every canonical quotient $\g_c$ is nilpotent of class at most two.

\subsection{Passage to the full completion}\label{sec:mhs-full}

Now lift the bound from the finite canonical quotients to the complete
Pronilpotent Lie algebra $\m_V$. For each $c\geq 1$, let $\pi_c:\m_V\to\g_c$.
By \eqref{eq:gamma3-zero}, $\gamma_3\m_V$ maps to $\gamma_3\g_c=0$ under
$\pi_c$. So
\[
\gamma_3\m_V\subseteq\bigcap_{c\geq 1}\ker\pi_c.
\tag{3.15}\label{eq:gamma3-in-kernels}
\]
The completion $\m_V$ is by construction the inverse limit
$\varprojlim_c\g_c$, so the intersection of all kernels is zero
(separatedness). Hence
\[
\gamma_3\m_V=0.
\tag{3.16}\label{eq:gamma3-mV-zero}
\]
The complete Lie algebra of the rational unipotent completion is
nilpotent of class at most two.

Two corollaries of this. First, because $\gamma_3\m_V=0$, the canonical map
$\m_V\to\g_2$ is an isomorphism. The tower $\g_1\to\g_2\to\g_3\to\cdots$ of
canonical quotients stabilizes at $c=2$, so the full completion
$\m_V=\g_2$ is finite-dimensional. Second, the abelian structure of $\m_V$
is now pinned down: $\m_V/[\m_V,\m_V]=H_1(V,\Q)$ by \eqref{eq:hurewicz}, and
the commutator $[\m_V,\m_V]$ is a quotient of $\wedge^2 H_1(V,\Q)$.

Translating to the unipotent algebraic group $U=\exp(\m_V)$: using the
Baker-Campbell-Hausdorff correspondence, $U$ is a unipotent algebraic group
of class at most two, finite-dimensional.

\subsection{Complex unipotent representations}\label{sec:mhs-complex}

So far everything has been over $\Q$. Now extend scalars to $\C$. For a
finitely generated group $\Gamma$, unipotent completion commutes with
extension of the characteristic-zero ground field
\cite[\S4]{Hain2015}: the completion over $\C$ is
$U_\C=U\times_\Q\C=\exp(\m_V\otimes\C)$ and satisfies the same universal
property for homomorphisms into complex unipotent algebraic groups.

Let $\rho:\Gamma\to V$ be any finite-dimensional unipotent representation
over $\C$; here $V$ is a unipotent algebraic group over $\C$. By the
universal property of the completion, $\rho$ factors through $U_\C$:
\[
\rho:\Gamma\longrightarrow U_\C\longrightarrow V.
\]
The middle term $U_\C$ has class at most two by Section~\ref{sec:mhs-full}
(tensored up by $-\otimes_\Q\C$, which preserves nilpotency class), so the
image $\im(\rho)$ is a quotient of a class-two unipotent group, hence has
class at most two. And $\im(\rho)$ is finite-dimensional because
$U_\C$ is finite-dimensional. This completes the proof of
Theorem~\ref{thm:completion}.

\subsection{Remarks on the proof}

\paragraph{Proper fibres give class one.} If $H_1(F,\Q)$ is pure of weight
$-1$ (for example, if $F$ is proper), the same argument gives $B_c=0$.
Indeed, purity gives $W_{-2}B_c=0$ directly from \eqref{eq:fibre-h1-onto};
combined with $W_{-2}B_c=B_c$ from \eqref{eq:Bc-above-minus-2}, this forces
$B_c=0$. A nilpotent Lie algebra with zero abelianization is zero, so
$\kk_c=0$ and $U_c$ is abelian. So the entire completion is abelian (class
one). This reproduces, structurally, the abelianity theorem of Aguilar
Aguilar--Campana \cite[arXiv version, Corollary 2(2)]{AguilarAguilarCampana2025}
for smooth quasi-projective varieties whose quasi-Albanese map is proper
and surjective.

\paragraph{Degenerate cases.} If $A$ is a point, then $H_1(V,\Q)=0$ by
\eqref{eq:alb-h1}, so $\g_c=0$ for all $c$ and the completion is zero. If
$F$ is a point, then exactness identifies $\Gamma$ with the free abelian
group $\Lambda$, and the conclusion (class at most two) is immediate. In
particular no division by a positive Betti number is ever required.

\paragraph{What was not used.} Nowhere did the proof assume that
$\pi_1(F,v)\to\Gamma$ is injective; only that the sequence is exact.
Nowhere did the proof assume that $\Gamma$ is nilpotent; only
finitely generated. And nowhere did the proof assume specialness of $V$ or
of $F$: the five inputs listed at the start of Section~\ref{sec:mhs-proof}
are purely structural. Specialness of $V$ will enter only in
Section~\ref{sec:linear} when we apply the CDY geometric inputs to reduce
from general linear representations to the unipotent case.

\section{From completion to complex linear monodromy}\label{sec:linear}

We now prove Theorem~\ref{thm:linear}. The strategy is to replace $V$ by a
suitable finite \'etale cover on which the linear image is unipotent, and
then apply Theorem~\ref{thm:completion} on that cover. The whole content of
the argument that "specialness bounds general linear images" is supplied by
Cadorel--Deng--Yamanoi (CDY); our contribution is the assembly, which uses
Theorem~\ref{thm:completion} in a place where the CDY paper used a weaker
statement.

\subsection{The three CDY inputs}

We use three results of \cite{CDY2025}. Each is a deep result; we do not
reprove them.

\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2em]
\item \emph{Zariski closure decomposition.} By \cite[Theorem A, equivalently
      Theorem 4.1]{CDY2025}, if $V$ is smooth special quasi-projective and
      $\rho:\pi_1(V)\to\GL_d(\C)$ is a representation, then the identity
      component $L^0$ of the Zariski closure $L=\overline{\rho(\pi_1(V))}$
      is a direct product $L^0\simeq U\times T$, where $U$ is unipotent
      and $T$ is a torus. The Zariski closure has finitely many connected
      components, by algebraicity.
\item \emph{quasi-Albanese exactness.} By \cite[Lemma 4.4 and Proposition
      4.13]{CDY2025}, for a smooth special quasi-projective variety $V$, the
      algebraic quasi-Albanese map $a_V:V\to\Alb(V)$ is dominant, has
      connected general fibre, and satisfies the ordinary exact sequence
      \[
      \pi_1(F,v)\to\pi_1(V,v)\xrightarrow{(a_V)_*}\pi_1(\Alb(V),a_V(v))\to 1
      \]
      of Theorem~\ref{thm:completion} with a smooth general fibre $F$.
\item \emph{finite \'etale stability.} By \cite[Theorem 3.1]{CDY2025}, which
      attributes the statement to \cite[Proposition 10.11]{Campana2011}, a
      connected finite \'etale cover $V_0\to V$ of a smooth special
      quasi-projective variety $V$ is itself smooth special quasi-projective.
\end{enumerate}

The specialness convention in (i)-(iii) is the open-variety convention
of \cite[Definition 2.1]{CDY2025}. We will show after
Definition~\ref{def:pair-special} that the orbifold-pair specialness
verified for the surface example in Theorem~\ref{thm:example}(i) implies
this open-variety convention. So the whole chain of implications remains
intact for the surface.

