Campana's theory of special varieties relates the absence of fibrations
of general type to restrictions on topology and monodromy. Let $X$ be a
connected smooth complex quasi-projective variety with a smooth
projective compactification $\bar X$ whose boundary $D=\bar X - X$ is a
reduced simple normal crossing divisor. Let $f : \bar X \to Z$ be a
fibration, meaning a surjective algebraic map with connected general
fibre. For a prime divisor $Q$ in the smooth projective base $Z$, the
orbifold multiplicity is
$$ m_f(Q) = \inf_{f(P)=Q} r(f,P) m_P(D), $$
where the infimum ranges over prime divisors $P$ of $\bar X$ that map onto
$Q$, the integer $r(f,P)$ is the multiplicity of $P$ in the divisor
$f^*Q$, and $m_P(D)$ is the multiplicity of $P$ in the boundary. Removed
components carry $m_P(D) = \infty$; components outside the boundary carry
$m_P(D) = 1$. The orbifold base divisor on $Z$ is
$$ \Delta(f, D) = \sum_Q (1 - 1/m_f(Q))\, Q. $$
The quotient $1/\infty$ is 0, so boundary components contribute coefficient
one. A fibration is of general type when the Kodaira dimension of
$K_Z + \Delta(f,D)$ equals $\dim Z$. The pair $(\bar X, D)$ is special when
every fibration from every smooth projective birational model of
$(\bar X, D)$, with the same boundary and every allowed elementary
birational orbifold morphism, has orbifold base of Kodaira dimension less
than the base dimension. This pair convention and the open-variety
convention of Cadorel, Deng, and Yamanoi are compatible on smooth
compactifications of the same open variety.
./pdf/campana-2004-orbifolds-special-classification.pdf
./pdf/campana-2011-orbifoldes-geometriques-speciales.pdf

The linear monodromy of the ordinary topological fundamental group of a
special quasi-projective variety is bounded at nilpotency class two. For a
group $G$ put $\gamma_1 G = G$ and $\gamma_{r+1} G = [G, \gamma_r G]$. The
class is at most two when $\gamma_3 G = 1$, equivalently when every
commutator $[a,b] = a b a^{-1} b^{-1}$ is central. A property holds
virtually for $G$ when it holds in a subgroup of finite index.

Theorem 1 (two-step linear monodromy). For every connected smooth special
complex quasi-projective variety $V$, every positive integer $d$, and
every representation $\rho : \pi_1(V) \to \mathrm{GL}_d(\mathbb{C})$, the
image $\rho(\pi_1(V))$ has a subgroup of finite index and nilpotency class
at most two.

Cadorel, Deng, and Yamanoi showed that the identity component of the
Zariski closure of $\rho(\pi_1(V))$ is a product of a unipotent group and
a torus, hence virtually nilpotent, but left the class of the unipotent
part unconstrained. The class-two bound was subsequently announced by Cao,
Deng, Hacon, and Paun on the strength of a different method. The
argument uses a structural completion theorem that does not depend on
specialness of the fibre.
./pdf/cdy-2025-hyperbolicity-fundamental-groups-III.pdf
./pdf/cdhp-2026-two-step-nilpotent-monodromy.pdf

For contrast, in the compact Kahler case the abelianity conjecture of
Campana predicts virtual abelianity, not just class two; the conjecture
is Campana's [Conjecture 7.1, p. 595]. Campana and Claudon proved it for
special compact Kahler threefolds. The open quasi-projective setting
admits genuinely nonabelian two-step nilpotent groups, and Cadorel, Deng,
and Yamanoi exhibit one: the complement of the zero section in a
degree-one line bundle on an elliptic curve has a Heisenberg central
extension as its fundamental group, with $K+D = 0$ on the ruled
compactification.
./pdf/campana-2004-orbifolds-special-classification.pdf
./pdf/campana-claudon-2014-abelianity-threefolds.pdf
./pdf/cdy-2025-hyperbolicity-fundamental-groups-III.pdf

Theorem 2 (completion under quasi-Albanese exactness). Let $V$ be a
connected smooth complex quasi-projective variety with base point $v$ and
algebraic quasi-Albanese map $a : V \to A = \mathrm{Alb}(V)$. Suppose $a$
is dominant. Choose a smooth connected general fibre $F$ through $v$, and
suppose the sequence of ordinary fundamental groups
$$ \pi_1(F,v) \longrightarrow \pi_1(V,v) \xrightarrow{a_*} \pi_1(A,a(v))
\longrightarrow 1 $$
is exact. Then every canonical rational unipotent quotient of
$\pi_1(V,v)$ has nilpotency class at most two; the full rational unipotent
completion is finite-dimensional and of class at most two; and every
finite-dimensional complex unipotent representation has image of class at
most two.

The quasi-Albanese map is the universal algebraic morphism from $V$ to a
semiabelian variety, in the sense of Fujino's construction. Exactness at
the first arrow identifies the image of the fibre group with the entire
kernel of $a_*$; it does not require the fibre group to inject. No
specialness of $V$, no specialness of $F$, and no prior nilpotence of
$\pi_1(V)$ is assumed. The proof uses two geometric facts: the
quasi-Albanese map induces an isomorphism $H_1(V,\mathbb{Q}) \cong
H_1(A,\mathbb{Q})$ modulo torsion, and the algebraic inclusion of the
smooth fibre $F \hookrightarrow V$ is a morphism of pointed mixed Hodge
structures.
./pdf/fujino-2024-quasi-albanese-maps.pdf

A second consequence, when the logarithmic Kodaira dimension of $V$
vanishes, is virtual class-two monodromy for every linear representation.

A different question concerns the general Albanese fibre of a compact
special variety: Campana asked whether it is special, and Campana and
Claudon answered in the affirmative for smooth projective varieties.
With reduced open boundary the analogous statement fails, and the
following open surfaces exhibit the failure.