\subsection{Proof of Theorem~\ref{thm:linear}}

Fix $V$ smooth special quasi-projective, and $\rho:\pi_1(V)\to\GL_d(\C)$.
Let $L=\overline{\rho(\pi_1(V))}$ be the Zariski closure and $L^0$ its
identity component.

\begin{proof}[Proof of Theorem~\ref{thm:linear}]
Let $\Gamma=\pi_1(V)$, and let $\Gamma_0=\rho^{-1}(L^0)$ be the preimage
in $\Gamma$ of the identity component. Since $L/L^0$ is finite, the index
$[\Gamma:\Gamma_0]=[L:L^0]$ is finite. By the algebraic Riemann existence
theorem \cite[Expose XII, Theoreme 5.1, p.~251]{SGA1}, the subgroup
$\Gamma_0$ corresponds to a connected finite \'etale cover
$V_0\to V$, with $\pi_1(V_0)\simeq\Gamma_0$.

By (iii), $V_0$ is smooth special quasi-projective. By (i) applied to $V$
(special), $L^0\simeq U\times T$ with $U$ unipotent and $T$ a torus.
The representation $\rho|\Gamma_0$ has image in $L^0=U\times T$; write
$\rho_U:\Gamma_0\to U$ for the projection.

Apply (ii) to $V_0$. The quasi-Albanese map $a_{V_0}:V_0\to\Alb(V_0)$ is
dominant, its general fibre $F_0$ is smooth and connected, and the
ordinary sequence in the sense of Theorem~\ref{thm:completion} is exact
on $V_0$. All three hypotheses of Theorem~\ref{thm:completion} therefore
hold for $V_0$. The conclusion is that the unipotent image $\rho_U(\Gamma_0)$
has nilpotency class at most two. The projection to $T$ is abelian
because $T$ is a torus.

Write any $\gamma\in\Gamma_0$ as $\rho(\gamma)=(u_\gamma,t_\gamma)\in U\times T$.
The triple commutator $[[\gamma_1,\gamma_2],\gamma_3]$ in $\Gamma_0$ maps
under $\rho$ to $[[(u_1,t_1),(u_2,t_2)],(u_3,t_3)]$, which projects to
$[[u_1,u_2],u_3]$ in $U$ and to $[[t_1,t_2],t_3]=e$ in $T$. The first
factor is trivial because $U$ is class at most two, and the second factor
is trivial because $T$ is abelian. So $\rho(\Gamma_0)$ is class at most
two. Since $\Gamma_0$ has finite index in $\Gamma$, this proves
Theorem~\ref{thm:linear}.
\end{proof}

\subsection{Remarks on the proof}

\paragraph{Why we reapply exactness on $V_0$.} We do not simply apply the
definition of $U$ on the original $V$; we re-establish it on the cover.
This matters because unipotent completion need not be unchanged under
passing to a finite-index subgroup of $\Gamma$: passing to a finite-index
subgroup can change the rational unipotent completion. Theorem~\ref{thm:completion}
must be applied to the group $\Gamma_0=\pi_1(V_0)$ where $\rho|\Gamma_0$ is
actually unipotent-valued. It is not enough to know that $\rho(\Gamma)$'s
Zariski closure is a product.

\paragraph{What is used and what is not.} The proof uses (i) and (ii) and
(iii) of CDY, plus Theorem~\ref{thm:completion}. It uses no fibre-specialness
assertion. It uses no mixed Hodge structure on an arbitrary $d$-dimensional
linear quotient; the mixed Hodge argument was carried out entirely on the
canonical nilpotent quotients in Section~\ref{sec:mhs-proof}. It uses no
properness or compactness of $V$. It does not need the ordinary group
$\Gamma$ itself to embed in its completion.

\subsection{The logarithmic-Kodaira-zero corollary}

The same inputs give a clean corollary at logarithmic Kodaira dimension
zero, where Campana's theorem upgrades specialness to a form the completion
theorem can use directly.

\begin{corollary}[Log-Kodaira-zero monodromy]\label{cor:log-kodaira-zero}
Let $V$ be a connected smooth complex quasi-projective variety with
logarithmic Kodaira dimension $\bar\kappa(V)=0$, and put
$\Gamma=\pi_1(V)$. For every positive integer $d$ and every
representation $\rho:\Gamma\to\GL_d(\C)$, the image $\rho(\Gamma)$ has a
subgroup of finite index and nilpotency class at most two. The full
rational unipotent completion of $\Gamma$ is a finite-dimensional unipotent
algebraic group of class at most two. Every finite-dimensional complex
unipotent representation of $\Gamma$ has image of class at most two.
\end{corollary}

\begin{proof}
Choose a smooth projective compactification $Y$ of $V$ with reduced
simple-normal-crossing boundary $D=Y\setminus V$. Then
$\bar\kappa(V)=\kappa(Y,K_Y+D)=0$ by definition of the logarithmic
Kodaira dimension. Campana's theorem
\cite[arXiv version, Theorem 6.7 and Corollary 4.11]{Campana2011} applies
to smooth projective pairs with reduced simple normal crossing boundary
and gives that $(Y,D)$ is special. By the comparison of conventions after
Definition~\ref{def:pair-special} below, $V$ is special in the open-variety
convention of CDY.

Theorem~\ref{thm:linear} then gives the linear-image assertion. For the
completion assertions, note that by the CDY quasi-Albanese input (ii) above,
applied directly to $V$ (special), the quasi-Albanese map $a_V$ is
dominant, has a smooth connected general fibre, and satisfies the ordinary
exact sequence of \eqref{eq:exact-ref}. Theorem~\ref{thm:completion}
therefore gives the full rational unipotent completion and its finite-
dimensional class-two behaviour, and every complex unipotent image.
\end{proof}

The finite-index subgroup in Corollary~\ref{cor:log-kodaira-zero} may
depend on the representation. The corollary does not assert virtual two-step
nilpotence of the ordinary discrete group $\Gamma$ itself; that is exactly
the open question of Section~\ref{sec:ordinary}. The faithfulness issue --
the possible kernel of $\Gamma\to\Gamma^{\mathrm{un}}$ -- is genuine and
unaddressed by any completion argument.

\section{The surface and the orbifold machinery}\label{sec:definitions}

We now construct the surface of Theorem~\ref{thm:example}, describe its two
natural projections, and set up the orbifold conventions needed for the
specialness proof. This is the longest section of formal definitions in the
paper; a reader willing to accept the definitions can skip to
Section~\ref{sec:specialness} on first pass.

\subsection{The blowup and its two projections}

Let $E$ be a complex elliptic curve, let $q_1,\ldots,q_n$ be distinct
points of $\PP^1$ with $n\geq 3$, and fix arbitrary points
$e_1,\ldots,e_n\in E$. Set
\[
S=E\times\PP^1,\qquad H_i=E\times\{q_i\},\qquad p_i=(e_i,q_i).
\tag{5.1}\label{eq:surface-setup}
\]
Let
\[
b:Y=\Bl_{\{p_1,\ldots,p_n\}}S\longrightarrow S
\tag{5.2}\label{eq:blowup}
\]
be the blowup of $S$ at the $n$ distinct points $p_i$. Because the $q_i$
are distinct, the points $p_i$ are distinct, and the blowup is a standard
smooth birational morphism. Let
\[
A_i=b^{-1}(p_i)\subseteq Y
\]
be the $i$-th exceptional divisor. Each $A_i$ is isomorphic to $\PP^1$.
Let $D_i$ be the strict transform of the elliptic curve $H_i=E\times\{q_i\}$.
We set
\[
D=\sum_{i=1}^n D_i,\qquad X=Y\setminus D.
\tag{5.3}\label{eq:D-X}
\]
The curves $D_i$ are pairwise disjoint (because the $H_i$ are distinct
horizontal slices), and each $A_i$ meets $D_i$ in exactly one point,
transversely. Away from that point, $A_i$ does not meet $D$.

Two projections matter. First, set
\[
\bar a=\operatorname{pr}_E\circ b:Y\to E,\qquad
a=\bar a|_X:X\to E.
\tag{5.4}\label{eq:a-and-bara}
\]
Second, set
\[
j=\operatorname{pr}_{\PP^1}\circ b:Y\to\PP^1.
\tag{5.5}\label{eq:j-projection}
\]
The projection $a$ will turn out to be the quasi-Albanese map. The
projection $j$ will turn out to be the logarithmic Iitaka fibration; its
general fibre is a complete elliptic curve, and its fibre over $q_i$ has two
irreducible components.

Two basic intersection facts follow from the blowup construction. For all
$i,j$,
\[
A_i\simeq\PP^1,\qquad D_i\simeq E,\qquad D_i\cdot A_j=\delta_{ij},
\qquad A_i^\circ:=A_i\cap X\simeq\AA^1.
\tag{5.6}\label{eq:intersection-facts}
\]
The last equality is because $A_i$ is a $\PP^1$ meeting $D$ in a single
point, so $A_i\setminus D$ is a copy of $\AA^1$; call its one-point
compactification point $A_i\cap D_i$ the "deleted point" of the exceptional
curve.

\subsection{Fibres of the two projections}

We describe the fibres of $\bar a$ and of $j$.

\paragraph{Fibres of $\bar a$ over $E$.} For $e\in E$, the fibre of
$\operatorname{pr}_E:S\to E$ over $e$ is the slice $\{e\}\times\PP^1$, a
$\PP^1$. Blowing up at a point of this slice turns it into the union of the
exceptional curve and the strict transform. Precisely, set
$I(e)=\{i:e_i=e\}$, so $|I(e)|$ is the number of blown-up points on this
slice. Let $B_e$ be the strict transform of $\{e\}\times\PP^1$ in $Y$. Then
the scheme fibre of $\bar a$ over $e$ is
\[
\bar a^{*}(e)=B_e+\sum_{i\in I(e)}A_i,
\tag{5.7}\label{eq:a-fibre-scheme}
\]
all components with multiplicity one, and the $A_i$ (for $i\in I(e)$) are
mutually disjoint $\PP^1$'s each meeting $B_e$ in one point. Restricting to
the open surface $X$:
\[
B_e\cap X\simeq\PP^1\setminus\{q_j:j\notin I(e)\},\qquad
A_i^\circ=A_i\cap X\simeq\AA^1\quad(i\in I(e)).
\tag{5.8}\label{eq:a-fibre-open}
\]
Each $A_i^\circ$ is attached to $B_e\cap X$ at the point corresponding to
$q_i$ (the "vertical tangent direction" at $p_i$ in the blowup); the
"horizontal tangent direction" at $p_i$, which is the deleted point
$A_i\cap D_i$, is removed from $X$. So every fibre of $a:X\to E$ is
connected. For $e\notin\{e_1,\ldots,e_n\}$ the set $I(e)$ is empty and
\[
a^{-1}(e)\simeq\PP^1\setminus\{q_1,\ldots,q_n\}.
\tag{5.9}\label{eq:general-a-fibre}
\]
This is the general fibre $F$.

\paragraph{Fibres of $j$ over $\PP^1$.} For $t\notin\{q_1,\ldots,q_n\}$, the
slice $E\times\{t\}$ is a complete elliptic curve disjoint from all blowup
centres and disjoint from $D$; it is contained entirely in $X$. So the fibre
of $j$ over $t$ is a complete elliptic curve. At $t=q_i$, the fibre of
$\operatorname{pr}_{\PP^1}:S\to\PP^1$ is the curve $H_i$, on which we have
blown up one point $p_i$. The scheme fibre of $j$ over $q_i$ in $Y$ is the
union
\[
j^*(q_i)=D_i+A_i,
\tag{5.10}\label{eq:j-fibre-scheme}
\]
with both components of multiplicity one. (The strict transform $D_i$ comes
from the horizontal direction, and $A_i$ from the vertical direction at
$p_i$.) In particular all fibres of $j$ are connected, and
$j_*\OO_Y=\OO_{\PP^1}$.

\subsection{The logarithmic canonical divisor}

The strict-transform and adjunction formulas give a clean description of
the log-canonical divisor $K_Y+D$.