Theorem 3 (special surface with nonspecial fibre). Let $E$ be a complex
elliptic curve, let $q_1,\dots,q_n \in \mathbb{P}^1$ be distinct with
$n \geq 3$, and choose arbitrary points $e_1,\dots,e_n \in E$. Let
$$ b : Y = \mathrm{Bl}_{\{(e_i,q_i)\}}(E \times \mathbb{P}^1)
\longrightarrow E \times \mathbb{P}^1 $$
be the blowup at the $n$ distinct points. Let $D_i$ be the strict
transform of $E \times \{q_i\}$, let $D = \sum_i D_i$, and let
$X = Y - D$. Then:
(i) the pair $(Y,D)$ is special in the orbifold-pair sense: on every
smooth projective birational model carrying the same reduced boundary,
every fibration has orbifold base of Kodaira dimension less than the
base dimension;
(ii) the morphism $a = (\mathrm{pr}_E \circ b)|_X : X \to E$ is the
algebraic quasi-Albanese map after choosing origins, every fibre of $a$ is
connected, and the general fibre is $F \simeq \mathbb{P}^1 -
\{q_1,\dots,q_n\}$;
(iii) the induced map $a_* : \pi_1(X) \to \pi_1(E)$ is an isomorphism, so
$\pi_1(X) \simeq \mathbb{Z}^2$ and the map $\pi_1(F) \to \pi_1(X)$ is the
zero map.

The general fibre $F$ has logarithmic canonical degree $n - 2 > 0$, so $F$
is not special. By (i) and the pair-to-open comparison, the total space
$X$ is special. A special smooth quasi-projective surface therefore has a
general quasi-Albanese fibre that is not special, and the fibre-
specialness assertion of Cao, Deng, Hacon, and Paun fails on this example.
The fibre's ordinary group is free of rank $n-1$, but its image in
$\pi_1(X) = \mathbb{Z}^2$ is trivial, so the total space still meets the
monodromy bound of Theorem 1.

The geometric mechanism behind the failure lies in the orbifold
multiplicities. At each $q_i$ the complete scheme fibre of
$j = \mathrm{pr}_{\mathbb{P}^1} \circ b$ is the union $D_i + A_i$ with
both components of multiplicity one; the curve $A_i$ is the exceptional
$\mathbb{P}^1$ introduced by the blowup. The retained $A_i$ makes the
direct orbifold multiplicity at $q_i$ equal to one, whereas contracting
$A_i$ first discards its contribution. Campana's composition rule records
exactly the exceptional-divisor hypothesis under which the direct and
iterated bases agree. The compound passage in the CDHP proof of Claim 7.17
treats flatness over the Iitaka base $J$ as flatness over a contracted
intermediate space, and it is the latter that fails.

The proof of Theorem 2 is a mixed-Hodge-theoretic weight argument on the
canonical quotients of the rational unipotent completion, and the proof of
Theorem 1 assembles the Cadorel-Deng-Yamanoi geometric inputs with
Theorem 2 on a finite-etale cover.

Proof of Theorem 2. Set $\Gamma = \pi_1(V,v)$, $\Lambda = \pi_1(A,a(v))$,
and let $N$ be the image of $\pi_1(F,v)$ in $\Gamma$. Exactness of the
sequence in Theorem 2 identifies $N$ with the entire kernel of $a_*$. The
quasi-Albanese map induces an isomorphism
$$ H_1(V,\mathbb{Z}) / \mathrm{Tor}\, H_1(V,\mathbb{Z})
\xrightarrow{\ \sim\ } H_1(A,\mathbb{Z}), $$
so $\Lambda$ is free abelian and equals $H_1(A,\mathbb{Z})$. Two
consequences follow: every commutator of $\Gamma$ lies in $N$, and the
quotient $N/[\Gamma,\Gamma] \simeq \mathrm{Tor}\, H_1(V,\mathbb{Z})$ is
finite. Consequently every element of $N$ has a positive power in
$[\Gamma,\Gamma]$; when the torsion vanishes, $N = [\Gamma,\Gamma]$.
./pdf/fujino-2024-quasi-albanese-maps.pdf

Let $\mathfrak{m}_V$ be the complete pronilpotent Lie algebra of the
rational unipotent completion of $\Gamma$. For $c \geq 1$, the canonical
$c$-step quotient is
$$ \mathfrak{g}_c = \mathfrak{m}_V / \overline{\gamma_{c+1} \mathfrak{m}_V},
\qquad U_c = \exp(\mathfrak{g}_c), $$
where $\gamma_1 = \mathfrak{m}_V$, $\gamma_{r+1} = [\mathfrak{m}_V,
\gamma_r]$, and closure is taken in the inverse-limit topology. Since
$\Gamma$ is finitely generated, $\mathfrak{g}_c$ is finite-dimensional,
the image of $\Gamma$ in $U_c$ is Zariski dense, and
$\mathfrak{g}_c / [\mathfrak{g}_c, \mathfrak{g}_c] \simeq H_1(V,
\mathbb{Q})$. All these structures are canonical and are compatible with
morphisms induced by based algebraic maps of the underlying varieties.
./pdf/hain-1987-derham-homotopy-I.pdf

Recall the terminology of mixed Hodge structures. A rational mixed Hodge
structure is a finite-dimensional $\mathbb{Q}$-vector space with
increasing weight filtration $W_\bullet$ and decreasing Hodge filtration
on the complexification, whose graded pieces are pure. It has weights at
most $r$ when $W_r$ is the entire space; it is pure of weight $r$ when
additionally $W_{r-1} = 0$. Morphisms are strict: the image of a morphism
receives the filtration it has as a subquotient of the source. Tensor
products add weights. For smooth quasi-projective $V$, the first homology
$H_1(V,\mathbb{Q})$ carries weights $-1$ and $-2$, and for the smooth
general fibre $F$ the same bound holds; weights add under tensor product.
./pdf/deligne-1971-theorie-de-hodge-II.pdf