Let $K_S=\operatorname{pr}_E^*K_E+\operatorname{pr}_{\PP^1}^*K_{\PP^1}$, so
$K_S$ has degree $0$ on the elliptic direction and
$\operatorname{pr}_{\PP^1}^*K_{\PP^1}$ has degree $-2$ on the $\PP^1$
direction. The strict-transform formula for a single blowup at a point of a
smooth surface says $K_Y=b^*K_S+\sum_iA_i$ (one exceptional curve per point
blown up). The strict transform of $H_i=\operatorname{pr}_{\PP^1}^*q_i$ is
$D_i$, and $b^*H_i=D_i+A_i$, so $D_i=b^*H_i-A_i$. Therefore
\[
D=\sum_iD_i=b^*\Bigl(\sum_iH_i\Bigr)-\sum_iA_i,
\]
and
\[
K_Y+D=b^*K_S+\sum_iA_i+b^*\sum_iH_i-\sum_iA_i
=b^*\bigl(K_S+\sum_iH_i\bigr).
\]
But $K_S+\sum_iH_i=\operatorname{pr}_E^*K_E+\operatorname{pr}_{\PP^1}^*K_{\PP^1}
+\operatorname{pr}_{\PP^1}^*\sum_iq_i=\operatorname{pr}_{\PP^1}^*\bigl(K_{\PP^1}
+\sum_iq_i\bigr)$, and
$K_{\PP^1}+\sum_iq_i$ has degree $-2+n=n-2$ on $\PP^1$. Hence
\[
K_Y+D\sim\bigl(j^*\OO_{\PP^1}\bigr)\otimes\OO_{\PP^1}(n-2)
=j^*\OO_{\PP^1}(n-2).
\tag{5.11}\label{eq:log-canonical}
\]
Taking global sections and using $j_*\OO_Y=\OO_{\PP^1}$ and the projection
formula,
\[
H^0(Y,m(K_Y+D))=H^0(\PP^1,\OO_{\PP^1}(m(n-2)))\qquad(m\geq1).
\tag{5.12}\label{eq:pluricanonical}
\]
So
\[
\kappa(Y,K_Y+D)=1,
\]
and the pluricanonical sections factor through $j$; the projection $j$ is
the logarithmic Iitaka fibration of $(Y,D)$. Because $n\geq 3$, $n-2>0$, so
the general fibre of $j$ has log-canonical degree $n-2>0$ and is not special
(Section~\ref{sec:specialness} makes this precise).

\subsection{Orbifold pairs and the infimum convention}\label{sec:orbifold}

A \emph{smooth orbifold pair} is a smooth variety $Y$ together with a
$\Q$-divisor
\[
D=\sum_P\Bigl(1-\frac{1}{m_P(D)}\Bigr)P,\qquad
m_P(D)\in\Z_{\geq 1}\cup\{\infty\},
\tag{5.13}\label{eq:orbifold-divisor}
\]
with only finitely many nonzero coefficients. The convention is
$1/\infty=0$: a component of $D$ with multiplicity $\infty$ has
coefficient one (a "reduced" component), and a prime divisor outside the
support of $D$ has multiplicity $1$, hence contributes coefficient zero.

A \emph{fibration} is a surjective morphism with connected general fibre.
Let $f:Y\to Z$ be a fibration with $Y$ smooth projective and $Z$ smooth,
and let $D$ be an orbifold divisor on $Y$. For a prime divisor $Q\subset Z$
and a prime divisor $P\subset Y$ mapping onto $Q$, write $r(f,P)$ for the
multiplicity of $P$ in $f^*Q$ (the "orbifold ramification index" of $f$ at
$P$ over $Q$). The \emph{orbifold base divisor} is
\[
\Delta(f,D)=\sum_Q\Bigl(1-\frac{1}{m_f(Q)}\Bigr)Q,\qquad
m_f(Q)=\inf_{f(P)=Q}r(f,P)\,m_P(D).
\tag{5.14}\label{eq:infimum}
\]
This is the infimum convention of \cite[Definition 2.1]{CDHP2026}, not a
greatest-common-divisor convention. Only prime divisors $P$ that map onto
$Q$ (scheme-theoretically) enter the infimum; in particular a divisor $P$
with $\codim_Z f(P)\geq 2$ is \emph{$f$-exceptional} and is not counted at
this step. It may enter the infimum of a \emph{composition} $f'\circ f$;
that is exactly the subtlety exploited in Section~\ref{sec:composition}.

For the surface $(Y,D)$, \eqref{eq:j-fibre-scheme} gives
\[
m_j(q_i)=\min\{1\cdot\infty,1\cdot 1\}=1,\qquad
\Delta(j,D)=0,
\tag{5.15}\label{eq:j-base-zero}
\]
the two terms coming from the two components $D_i$ (multiplicity $\infty$
in $D$) and $A_i$ (multiplicity $1$). For $t\notin\{q_i\}$ the unchanged
elliptic fibre gives multiplicity one as well. So the orbifold base of $j$
is $(\PP^1,0)$: logarithmic Kodaira dimension one on the source, orbifold
base divisor zero on the base.

A \emph{curve criterion}: for a smooth projective curve $C$ and a divisor
$K_C+\Delta$, with $\Delta$ an orbifold divisor, the divisor is of general
type precisely when its degree is positive. For $C=\PP^1$ we then have
$\deg K_{\PP^1}=-2$, so achieving positive degree requires enough orbifold
multiplicity. In the surface example, at $q_i$ the base sees coefficient
$0$ (both components give multiplicity $1$), not the "general-type"
coefficient $1-1/1=0$ from a different computation. Section~\ref{sec:composition}
explains why the "two-step" orbifold base computed in the CDHP paper differs
at this point.

\subsection{Elementary birational orbifold morphisms and upper models}

A dominant morphism $u:(Y',D')\to(Y,D)$ of smooth orbifold pairs is an
\emph{orbifold morphism} if, for every prime divisor $P\subset Y$ and every
prime divisor $P'$ of the preimage $u^{-1}(P)$ with coefficient $r=r(u,P')$
in the pullback $u^*P$,
\[
r\,m_{P'}(D')\geq m_P(D).
\tag{5.16}\label{eq:orbifold-morphism}
\]
It is an \emph{elementary birational orbifold morphism} if in addition
$u$ is birational and pushes boundaries forward: $u_*D'=D$. When $D$ is
reduced, every component of $u^{-1}(D)$ must lie in the coefficient-one
part of $D'$; equality of pushforward divisors alone is weaker.

An \emph{upper model} of $(Y,D)$ is any smooth projective orbifold pair
$(Y',D')$ equipped with an elementary birational orbifold morphism
$u:(Y',D')\to(Y,D)$. Upper models are the "proper birational modifications"
over which the specialness condition quantifies.

A fibration $f:(Y,D)\to Z$ of an orbifold pair is \emph{neat} if it fits
into a commutative diagram
\[
\begin{array}{ccc}
(Y,D)&\xrightarrow{f}&Z\\
\downarrow{\scriptstyle w}&&\downarrow{\scriptstyle v}\\
(Y_0,D_0)&\xrightarrow{f_0}&Z_0
\end{array}
\]
with all spaces smooth, the horizontal maps fibrations, $w$ an elementary
birational orbifold morphism, and every $f$-exceptional divisor also
$w$-exceptional. It is \emph{high} if the morphism to the base, base
equipped with its induced orbifold divisor $\Delta(f,D)$, is an orbifold
morphism.