Hain's construction assigns to $\mathfrak{m}_V$ a pro-mixed Hodge
structure compatible with the bracket, in such a way that the closed
lower central ideals $\overline{\gamma_r \mathfrak{m}_V}$ are subobjects
and every $\mathfrak{g}_c$ inherits an ordinary mixed Hodge structure.
Based algebraic morphisms induce morphisms of these structures: in
particular the algebraic inclusion $F \hookrightarrow V$ yields a
morphism $\mathfrak{g}_c(F) \to \mathfrak{g}_c(V)$. Morgan's foundational
version had a functoriality assertion that was later corrected; the
pointed statement required is due to Hain.
./pdf/morgan-1978-alg-topology-smooth-alg-var.pdf
./pdf/morgan-1986-corrections.pdf

For fixed $c$, let $K_c \subseteq U_c$ be the Zariski closure of the
image of $N$, and let $\mathfrak{k}_c = \mathrm{Lie}\, K_c$. The morphism
$\mathfrak{g}_c(F) \to \mathfrak{g}_c(V)$ has image exactly
$\mathfrak{k}_c$: the image of a morphism of unipotent algebraic groups
is closed, and $\pi_1(F,v)$ is Zariski dense in $U_c(F)$.

Lemma. $K_c = [U_c,U_c]$ and $\mathfrak{k}_c = [\mathfrak{g}_c,
\mathfrak{g}_c]$.

Proof. Let $n \in N$. Some positive power $n^k$ lies in $[\Gamma,\Gamma]$.
The abelianization $U_c / [U_c,U_c]$ is an additive vector group and
carries no nontrivial torsion, so every $n$ maps to the identity; hence
$K_c \subseteq [U_c,U_c]$. Conversely $N$ is normal in $\Gamma$, and
$\Gamma \to U_c$ has Zariski-dense image, so $K_c$ is normal in $U_c$.
The image of $\Gamma$ in $U_c/K_c$ is dense and abelian because every
commutator of $\Gamma$ lies in $N$. The commutator morphism of $U_c/K_c$
therefore vanishes on a dense subset of its product and so vanishes
identically; hence $[U_c,U_c] \subseteq K_c$. Both inclusions give
$K_c = [U_c,U_c]$, and passing to Lie algebras yields the claim.

The first weight bound comes from the lower central series. For $r \geq
1$ the $r$-fold iterated bracket gives a surjective morphism of mixed
Hodge structures
$$ H_1(V,\mathbb{Q})^{\otimes r} \longrightarrow
\gamma_r \mathfrak{g}_c / \gamma_{r+1} \mathfrak{g}_c. $$
The source has weights at most $-r$; hence the target does as well. By
downward induction along the lower central filtration, $\gamma_r
\mathfrak{g}_c$ itself has weights at most $-r$. In particular $W_{-1}
\mathfrak{g}_c = \mathfrak{g}_c$ and, using $[\mathfrak{g}_c,
\mathfrak{g}_c] = \mathfrak{k}_c$, also $\mathfrak{k}_c \subseteq
W_{-2} \mathfrak{g}_c$.

The second bound uses $B_c = \mathfrak{k}_c / [\mathfrak{k}_c,
\mathfrak{k}_c]$. The composition $\mathfrak{g}_c(F) \to \mathfrak{k}_c
\to B_c$ factors through the abelianization and gives a surjective
mixed-Hodge morphism
$$ H_1(F,\mathbb{Q}) \longrightarrow B_c. $$
Strictness gives $W_{-3} B_c = 0$, while $B_c$ is a quotient of
$\mathfrak{k}_c \subseteq W_{-2} \mathfrak{g}_c$, so $W_{-2} B_c = B_c$.
Therefore $B_c$ is pure of weight $-2$.

The bracket followed by projection to $B_c$,
$$ \mathfrak{g}_c \otimes \mathfrak{k}_c \longrightarrow
\mathfrak{k}_c \longrightarrow B_c, $$
is a morphism of mixed Hodge structures. The source has weights at most
$-3$ and the target has $W_{-3} = 0$, so the morphism vanishes. Hence
$[\mathfrak{g}_c, \mathfrak{k}_c] \subseteq [\mathfrak{k}_c,
\mathfrak{k}_c]$. Combining with the lemma and the estimate
$[\gamma_2, \gamma_2] \subseteq \gamma_4 \subseteq \gamma_3 =
[\mathfrak{g}_c, \gamma_2]$,
$$ \gamma_3 \mathfrak{g}_c = [\mathfrak{g}_c, \mathfrak{k}_c]
\subseteq [\mathfrak{k}_c, \mathfrak{k}_c]
\subseteq \gamma_4 \mathfrak{g}_c \subseteq \gamma_3 \mathfrak{g}_c, $$
so $\gamma_3 \mathfrak{g}_c = \gamma_4 \mathfrak{g}_c$. Bracketing
further with $\mathfrak{g}_c$ propagates the equality along the lower
central series: $\gamma_k \mathfrak{g}_c = \gamma_3 \mathfrak{g}_c$ for
every $k \geq 3$. The $c$-step quotient satisfies $\gamma_{c+1}
\mathfrak{g}_c = 0$ by construction, so $\gamma_3 \mathfrak{g}_c =
\gamma_{c+1} \mathfrak{g}_c = 0$.

Now pass to $\mathfrak{m}_V = \varprojlim_c \mathfrak{g}_c$. Every
element of $\gamma_3 \mathfrak{m}_V$ maps to $\gamma_3 \mathfrak{g}_c =
0$ for every $c$, hence lies in the intersection of all kernels, which
vanishes by separatedness. So $\gamma_3 \mathfrak{m}_V = 0$. The
canonical tower stabilizes at $c = 2$, $\mathfrak{m}_V \cong \mathfrak{g}_2$
is finite-dimensional, its abelianization is $H_1(V,\mathbb{Q})$, and
its commutator is a quotient of $\wedge^2 H_1(V,\mathbb{Q})$.
Baker-Campbell-Hausdorff carries the conclusion to the unipotent
algebraic group $U = \exp(\mathfrak{m}_V)$, which is finite-dimensional
and of class at most two.

Complex unipotent representations: for a finitely generated $\Gamma$,
unipotent completion commutes with extension of the ground field from
$\mathbb{Q}$ to $\mathbb{C}$. Any finite-dimensional complex unipotent
representation $\rho : \Gamma \to V'$ factors uniquely through
$U_\mathbb{C} = U \otimes_\mathbb{Q} \mathbb{C}$, and the image of $\rho$
is a quotient of a class-two unipotent group, hence of class at most two.
./pdf/hain-2015-genus-3-mcg-not-kahler.pdf

Two remarks. If $H_1(F,\mathbb{Q})$ is pure of weight $-1$, as happens
when $F$ is proper, the same argument gives $B_c = 0$, hence
$\mathfrak{k}_c = 0$, and the completion is abelian; this recovers the
abelianity theorem of Aguilar Aguilar and Campana for quasi-Albanese maps
that are proper and surjective. If $A$ is a point or $F$ is a point, the
conclusion holds without division by any positive Betti number.
./pdf/aguilar-campana-2025-nilpotent-quasi-albanese.pdf

In the compact Kahler case, Campana proved by a homological criterion of
Stallings that surjectivity of the Albanese map forces every torsion-free
nilpotent quotient of the fundamental group to be abelian; the passage
from compact to open pulls in a boundary contribution of weight $-2$
that produces the class-two conclusion and no higher. Rogov
obtained a companion result in the direction of higher Albanese
manifolds, starting from dominance of the $s$th higher Albanese map for
some $s \geq 3$ or from a definable biholomorphism, and proving
stabilization at level two; the argument for Theorem 2 starts instead
from ordinary fundamental-group exactness for the classical quasi-Albanese
map.
./pdf/campana-1995-kahler-nilpotents.pdf
./pdf/rogov-2026-o-minimal-higher-albanese.pdf

Proof of Theorem 1. Let $L$ be the Zariski closure of $\rho(\pi_1(V))$
and $L^0$ its identity component. The subgroup
$\Gamma_0 = \rho^{-1}(L^0)$ has finite index in $\pi_1(V)$. By the
algebraic form of the Riemann existence theorem, $\Gamma_0$ corresponds
to a connected finite etale cover $V_0 \to V$ with $\pi_1(V_0) =
\Gamma_0$.
./pdf/sga1-revetements-etales-groupe-fondamental-2004.pdf

By the Cadorel-Deng-Yamanoi decomposition, $L^0 = U \times T$ with $U$
unipotent and $T$ a torus. The restricted representation
$\rho|_{\Gamma_0}$ takes values in $U \times T$; projection to $U$ gives
a unipotent representation $\rho_U : \Gamma_0 \to U$. A connected finite
etale cover of a smooth special quasi-projective variety is smooth,
special, and quasi-projective, so the Cadorel-Deng-Yamanoi geometric
inputs apply to $V_0$ directly. Its quasi-Albanese map is dominant, has
smooth connected general fibre, and satisfies the ordinary exact
sequence of Theorem 2. Theorem 2 then gives that $\rho_U(\Gamma_0)$ has
class at most two. The projection to $T$ is abelian, so every triple
commutator of $\rho(\Gamma_0)$ is trivial in both factors of $U \times
T$, and is itself trivial. Since $\rho(\Gamma_0)$ has finite index in
$\rho(\pi_1(V))$, the conclusion of Theorem 1 follows.
./pdf/cdy-2025-hyperbolicity-fundamental-groups-III.pdf

Reapplying exactness on $V_0$ rather than on $V$ is essential: unipotent
completion need not be unchanged under passage to a finite-index
subgroup, so Theorem 2 has to be applied directly to the group
$\Gamma_0$ on which $\rho$ is unipotent-valued. The proof uses no
fibre-specialness and no mixed Hodge structure on an arbitrary linear
quotient; the weight argument was carried out on the canonical quotients
in the proof of Theorem 2.

Log-Kodaira-zero corollary. Let $V$ be connected smooth complex
quasi-projective with $\bar\kappa(V) = 0$ and $\rho : \pi_1(V) \to
\mathrm{GL}_d(\mathbb{C})$ any representation. The same three
conclusions hold: $\rho(\pi_1(V))$ is virtually of class at most two;
the full rational unipotent completion of $\pi_1(V)$ is
finite-dimensional and of class at most two; and every
finite-dimensional complex unipotent representation has image of class
at most two.

Proof. Compactify $V$ to a smooth projective $Y$ with reduced normal-
crossing $D$. By the definition of $\bar\kappa$, $\kappa(Y, K_Y + D) =
0$. Campana's theorem on orbifold pairs, applied to the reduced pair
$(Y,D)$, gives specialness, and the comparison of conventions
establishes the open-variety specialness of $V$. Theorem 1 gives the
linear-image assertion. The Cadorel-Deng-Yamanoi quasi-Albanese inputs
applied directly to $V$ give the exactness hypothesis of Theorem 2,
which yields the completion assertions and every complex unipotent
image.
./pdf/campana-2011-orbifoldes-geometriques-speciales.pdf

An orbifold pair packages a smooth variety together with a $\mathbb{Q}$-
divisor
$$ D = \sum_P (1 - 1/m_P(D))\, P, \qquad
m_P(D) \in \mathbb{Z}_{\geq 1} \cup \{\infty\}, $$
with only finitely many nonzero coefficients. The convention
$1/\infty = 0$ gives a component of infinite multiplicity boundary
coefficient one and a component outside the boundary zero. For a
fibration $f : Y \to Z$ with $Y$ smooth projective and $Z$ smooth, and a
prime divisor $Q \subset Z$, the orbifold multiplicity and base divisor
are
$$ m_f(Q) = \inf_{f(P)=Q} r(f,P)\, m_P(D), \qquad
\Delta(f,D) = \sum_Q (1 - 1/m_f(Q))\, Q, $$
where $r(f,P)$ is the multiplicity of $P$ in $f^* Q$. A divisor $P$
with $\mathrm{codim}_Z f(P) \geq 2$ is $f$-exceptional and is omitted
from the immediate infimum, though it may enter a later composition. The
infimum convention is the one fixed by Cao, Deng, Hacon, and Paun.
./pdf/cdhp-2026-two-step-nilpotent-monodromy.pdf

For the specific surface, the fibre identification $j^*(q_i) = D_i + A_i$
gives
$$ m_j(q_i) = \min(1 \cdot \infty, 1 \cdot 1) = 1, $$
the first term from the deleted component $D_i$ with multiplicity
$\infty$ in $D$ and the second from the retained exceptional $A_i$ with
multiplicity $1$. Similarly every $t \notin \{q_i\}$ receives
multiplicity one from the unchanged complete elliptic fibre. Hence
$$ \Delta(j, D) = 0, $$
so the orbifold base of $j$ is $(\mathbb{P}^1, 0)$ even though $\kappa(Y,
K_Y + D) = 1$.