The \emph{canonical dimension} of a fibration is the infimum of the
Kodaira dimensions of its orbifold bases over all birational models, where
the Kodaira dimension of a $\Q$-divisor is the growth rate of its
plurisection spaces (value $-\infty$ if all vanish). For a neat fibration
$f:(Y,D)\to Z$, \cite[Lemma 2.7]{CDHP2026}, attributed there to
\cite[Corollaire 5.11(3)]{Campana2011}, gives the equality
\[
\kappa(f,D)=\kappa\bigl(Z,K_Z+\Delta(f,D)\bigr).
\tag{5.17}\label{eq:neat-kappa}
\]

\subsection{Specialness: pair convention and open convention}

\begin{definition}[Pair specialness; CDHP convention]\label{def:pair-special}
A smooth projective orbifold pair $(Y,D)$ is \emph{special} if for every
elementary birational orbifold morphism $u:(Y',D')\to(Y,D)$ with
$(Y',D')$ smooth projective, and for every neat fibration
$g:(Y',D')\to Z$ with $\dim Z>0$,
\[
\kappa\bigl(Z,K_Z+\Delta(g,D')\bigr)<\dim Z.
\tag{5.18}\label{eq:specialness-ineq}
\]
\end{definition}

A smooth open variety $X=Y\setminus D$ with reduced boundary $D$ is
\emph{logarithmically special} (equivalently, special in the open-variety
convention of \cite[Definition 2.1]{CDY2025}) when its log compactification
$(Y,D)$ has the specialness property \eqref{eq:specialness-ineq} over all
elementary birational orbifold modifications and all neat fibrations.

The CDY paper uses the second (open) convention throughout. To connect the
two conventions, and to justify using the pair condition of
Theorem~\ref{thm:example}(i) to feed the representational inputs of
Section~\ref{sec:linear}, we need:

\begin{proposition}[Pair specialness implies open specialness]\label{prop:pair-to-open}
Let $(Y,D)$ be a smooth projective orbifold pair whose boundary $D$ is a
reduced simple normal crossing divisor. If $(Y,D)$ is special in the sense
of Definition~\ref{def:pair-special}, then $X=Y\setminus D$ is special
in the open-variety convention of \cite[Definition 2.1]{CDY2025}.
\end{proposition}

\begin{proof}
Work with an open fibration $g:X\to B$ onto a positive-dimensional normal
quasi-projective base $B$, and a proper birational modification
$\tilde X\to X$. Compactify the modification over $(Y,D)$, resolve the
graph of the fibration, and choose a neat upper model $(Y',D')\to(Y,D)$ in
the sense of \cite[Proposition 2.8]{CDHP2026}. On this model, let
$D_{\mathrm{open}}$ be the reduced inverse image of $D$, and $D_{\mathrm{high}}$
the boundary of the upper orbifold pair. The orbifold-morphism condition
over the reduced divisor $D$ forces $D_{\mathrm{open}}\leq D_{\mathrm{high}}$
(componentwise multiplicities of the boundary). On the same resolved
fibration $g':(Y',D')\to B'$, formula~\eqref{eq:infimum} gives
\[
\Delta(g',D_{\mathrm{open}})\leq\Delta(g',D_{\mathrm{high}}).
\]
Kodaira dimension is monotone in the divisor: a smaller divisor has no
larger section growth. Therefore the Kodaira dimension of the former base
is at most that of the latter, which by Definition~\ref{def:pair-special}
is less than $\dim B'=\dim B$. The infimum over models allowed in the
open-variety definition is at most this value, so it too is $<\dim B$.
Hence $X$ is special in the open-variety convention.
\end{proof}

\section{The surface is special}\label{sec:specialness}

We prove Theorem~\ref{thm:example}(i): the pair $(Y,D)$ is special in the
sense of Definition~\ref{def:pair-special}, including every upper model.
Fix an arbitrary elementary birational orbifold morphism
$u:(Y',D')\to(Y,D)$ with $(Y',D')$ smooth projective, and a neat fibration
$g:(Y',D')\to Z$ with $\dim Z>0$. Because $u_*D'=D$ and $D$ is the
reduced sum $\sum_i D_i$, every component of $D'$ is either a strict
transform $D_i'$ of some $D_i$ (with multiplicity $\infty$, coefficient
one) or a $u$-exceptional curve. The images of the exceptional curves form
a finite subset of $Y$; call it $\Sigma$. Away from $\Sigma$, $u$ is an
isomorphism.

Let $A_i'$ be the strict transform of $A_i$ in $Y'$. Since $A_i$ is not a
component of $D$, the coefficient of $A_i'$ in $D'$ is zero (by $u_*D'=D$
and the fact that $A_i$ has coefficient zero in $D$). So $A_i'$ lies
outside the support of $D'$, and has multiplicity one there.

There are two cases for $\dim Z$.

\subsection{Two-dimensional base}

Since $\dim Y'=2$ and fibrations have connected general fibre,
$\dim Z=2$ forces $g$ to be birational. Compare $g$ with the identity
fibration $\operatorname{id}:(Y',D')\to(Y',D')$: both have the same source,
$g$ has base $Y'$ and source $Y'$ with the same identity source map. By
the definition of canonical dimension,
\[
\kappa(g,D')\leq\kappa(Y',K_{Y'}+D').
\tag{6.1}\label{eq:birational-kappa}
\]
So it suffices to bound $\kappa(Y',K_{Y'}+D')\leq 1$.

To do this we compare sections in $Y'$ and in $Y$. Choose $m\geq 1$ such
that $mD'$ is integral. Off the finite set $\Sigma$, $u$ is an
isomorphism. So the pushforward $u_*(m(K_{Y'}+D'))=m(K_Y+D)$ holds away
from $\Sigma$. A global holomorphic section of $m(K_{Y'}+D')$ restricts,
over $Y\setminus\Sigma$, to a holomorphic section of $m(K_Y+D)$ on that
same open set. A holomorphic section of a line bundle on a smooth surface
defined on the complement of a finite set of points extends uniquely
across those points: locally, after trivializing the line bundle, this is
the extension of a bounded holomorphic function across a point of
codimension two (the Hartogs phenomenon). We obtain an injection
\[
H^0(Y',m(K_{Y'}+D'))\lhook\joinrel\longrightarrow H^0(Y,m(K_Y+D)).
\tag{6.2}\label{eq:section-injection}
\]
The right-hand side grows polynomially in $m$ of degree one, by
\eqref{eq:pluricanonical} and \eqref{eq:log-canonical}. So the left-hand
side grows at most linearly in $m$, hence
\[
\kappa(Y',K_{Y'}+D')\leq 1.
\tag{6.3}\label{eq:kappa-at-most-1}
\]
By \eqref{eq:birational-kappa},
$\kappa(g,D')\leq 1<2=\dim Z$, so the specialness inequality
\eqref{eq:specialness-ineq} holds in dimension two. This does not require
birational invariance of log-Kodaira dimension for arbitrary changes of
the boundary; it uses only the one-sided passage of sections, and the
particular direction of the extension.