A dominant morphism $u : (Y', D') \to (Y, D)$ of smooth orbifold pairs is
an orbifold morphism when, for every prime divisor $P \subset Y$ and
every component $P'$ of $u^{-1}(P)$ with multiplicity $r$ in $u^* P$,
$$ r\, m_{P'}(D') \geq m_P(D). $$
It is an elementary birational orbifold morphism when additionally $u$ is
birational and $u_* D' = D$. An upper model of $(Y, D)$ is a smooth
projective orbifold pair $(Y', D')$ equipped with an elementary
birational orbifold morphism $u : (Y', D') \to (Y, D)$.

A fibration $f : (Y, D) \to Z$ is neat when it fits into a commutative
birational diagram with smooth spaces and horizontal fibrations, where
the source modification $w$ is an elementary birational orbifold
morphism and every $f$-exceptional divisor is also $w$-exceptional.
Neatness gives the equality
$$ \kappa(f, D) = \kappa(Z, K_Z + \Delta(f, D)), $$
which identifies the canonical dimension of $f$ with the Kodaira
dimension of the orbifold base $K_Z + \Delta(f, D)$.

Definition (pair specialness). A smooth projective orbifold pair $(Y, D)$
is special when, for every elementary birational orbifold morphism $u :
(Y', D') \to (Y, D)$ with $(Y', D')$ smooth projective, and every neat
fibration $g : (Y', D') \to Z$ with $\dim Z > 0$,
$$ \kappa(Z, K_Z + \Delta(g, D')) < \dim Z. $$

The pair convention implies the open-variety convention of Cadorel,
Deng, and Yamanoi. Start with an open fibration $g : X \to B$ and a
proper birational modification allowed in that convention. Compactify
over $(Y, D)$, resolve the graph of the fibration, and choose a neat
upper model $(Y', D') \to (Y, D)$. 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 $D_{\mathrm{open}} \leq D_{\mathrm{high}}$ on components, and
the infimum formula then gives
$$ \Delta(g', D_{\mathrm{open}}) \leq \Delta(g', D_{\mathrm{high}}) $$
on the same resolved fibration $g' : Y' \to B'$. Kodaira dimension is
monotone in the divisor, so
$$ \kappa(B', K_{B'} + \Delta(g', D_{\mathrm{open}})) \leq
\kappa(B', K_{B'} + \Delta(g', D_{\mathrm{high}})) < \dim B' $$
by pair specialness. The infimum over models in the open-variety
definition is at most this value.
./pdf/cdy-2025-hyperbolicity-fundamental-groups-III.pdf

For a smooth quasi-projective curve $C = \bar C - B$, specialness
reduces to $\deg(K_{\bar C} + B) = 2g(\bar C) - 2 + \#B \leq 0$. The
general fibre $F \simeq \mathbb{P}^1 - Q$ of $a$ has $g(\bar F) = 0$ and
$\#Q = n$, so $\deg(K_{\bar F} + Q) = n - 2$. For $n \geq 3$ this is
positive, and $F$ is not special.

Specialness on every upper model. Fix an 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 $\dim Z > 0$. Since
$u_* D' = D$ 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
$\infty$) or a $u$-exceptional curve. Let $\Sigma \subset Y$ be the
finite set of images of exceptional curves; on $Y' - u^{-1}(\Sigma)$ the
map $u$ restricts to an isomorphism onto $Y - \Sigma$. The strict
transform $A_i'$ of $A_i$ is not in the support of $D'$ and carries
multiplicity $1$.

Case 1, $\dim Z = 2$. Since the general fibre is connected, $g$ is
birational. Comparing $g$ with the identity fibration on $(Y', D')$,
$$ \kappa(g, D') \leq \kappa(Y', K_{Y'} + D'). $$
Choose $m \geq 1$ making $m D'$ integral. Pushforward along $u$
identifies $u_*(m(K_{Y'} + D'))|_{Y - \Sigma} = m(K_Y + D)|_{Y - \Sigma}$.
Restricting a global section $s \in H^0(Y', m(K_{Y'} + D'))$ to
$Y' - u^{-1}(\Sigma)$, pushing forward to $Y - \Sigma$, and extending
across the finite set $\Sigma$ by Hartogs (a section of a line bundle on
a smooth surface over the complement of a finite set extends), gives an
injection
$$ H^0(Y', m(K_{Y'} + D')) \lhook\joinrel\longrightarrow H^0(Y, m(K_Y + D)). $$
The right side grows linearly in $m$ by the identification $H^0(Y,
m(K_Y+D)) \simeq H^0(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}(m(n-2)))$.
The left side therefore grows at most linearly, so $\kappa(Y', K_{Y'} +
D') \leq 1$. Neatness gives $\kappa(Z, K_Z + \Delta(g, D')) =
\kappa(g, D') \leq 1 < 2 = \dim Z$.

Case 2, $\dim Z = 1$. Then $Z = C$ is a smooth projective curve, and it
suffices to rule out $\deg(K_C + \Delta(g, D')) > 0$. Choose $t \in
\mathbb{P}^1$ avoiding $\{q_i\}$ and the images under $j$ of the finitely
many exceptional curves of $u$. Then $C_t := (j \circ u)^{-1}(t)$ is a
smooth complete elliptic curve in $Y'$, disjoint from $\Supp D'$ and from
the exceptional locus of $u$.

Suppose $h = g|_{C_t} : C_t \to C$ is nonconstant. Then $h$ is finite of
degree $d > 0$. For each $c \in C$ let $m_c = m_g(c)$ be the orbifold
multiplicity. Every component $P$ of $g^* c$ meeting $C_t$ lies outside
$\Supp D'$, so $m_P(D') = 1$ and $r(g, P) \geq m_c$. For $x \in C_t$
above $c$, the ramification index $e_x \geq m_c$. If $m_c = \infty$, this
is impossible since $e_x$ is finite; hence every $m_c$ is finite under
nonconstancy. For finite $m_c$,
$$ e_x - 1 \geq e_x (1 - 1/m_c). $$
Summing over $c \in \Supp \Delta$ and applying Riemann-Hurwitz,
$$ 0 = \deg K_{C_t} = d \deg K_C + \sum_{x \in C_t} (e_x - 1) \geq
d \deg(K_C + \Delta). $$
Since $d > 0$, this forces $\deg(K_C + \Delta) \leq 0$, contradicting
positivity.

So $g$ is constant on the fibre of $j \circ u$ over every sufficiently
general $t$, hence on a dense union of complete fibres. Descent on
function fields gives a rational map $r : \mathbb{P}^1 \dashrightarrow
C$ with $g = r \circ j \circ u$ on a dense open. Since $\mathbb{P}^1$ is
smooth projective, $r$ extends to a morphism. Connectivity of the general
fibre of $g$ forces $\deg r = 1$, hence $C \simeq \mathbb{P}^1$ and $g =
j \circ u$ up to isomorphism.

The boundary of this possible curve base is computed on the arbitrary
model $Y'$. The strict transform $A_i'$ of $A_i$ has coefficient zero in
$D'$. Since $u$ is an isomorphism at the generic point of $A_i'$, the
multiplicity of $A_i'$ in $(j \circ u)^* q_i$ equals the multiplicity of
$A_i$ in $j^* q_i$, which is one. So
$$ m_{j \circ u}(q_i) \leq r(j \circ u, A_i')\, m_{A_i'}(D') =
1 \cdot 1 = 1. $$
The coefficient of $q_i$ in $\Delta(j \circ u, D')$ is $1 - 1/1 = 0$, and
for every $t \notin \{q_i\}$ the same computation with the strict
transform of $E \times \{t\}$ gives coefficient zero. Adding boundary on
further exceptional components can only decrease the infimum, which is
already one. Therefore
$$ \Delta(j \circ u, D') = 0 $$
and $\deg K_{\mathbb{P}^1} = -2 < 0$, contradicting positivity.

Both possible base dimensions are excluded, so $(Y, D)$ is special in the
pair sense on every upper model.

Quasi-Albanese identification. Choose $q_* \in \mathbb{P}^1 -
\{q_1,\dots,q_n\}$. The complete elliptic curve $E \times \{q_*\}$ is
disjoint from every $H_i$ and from every blowup centre, so it lies in $X$
and gives a section
$$ s : E \hookrightarrow X, \qquad a \circ s = \mathrm{id}_E. $$

Units lemma. Every invertible regular function on $X$ is constant.

Proof. A unit $v \in \Gamma(X, \mathcal{O}_X)^*$ has rational extension
to $Y$ whose divisor $\mathrm{div}_Y(v) = \sum_i b_i D_i$ is supported on
$D$. Intersecting with $A_j$ and using $D_i \cdot A_j = \delta_{ij}$,
$$ 0 = \mathrm{div}_Y(v) \cdot A_j = b_j $$
because $\mathrm{div}_Y(v)$ is principal and has degree zero on every
complete curve. Hence $\mathrm{div}_Y(v) = 0$, the function $v$ is
nowhere-vanishing on $Y$, and $v$ is constant on the connected
projective variety $Y$.

Fujino's quasi-Albanese universal property identifies $a$ with $a_X$.
Let $f : X \to G$ be an algebraic morphism to a semiabelian variety $G$
fitting into
$$ 1 \to T \to G \xrightarrow{\rho} B \to 1, \qquad
T \simeq \mathbb{G}_m^r, $$
with $B$ abelian. For $e \in E - \{e_1,\dots,e_n\}$ the fibre
$a^{-1}(e)$ is the punctured line $\mathbb{P}^1 - \{q_1,\dots,q_n\}$,
contained in the complete curve $\{e\} \times \mathbb{P}^1 \subset Y$. So
$\rho \circ f$ extends to a morphism $\mathbb{P}^1 \to B$ over this
fibre, and every such morphism is constant: an invariant one-form on $B$
pulls back to a holomorphic one-form on $\mathbb{P}^1$, the module
$\Omega^1(\mathbb{P}^1)$ has no nonzero sections, so the differential of
the extension vanishes and constancy follows in characteristic zero.
Hence $\rho \circ f$ is constant on each fibre $a^{-1}(e)$ with $e \notin
\{e_i\}$.

Comparing $\rho \circ f$ with $(\rho \circ f \circ s) \circ a$ on the
dense open set $X - \bigcup_i a^{-1}(e_i)$ gives equality on all of $X$:
$$ \rho \circ f = (\rho \circ f \circ s) \circ a. $$
The morphism $h(x) = f(x) - f(s(a(x))) \in G$ satisfies $\rho \circ h =
0$, so $h$ takes values in $T$. Each coordinate of $h : X \to
\mathbb{G}_m^r$ is invertible on $X$ and hence constant by the units
lemma. On $s(E)$ the composition $s \circ a \circ s = s$, so $h$
vanishes on $s(E)$ and therefore vanishes identically on $X$. This gives
the unique factorization
$$ f = (f \circ s) \circ a $$
of every algebraic morphism $f : X \to G$ through $a$. Uniqueness follows
by precomposing with $s$.