\subsection{Curve base}

Now suppose $Z=C$ is a smooth projective curve. Set
$\Delta=\Delta(g,D')$ and assume for contradiction that
\[
\deg(K_C+\Delta)>0.
\tag{6.4}\label{eq:positive-base}
\]
We will obtain a contradiction by restricting $g$ to a suitable complete
elliptic curve inside $(Y',D')$ and applying Riemann-Hurwitz.

\paragraph{Choice of a clean elliptic fibre.} Choose
$t\in\PP^1\setminus\{q_1,\ldots,q_n\}$ such that the fibre
$E\times\{t\}\subseteq X$ avoids the images under $j$ of the
$u$-exceptional curves. The set of bad $t$ is finite, so such $t$ exists.
Then the fibre
\[
C_t:=(j\circ u)^{-1}(t)\subseteq Y'
\tag{6.5}\label{eq:clean-fibre}
\]
is an unchanged complete elliptic curve: it is the strict transform of
$E\times\{t\}$ under any chain of modifications, but the modifications act
in codimension one away from the general elliptic fibre. So $C_t$ is
smooth, complete, elliptic, disjoint from $\Supp D'$, and disjoint from
the exceptional locus of $u$. In particular $C_t$ is disjoint from every
component $P$ of any $g^*c$ with $c\in C$.

\paragraph{Nonconstant restriction contradicts positivity.} Suppose
$h=g|_{C_t}:C_t\to C$ is nonconstant. Then $h$ is finite and surjective,
of some degree $d=\deg(h)>0$. For a point $c\in C$, let $m_c$ be its
orbifold multiplicity, i.e. $m_c=m_g(c)$ in the infimum \eqref{eq:infimum}.
Any component $P$ of $g^*c$ that meets $C_t$ has $m_P(D')=1$ (since $C_t$
avoids $\Supp D'$). By the infimum definition,
\[
r(g,P)\geq m_c.
\tag{6.6}\label{eq:ram-index-bound}
\]
For $x\in C_t$ above $c$, write $e_x$ for the ramification index of $h$ at
$x$, i.e. the intersection multiplicity $(C_t\cdot g^*c)_x$. This is a
sum of positive intersection orders $r(g,P)$ over the components $P$ of
$g^*c$ through $x$, so $e_x\geq m_c$. If $m_c=\infty$ (an infinite base
multiplicity), the inequality $e_x\geq m_c$ is impossible since $e_x$ is
finite and $h$ is surjective; so under $h$ nonconstant, all $m_c$ are
finite. For finite $m_c$,
\[
e_x-1\geq e_x-1=\frac{e_x\,m_c-e_x}{m_c}\geq\frac{e_x(m_c-1)}{m_c}=e_x\Bigl(1-\frac1{m_c}\Bigr).
\]
Summing over $c\in\Supp\Delta$ and over points of $C_t$ above $c$, and
using the Riemann-Hurwitz formula for $h:C_t\to C$ (a finite map of smooth
projective curves),
\[
0=\deg K_{C_t}=d\deg K_C+\sum_{x\in C_t}(e_x-1)
\geq d\deg K_C+d\deg\Delta=d\deg(K_C+\Delta).
\tag{6.7}\label{eq:riemann-hurwitz}
\]
Since $d>0$, this forces $\deg(K_C+\Delta)\leq 0$, contradicting
\eqref{eq:positive-base}.

\paragraph{Constant restriction propagates by descent.} So $h=g|_{C_t}$
is constant. That is, $g$ is constant on the fibre of $j\circ u$ over
$t$. Varying $t$ over a dense open subset of $\PP^1$, we see that $g$ is
constant on all sufficiently general fibres of $j\circ u$. Those general
fibres are connected (Section~\ref{sec:definitions}). A morphism constant
on each fibre of a fibration $j\circ u:Y'\to\PP^1$ descends to the base on
a dense open subset: it factors through a rational map
$r:\PP^1\dashrightarrow C$ with $g=r\circ j\circ u$ on the open. Since
$\PP^1$ is a smooth projective curve, $r$ extends to a regular morphism
$\PP^1\to C$.

If $\deg r>1$, then a general fibre of $g$ is the disjoint union of the
several fibres of $j\circ u$ over the distinct preimages of a general point
of $C$. This contradicts connectedness of the general fibre of $g$. So
$\deg r=1$, and $C\simeq\PP^1$, and $g=j\circ u$ up to isomorphism.

\paragraph{Boundary of the base under $g=j\circ u$.} Finally we compute
the orbifold divisor of this possible curve base. On the arbitrary model
$Y'$, let $A_i'$ be the strict transform of $A_i$. Its coefficient in
$D'$ is zero (as noted at the start of this section). Since $u$ is an
isomorphism at the generic point of $A_i'$, the multiplicity of $A_i'$ in
$(j\circ u)^*q_i$ is exactly the multiplicity of $A_i$ in $j^*q_i$, which
is one (from \eqref{eq:j-fibre-scheme}). So the term $A_i'$ alone already
gives $r(m\circ u,A_i')m_{A_i'}(D')=1\cdot 1=1$, hence
$m_{j\circ u}(q_i)=1$ and the coefficient of $q_i$ in
$\Delta(j\circ u,D')$ is $1-1/1=0$. For $t\notin\{q_i\}$, the strict
transform of the elliptic fibre again contributes multiplicity one with
boundary coefficient zero, giving coefficient zero at $t$. Adding boundary
on further exceptional curves can only decrease an infimum already equal
to one, so
\[
\Delta(j\circ u,D')=0.
\]
Thus $\deg(K_C+\Delta)=\deg K_{\PP^1}=-2$, contradicting
\eqref{eq:positive-base}.

Both cases of possible base dimension have been excluded, so
$(Y,D)$ is special in the sense of Definition~\ref{def:pair-special},
including every upper model. This proves Theorem~\ref{thm:example}(i).

\section{The quasi-Albanese map and the ordinary fundamental group}\label{sec:albanese}

We now prove Theorem~\ref{thm:example}(ii) and (iii). The fibre description
of Section~\ref{sec:definitions} already shows that $a:X\to E$ has connected
fibres; we identify $a$ by the quasi-Albanese universal property, and then
compute the ordinary fundamental group $\pi_1(X)$.