For the based universal property, take $s(0)$ as base point with $0 \in
E$ the group identity, and suppose $f(s(0)) = 0$. Then $g = f \circ s :
E \to G$ is a homomorphism of algebraic groups. Its projection $\rho
\circ g : E \to B$ is a homomorphism by the rigidity theorem for abelian
varieties. The group-law defect
$$ \Phi : E \times E \to G, \qquad
(e,e') \mapsto g(e+e') - g(e) - g(e') $$
takes values in $T$ because $\rho \circ \Phi = 0$, and is constant
because $E \times E$ is a connected projective variety and $T \simeq
\mathbb{G}_m^r$ is affine. Evaluating at $(0,0)$ gives $\Phi(0,0) =
-g(0) = 0$, so $\Phi \equiv 0$ and $g$ is a homomorphism. This completes
the identification
$$ \mathrm{Alb}(X) = E, \qquad a_X = a, $$
and hence proves part (ii) of Theorem 3.
./pdf/fujino-2024-quasi-albanese-maps.pdf

The ordinary fundamental group. Put $U = X - \bigcup_i A_i^\circ$. Since
$A_i^\circ = A_i \cap X$, the identity $X - A_i^\circ = X - A_i$ holds,
and
$$ U = X - \bigcup_i A_i = Y - D - \bigcup_i A_i
= Y - \bigcup_i b^{-1}(H_i) = E \times (\mathbb{P}^1 - Q), $$
with $Q = \{q_1,\dots,q_n\}$. The last step uses that $b$ restricts to
an isomorphism away from the blowup centres, all of which lie on the
$H_i$. Therefore
$$ \pi_1(U) = \pi_1(E) \times \pi_1(\mathbb{P}^1 - Q)
= \mathbb{Z}^2 \times F_{n-1}, $$
with $F_{n-1}$ free of rank $n - 1$.

Each $A_i^\circ$ is a smooth complex submanifold of $X$ of complex
codimension one. The inclusion $U \hookrightarrow X$ induces a surjection
on $\pi_1$ whose kernel is the normal closure of the meridians of
$A_i^\circ$: a loop in $X$ can be pushed off $A_i^\circ$, and a
null-homotopy of the pushed-off loop can be chosen transverse to
$A_i^\circ$, meeting it in finitely many points; a small disc around each
such point rewrites the boundary loop as a product of conjugates of
meridians.

The meridian is identified in the blowup chart. Take local coordinates
$(x, t)$ on $S$ near $p_i$ with $x = e - e_i$ and $t = q - q_i$, and use
the blowup chart $b(u, t) = (ut, t)$, in which $A_i$ is $\{t = 0\}$ with
coordinate $u \in \mathbb{P}^1$, $A_i^\circ$ is $\{t = 0, u \in
\mathbb{C}\}$, and the strict transform $D_i$ of $H_i$ meets the chart
only at $u = \infty$. For $u = 0$ the disc $\{u = 0, |t| \leq
\varepsilon\}$ lies entirely in $X$, since its only point with $t = 0$ is
the centre $u = 0$, $t = 0 \in A_i^\circ$; its boundary is the loop $t =
\varepsilon e^{2\pi i \theta}$. Under $b$, this loop maps to
$$ (e, q) = (e_i, q_i + \varepsilon e^{2\pi i \theta}), $$
a loop winding once around $q_i$ with $e$ fixed. Its class in $\pi_1(U)$
is therefore the pair (identity in $\mathbb{Z}^2$, $m_i$), where $m_i$
is the standard puncture loop around $q_i$ in $\pi_1(\mathbb{P}^1 - Q)$.
Since the loop bounds an actual disc in $X$, the puncture loop $m_i$ is
trivial in $\pi_1(X)$ for every $i$.

Therefore
$$ \pi_1(X) = \frac{\pi_1(U)}{\langle\!\langle m_1, \dots, m_n
\rangle\!\rangle} = \frac{\mathbb{Z}^2 \times F_{n-1}}{1 \times F_{n-1}}
\simeq \mathbb{Z}^2. $$
Under the identification $U \simeq E \times (\mathbb{P}^1 - Q)$ via $b$,
the map $a$ is the projection to $E$, so $a_* : \pi_1(X) \to \pi_1(E)$
is the resulting isomorphism; the section $s$ also gives a right inverse.

The general fibre $F = \mathbb{P}^1 - Q$ has $\pi_1(F) = F_{n-1}$ freely
generated by the $m_i$, so the map $\pi_1(F) \to \pi_1(X)$ is the zero
map. This proves part (iii) of Theorem 3.

The orbifold base of the composition. For a composition of fibrations
$Y \xrightarrow{c} \bar X \xrightarrow{\bar j} J$ in the orbifold
setting, two divisors have to be distinguished:
$$ \Delta(\bar j \circ c, D) \quad \text{and} \quad
\Delta(\bar j, \Delta(c, D)). $$
Campana proved that on smooth orbifold pairs
$$ \Delta(\bar j \circ c, D) \leq \Delta(\bar j, \Delta(c, D)), $$
with equality as soon as
$c : (Y, D) \to (\bar X, \Delta(c, D))$ is an orbifold morphism; a
$c$-exceptional divisor that dominates a divisor of $J$ enters the direct
infimum but not the iterated one.
./pdf/campana-2011-orbifoldes-geometriques-speciales.pdf

On the specific surface, take $c = b : Y \to S$ and $\bar j =
\text{pr}_{\mathbb{P}^1} : S \to \mathbb{P}^1$. The iterated base is
$$ \Delta(b, D) = \sum_i H_i, \qquad
\Delta(\text{pr}_{\mathbb{P}^1}, \sum_i H_i) = \sum_i q_i. $$
Indeed, for each $H_i$ the only $Y$-divisor mapping onto it is its strict
transform $D_i$ with $r(b, D_i) = 1$ and $m_{D_i}(D) = \infty$, so
$m_b(H_i) = 1 \cdot \infty = \infty$ and $H_i$ enters $\Delta(b, D)$ with
coefficient $1 - 1/\infty = 1$; and for each $q_i$ the only $S$-divisor
mapping onto it is $H_i$ with $r(\text{pr}_{\mathbb{P}^1}, H_i) = 1$ and
$m_{H_i}(\sum H_i) = \infty$, so $q_i$ enters $\Delta(\text{pr}_{\mathbb{P}^1},
\sum H_i)$ with coefficient one.