% ---------------------------------------------------------------
\subsection{A section of $a$ and the unit lemma}

Choose $q_*\in\PP^1\setminus\{q_1,\ldots,q_n\}$. The curve
$E\times\{q_*\}$ is a complete elliptic curve disjoint from every
horizontal boundary curve $H_i$ and disjoint from every blowup centre; it
is fully contained in the open $X$. It defines a section
\[
s:E\hookrightarrow X,\qquad a\circ s=\operatorname{id}_E.
\tag{7.1}\label{eq:section}
\]

\begin{lemma}[Units]\label{lem:units}
Every invertible regular function on $X$ is constant.
\end{lemma}
\begin{proof}
Let $v\in\Gamma(X,\OO_X)$ be a unit. Consider its rational extension
$v\in\C(Y)^{*}$ to the projective normal surface $Y$. Since $v$ is
invertible on the open set $X=Y-D$, its divisor
$\divisor_Y(v)$ on $Y$ is supported on the boundary $D=\sum_iD_i$. Write
$\divisor_Y(v)=\sum_i b_iD_i$ for integers $b_i\in\Z$.

We evaluate $b_j$ by intersecting with the exceptional curve $A_j$.
Because $\divisor_Y(v)$ is principal, its pushforward to any smooth
projective curve (namely $A_j\simeq\PP^1$) has degree zero:
\[
\deg\bigl(\divisor_Y(v)|_{A_j}\bigr)=0.
\tag{7.2}\label{eq:principal-zero}
\]
On the other hand, evaluating the intersection with $A_j$ in $Y$ using
$D_i\cdot A_j=\delta_{ij}$ from \eqref{eq:intersection-facts},
\[
\divisor_Y(v)\cdot A_j=\sum_i b_i(D_i\cdot A_j)=b_j.
\tag{7.3}\label{eq:intersection-principal}
\]
Combining \eqref{eq:principal-zero} and \eqref{eq:intersection-principal},
$b_j=0$ for every $j$. So $\divisor_Y(v)=0$.

A nonzero rational function on a projective variety with trivial divisor is
a nonzero regular function with no zeros on $Y$. Since $Y$ is
proper, any such global section is constant (max-modulus or Stein
argument: a regular function on a connected smooth proper complex
manifold is constant). Hence $v$ is constant on $Y$, and $v$ is constant
on $X$.
\end{proof}

% ---------------------------------------------------------------
\subsection{The quasi-Albanese universal property}

\begin{proposition}\label{prop:a-is-alb}
For every semiabelian variety $G$ and every algebraic morphism
$f:X\to G$, there exists a unique algebraic expression
$f=(f\circ s)\circ a$, where $f\circ s:E\to G$ is the pullback of the
section \eqref{eq:section}.
\end{proposition}

\begin{proof}
Write $G$ in an exact sequence
\[
1\longrightarrow T\longrightarrow G\xrightarrow{\rho}B\longrightarrow 1,
\qquad T\simeq\Gm^r,
\]
with $B$ an abelian variety.

\emph{Step 1: $\rho\circ f$ factors through $a$.} For a fibre
$a^{-1}(e)$ over $e\in E\setminus\{e_1,\ldots,e_n\}$, this fibre is the
punctured projective line $\PP^1\setminus\{q_1,\ldots,q_n\}$, which is
contained in an unchanged complete $\PP^1$ inside $Y$. So $\rho\circ f$ extends
over this open fibre to a morphism $\PP^1\to B$. But $B$ is an abelian
variety and every invariant holomorphic one-form on $B$ pulls back to a
holomorphic one-form on $\PP^1$; there are none ($\Omega^1(\PP^1)=0$). So
the differential of $\PP^1\to B$ vanishes, which in characteristic zero
forces the morphism to be constant. Hence $\rho\circ f(a^{-1}(e))$ is a
single point for $e\notin\{e_i\}$.

Comparing at the point provided by the section, the morphism
$\rho\circ f$ and $(\rho\circ f\circ s)\circ a$ agree on the dense open
subset $X\setminus\cup a^{-1}(e_i)$ of $X$. By continuity (dense open)
they agree everywhere:
\[
\rho\circ f=(\rho\circ f\circ s)\circ a.
\tag{7.4}\label{eq:rho-factor}
\]

\emph{Step 2: the affine defect vanishes.} Define
\[
h(x)=f(x)-f(s(a(x)))\in G.
\]
Under $\rho$, $h$ maps to $\rho\circ f(x)-\rho\circ f(s(a(x)))$, which
vanishes by \eqref{eq:rho-factor}: so $h(x)\in T$ for all $x$. Each of the
$r$ coordinates of $h:X\to T=\Gm^r$ is an invertible regular function on
$X$; by Lemma~\ref{lem:units} it is constant. On the image $s(E)$, we
have $s(a(s(e)))=s(e)$ and so $h(s(e))=0$ identically. Hence the constant
value is $0$, and $h\equiv 0$. Therefore
\[
f=(f\circ s)\circ a.
\tag{7.5}\label{eq:universal-factorization}
\]
Uniqueness follows by precomposition with $s$.
\end{proof}

% ---------------------------------------------------------------
\subsection{The based universal property}

Now refine Proposition~\ref{prop:a-is-alb} to a based statement, using a
section $s(0)$ with $0\in E$ the group identity.

\begin{proposition}\label{prop:based}
Let $f:X\to G$ be an algebraic morphism with $f(s(0))=0$. Then the map
$g=f\circ s:E\to G$ is a homomorphism of algebraic groups.
\end{proposition}

\begin{proof}
By rigidity of abelian varieties, the projection $\rho\circ g:E\to B$ is a
homomorphism of groups (rigidity theorem for abelian varieties: a morphism
of abelian varieties preserving the identity is a homomorphism after
translation, and here we have the based point). The group-law defect
\[
\Phi(g):E\times E\longrightarrow G,\quad(e,e')\mapsto g(e+e')-g(e)-g(e')
\tag{7.6}\label{eq:defect}
\]
lands in the affine group $T=\Gm^r$: after applying $\rho$, it vanishes
because $\rho\circ g$ is a homomorphism. The source $E\times E$ is a
connected projective variety, and $T\simeq\Gm^r$ is affine. Any morphism
from a connected projective variety to an affine torus is constant. So
$\Phi(g)$ is constant. Evaluating at $(0,0)$: $\Phi(g)(0,0)=g(0)-g(0)-g(0)=-g(0)$, and
$g(0)=f(s(0))=0$, so $\Phi(g)(0,0)=0$. Hence the constant is zero, so
$\Phi(g)\equiv 0$, and $g$ is a homomorphism.
\end{proof}

Combining Propositions~\ref{prop:a-is-alb} and~\ref{prop:based}, we obtain
\[
\Alb(X)=E,\qquad a_X=a.
\tag{7.7}\label{eq:alb-is-E}
\]
The general fibre in \eqref{eq:general-a-fibre} is the punctured projective
line $F\simeq\PP^1\setminus\{q_1,\ldots,q_n\}$. Its logarithmic canonical
degree (computed below in Section~\ref{sec:specialness-example}) is
$n-2>0$, so the identity fibration on $F$ is of general type; hence $F$ is
not special. Combined with connectedness of all fibres of $a$, this proves
Theorem~\ref{thm:example}(ii).

% ---------------------------------------------------------------
\subsection{The ordinary fundamental group of $X$}

We now compute $\pi_1(X)$. The key geometric input is the retained
exceptional affine lines $A_i^\circ\simeq\AA^1$. Rather than being part of
the boundary, they sit inside $X$ and their local neighborhoods supply
small discs bounding the puncture loops of every fibre.

\paragraph{Removing the exceptional curve.} Set
\[
U=X\setminus\bigcup_{i=1}^nA_i^\circ.
\tag{7.8}\label{eq:U}
\]
Because $A_i\cap D_i$ is removed from $X$ (it lies in $D$), we have
$A_i^\circ=A_i-(A_i\cap D_i)$, and
\[
Y-D-\bigcup_iA_i=Y-\Bigl(\sum_iD_i+\sum_iA_i\Bigr)
=Y-\bigcup_ib^{-1}(H_i).
\tag{7.9}\label{eq:Y-minus}
\]
The morphism $b:Y\to S$ is an isomorphism away from the blowup centres, and
the centres lie on the curves $H_i$. So $b$ restricts to an isomorphism
\[
U\xrightarrow{\ \sim\ }S\setminus\bigcup_iH_i
=E\times(\PP^1\setminus Q),\qquad Q=\{q_1,\ldots,q_n\}.
\tag{7.10}\label{eq:product-open}
\]

\paragraph{Meridians of the retained curves.} Each $A_i^\circ$ is a closed
smooth complex subvariety of $X$, of complex codimension one (real
codimension two). Standard topology gives: the inclusion $U\hookrightarrow X$
induces a surjection on fundamental groups, and its kernel is the normal
closure of the meridians around the $A_i^\circ$ (loops in the boundary of a
small tubular neighborhood that link the divisor once).
For completeness we recall the argument. A loop in $X$ can be perturbed off
$A_i^\circ$ (a real codimension-two submanifold); a null-homotopy of the
perturbed loop can be made transverse to $A_i^\circ$, relative to the
boundary of its parameter disc, so it meets each $A_i^\circ$ in finitely
many interior points. Removing a small disc around each such point rewrites
the loop as a product of conjugates of meridians. So the kernel is as
stated; this uses only compactness of the parameter disc, not compactness
of the subvariety.

\paragraph{The meridian is a puncture loop.} To identify the meridians
explicitly, choose local coordinates on $S$ near $(e_i,q_i)$: on the
neighborhood $S\supset\{|e-e_i|<\varepsilon\}\times\{|q-q_i|<\varepsilon\}$,
set $x=e-e_i$ and $t=q-q_i$. The centre $p_i$ is $(x,t)=(0,0)$. In the
blowup chart $u=x/t$, the point of $A_i^\circ$ corresponds to $(u,t)=(u,0)$,
and
\[
b(u,t)=(ut,t),\qquad A_i=\{(u,0):u\in\PP^1\},\quad
A_i^\circ=\{u\in\AA^1,t=0\}.
\tag{7.11}\label{eq:blowup-chart}
\]
The strict transform $D_i$ of $H_i=\{t=0\}$ does not meet this chart (it
sits in the complementary chart, meeting $A_i$ at the point $u=\infty$).
For any fixed $u_0\in\AA^1$ and small |t|, the disc
$\{u=u_0,|t|\leq\varepsilon\}$ is a disc transverse to $A_i^\circ$ and
contained in $X$. Its boundary loop $t=\varepsilon e^{2\pi i\theta}$ is a
meridian of $A_i^\circ$. Under \eqref{eq:product-open} it has fixed elliptic
coordinate $u_0\varepsilon e^{2\pi i\theta}$ and winds once around $q_i$
in the second factor. This is a conjugate of the standard puncture loop
$m_i$ of $\PP^1\setminus Q$.

\paragraph{The fundamental group.} Puncture loops $m_1,\ldots,m_n$ of
$\PP^1\setminus Q$ generate its fundamental group with the single relation
$m_1\cdots m_n=1$ (as an oriented positively-configured set around the
punctures). Using the product formula \eqref{eq:product-open},
\[
\pi_1(U)=\pi_1(E)\times\pi_1(\PP^1\setminus Q)
=\Z^2\times F_{n-1},
\tag{7.12}\label{eq:pi1-U}
\]
where $F_{n-1}$ is free of rank $n-1$. Each $m_i$ generates a free summand
of the second factor; killing all conjugates of $m_1,\ldots,m_n$ kills
$F_{n-1}$ entirely. Therefore
\[
\pi_1(X)=\frac{\pi_1(U)}{\langle\!\langle m_1,\ldots,m_n\rangle\!\rangle}
=\frac{\Z^2\times F_{n-1}}{1\times F_{n-1}}\simeq\Z^2.
\tag{7.13}\label{eq:pi1-X}
\]
Under the product identification \eqref{eq:product-open}, the map $a$ is
the projection to the elliptic factor, so its induced map on $\pi_1$ is
the canonical projection $\Z^2\to\Z^2$, i.e. an isomorphism. The section
\eqref{eq:section} also shows directly that $a_*$ is split injective.

\paragraph{The fibre's group dies.} The general fibre
$F=\PP^1\setminus Q$ has fundamental group free of rank $n-1$. Every
generator of $\pi_1(F)$ is a word in the puncture loops $m_1,\ldots,m_n$,
and every puncture loop is trivial in $\pi_1(X)$ by \eqref{eq:pi1-X}. So
the map $\pi_1(F)\to\pi_1(X)$ is the zero map. This proves
Theorem~\ref{thm:example}(iii).

The example also makes clear what completion alone controls. The group
$\pi_1(F)$ is the nonabelian free group $F_{n-1}$, but its image in
$\pi_1(X)=\Z^2$ is trivial, and $\pi_1(X)$ satisfies the class-two bound.
The completion theorem addresses the image in each canonical unipotent
quotient; control of the completion's kernel, invisible to all such
quotients, is what remains open in Section~\ref{sec:ordinary}.

\subsection{The general fibre is not special}\label{sec:specialness-example}

For a smooth quasi-projective curve $C=\overline C\setminus B$ (with
$\overline C$ a smooth projective curve and $B$ a finite set of punctures),
Campana's criterion says specialness is equivalent to
\[
\deg(K_{\overline C}+B)\leq 0.
\tag{7.14}\label{eq:curve-criterion}
\]
For the general fibre of $a$, we have $\overline F=\PP^1$ and $B=Q$ with
$\#Q=n\geq 3$. So
\[
\deg(K_{\PP^1}+Q)=-2+n=n-2>0.
\tag{7.15}\label{eq:F-degree}
\]
By \eqref{eq:curve-criterion}, $F$ is not special. This completes the
statement of Theorem~\ref{thm:example}: a smooth special quasi-projective
surface whose general quasi-Albanese fibre is not special.