The direct base is computed on $Y$ directly from $j = \text{pr}_{\mathbb{P}^1}
\circ b$. The fibre over $q_i$ is $D_i + A_i$, so
$$ m_j(q_i) = \min(r(j, D_i) m_{D_i}(D), r(j, A_i) m_{A_i}(D))
= \min(1 \cdot \infty, 1 \cdot 1) = 1, $$
the second term from the retained exceptional $A_i$ outside the support of
$D$. Similarly every $t \notin \{q_i\}$ has $m_j(t) = 1$ from the
unchanged complete elliptic fibre. Hence
$$ \Delta(j, D) = 0. $$
The two answers differ at every $q_i$: the direct boundary is trivial,
whereas the iterated boundary is $\sum q_i$ with $\deg(K_{\mathbb{P}^1} +
\sum q_i) = n - 2 > 0$, hence general type.

The CDHP proof asserts equality $\Delta(\bar j \circ c, D) = \Delta(\bar
j, \Delta(c, D))$ (Claim 7.17). Its justification treats every
$\widetilde c$-exceptional divisor as having infinite boundary
multiplicity, on the strength of flatness of $(Y \times_J J')_{\text{main}}
\to \bar X'$. The preceding construction arranges flatness over $J'$, not
over $\bar X'$; these are different morphisms. In the instantiated case,
$\widetilde c = b : Y \to S$ is the blowdown, whose fibres are points away
from the centres and curves over the centres, so $b$ is not flat. The
divisors $A_i$ are exceptional for $\widetilde c = b$ but not for
$\nu = \text{id}_Y$, and carry boundary multiplicity $1$, not $\infty$.
Omitting $A_i$ from the direct infimum is what turns the true direct value
$\Delta(\text{pr}_{\mathbb{P}^1} \circ b, D) = 0$ into the spurious value
$\sum q_i$.
./pdf/cdhp-2026-two-step-nilpotent-monodromy.pdf

Neither obvious repair succeeds. Adding the $A_i$ to the boundary replaces
$X$ by its blowdown $E \times (\mathbb{P}^1 - Q)$, which is not special
(its quasi-Albanese base $\mathbb{P}^1 - Q$ has general type for $n \geq
3$) and whose fundamental group is $\mathbb{Z}^2 \times F_{n-1}$, not
$\mathbb{Z}^2$. Making the intermediate target $\bar X'$ remember the
blowups retains the reducible fibres $D_i + A_i$ and restores the zero
orbifold base, but then the general-type base computed on the contracted
model is lost. Neither repair yields the fibration-specialness conclusion.

The CDHP Theorem A, on the class-two bound for linear images, is
untouched: the counterexample concerns the fibre-specialness assertion
(Theorem E), and Theorem 1 is proved independently of Claim 7.17
and of any fibre-specialness input.

Ordinary fundamental groups. The class-two bound of Theorem 1 is
optimal even for the ordinary (topological) fundamental group: CDY
construct a smooth special quasi-projective surface whose fundamental
group is a Heisenberg central extension -- two-step nilpotent and
nonabelian. Whether every smooth special quasi-projective variety has
virtually nilpotent $\pi_1$ is open.
./pdf/cdy-2025-hyperbolicity-fundamental-groups-III.pdf

Proposition. If $\pi_1(V)$ is virtually solvable for $V$ smooth
quasi-projective, then $\pi_1(V)$ is virtually nilpotent.
./pdf/rogov-2025-bns-quasi-projective.pdf

Proof. Rogov proves that for normal quasi-projective $V$ a virtually
solvable quotient of $\pi_1(V)$ forces $V$ to admit a fibration over a
quasi-projective hyperbolic orbicurve, and derives as corollary that
$\pi_1(V)$ is virtually solvable if and only if it is virtually
nilpotent. Mal'cev's theorem then provides a torsion-free nilpotent
subgroup of finite index, which admits a faithful rational unipotent
representation (Segal, Polycyclic Groups, Cambridge Tracts in
Mathematics 82, 1983, Chapters 1 and 5, Chapter 6 p. 100).

Consequently the class-two bound of Theorem 1 is not accessible from
virtual nilpotence of $\pi_1(V)$ alone: virtual solvability must come
from somewhere, and it is not a consequence of specialness.

Theorem 3 shows that virtual nilpotence of $\pi_1(V)$ cannot be derived
by induction on the quasi-Albanese fibration. On the surface $X$ the
group $\pi_1(X) = \mathbb{Z}^2$ is abelian, hence virtually nilpotent;
the general fibre $F = \mathbb{P}^1 - Q$ has $\pi_1(F) = F_{n-1}$, the
free group of rank $n - 1$. The map $\pi_1(F) \to \pi_1(X)$ is trivial
though the source is not nilpotent. This is precisely where control
over the image of $\pi_1(F)$ in $\pi_1(V)$ and control over $\pi_1(F)$
itself diverge.

The completion theorem addresses the image in every canonical unipotent
quotient of $\pi_1(V)$. Control over the kernel of the completion map --
the subgroup of $\pi_1(V)$ that no unipotent representation sees --
remains necessary for a proof of virtual nilpotence of $\pi_1(V)$
itself, and is not addressed by the completion theorem. The kernel is
not vacuous: a finitely generated group with trivial abelianization has
trivial rational unipotent completion, since a nonzero nilpotent Lie
algebra has nonzero abelianization. The entire group can therefore be
invisible to rational unipotent completion. Specialness of the total
space does not eliminate this phenomenon.

Comparison with the compact case. Campana asked whether the general
Albanese fibre of a compact special variety is special, and Campana and
Claudon answered affirmatively for smooth projective varieties (their
Theorem 2.4). For quasi-projective varieties with reduced boundary,
Theorem 3 shows that the analogue fails: a smooth special quasi-
projective surface has a general quasi-Albanese fibre that is not
special.
./pdf/campana-2004-orbifolds-special-classification.pdf
./pdf/campana-claudon-2016-quelques-proprietes-stabilite.pdf