The structural completion theorem

A semiabelian variety is a connected algebraic group fitting into an exact sequence \[1\longrightarrow\mathbf G_m^r\longrightarrow A\longrightarrow B \longrightarrow1,\] where \(B\) is an abelian variety. For a connected smooth quasi-projective variety \(V\) with a chosen base point, the algebraic quasi-Albanese map \(a_V:V\to\mathop{\mathrm{Alb}}(V)\) is the universal based algebraic morphism to a semiabelian variety. This is the open analogue of the Albanese map. We use the construction and universal property recalled by Fujino (Fujino 2025, Definition 2.18 and Theorem 3.16); universality concerns algebraic morphisms, not arbitrary holomorphic maps.

The rational unipotent completion of a group \(\Gamma\) is the universal pro-unipotent algebraic group receiving a homomorphism from \(\Gamma\). Here pro-unipotent means an inverse limit of unipotent algebraic groups, and universality means that every homomorphism from \(\Gamma\) to a finite-dimensional unipotent algebraic group over \(\mathbf Q\) factors uniquely through the completion. For the finitely generated groups considered here and each integer \(c\geq1\), its canonical \(c\)-step quotient is the universal unipotent quotient of nilpotency class at most \(c\): every homomorphism to a unipotent algebraic group of class at most \(c\) factors uniquely through it. Section 2 gives its Lie-algebra construction.

Exactness in (1) identifies the image of the fibre group with the whole kernel of \(a_*\); the first arrow need not be injective. The theorem assumes neither specialness of \(V\) or \(F\) nor initial nilpotence of the discrete group \(\Gamma\). Its geometric hypotheses are useful because the quasi-Albanese map identifies first homology, and the algebraic fibre inclusion gives a morphism of pointed mixed Hodge structures. These two facts connect the actual fibre image to the commutator of each canonical unipotent quotient.

The proof uses Deligne’s mixed Hodge theory for smooth open varieties (Deligne 1971), Morgan’s foundational work on nilpotent homotopy data (Morgan 1978, 1986), and Hain’s canonical pointed construction (Hain 1987). At each finite stage, let \(\mathfrak g\) be the canonical nilpotent Lie algebra and \(\mathfrak k\) the image of the fibre Lie algebra. Exactness and the quasi-Albanese first-homology isomorphism imply \(\mathfrak k=[\mathfrak g,\mathfrak g]\). The abelianization of \(\mathfrak k\) is a quotient of \(H_1(F,\mathbf Q)\), with weights \(-1\) and \(-2\), while its position in a commutator forces weights at most \(-2\). It is therefore pure of weight \(-2\). The conjugation bracket has source of weights at most \(-3\), so it vanishes in this abelianization. This yields \(\gamma_3\mathfrak g\subseteq\gamma_4\mathfrak g\), and nilpotence forces both terms to vanish. We first prove the bound on canonical quotients carrying mixed Hodge structures, then pass to arbitrary unipotent representations by universality.

To deduce Theorem 1, we apply this structural result after a finite étale cover. Such covers preserve specialness, and the quasi-Albanese dominance and ordinary exactness results of Cadorel–Deng–Yamanoi apply to the cover itself (Cadorel et al. 2025, Theorem 3.1, Lemma 4.4 and Proposition 4.13). The resulting unipotent image has class at most two, and the torus factor in the Zariski closure is abelian. This also explains why no specialness assertion about the general fibre is needed.

There are several predecessors for the relation between Albanese maps and nilpotent quotients. In the compact Kähler case, Campana proved that surjectivity of the Albanese map makes every torsion-free nilpotent quotient abelian (Campana 1995, Corollary 3.1). His argument uses a homological criterion of Stallings. Aguilar Aguilar–Campana obtained the same abelianity conclusion for smooth quasi-projective varieties whose quasi-Albanese map is proper and surjective (Aguilar Aguilar and Campana 2025, arXiv version, Corollary 2(2)). Theorem 2 permits a nonproper map; the weight-\(-2\) contribution from the fibre boundary accounts for its class-two conclusion. When \(H_1(F,\mathbf Q)\) is pure of weight \(-1\), the proof instead gives an abelian completion (Remark 4).

Rogov obtained another completion theorem through higher Albanese manifolds (Rogov 2026, Theorem B, equivalently Theorem 7.9 and Corollary 7.10). For a normal quasi-projective variety and a level \(s\geq3\), he assumes either dominance of its \(s\)th higher Albanese map or a definable biholomorphism between that higher Albanese manifold and the definable analytification of a quasi-projective variety. He proves stabilization at level two. Theorem 2 instead starts from ordinary fundamental-group exactness for the classical quasi-Albanese map. Ordinary dominance alone is not substituted for either set of hypotheses.

A special surface with nonspecial quasi-Albanese fibre

Campana asked whether general Albanese fibres of compact special varieties are special (Campana 2004, Question 5.4, p. 577). Campana–Claudon answered this for smooth projective varieties (Campana and Claudon 2016, Theorem 2.4). The following open surfaces show that the analogous assertion fails with reduced boundary. The construction retains exceptional affine lines after deleting the strict transforms of horizontal elliptic curves.

The general fibre has logarithmic canonical degree \(n-2>0\), and so is not special. This contradicts the fibre-specialness assertion of (Cao et al. 2026, Theorem E, equivalently Theorem 7.15) in the fixed version arXiv:2603.14539v2. The fibre’s ordinary group is free of rank \(n-1\), but its image in the total group is trivial. Thus the example identifies the difference between the fibre and the image controlled by Theorem 2; the total group \(\pi_1(X)=\mathbf Z^2\) satisfies the monodromy bound.

The geometric mechanism also explains the relevant orbifold calculation. Above each \(q_i\), the complete fibre has one deleted component and one retained component, both of multiplicity one. The retained component makes the direct orbifold multiplicity equal to one. Contracting it before calculating the base forgets its contribution. Campana’s composition rule records precisely the exceptional-divisor hypotheses needed for equality of these two bases (Campana 2011, arXiv version, Proposition 3.13). Section 7 identifies the resulting failure in (Cao et al. 2026, Claim 7.17), where flatness over the Iitaka base is used as flatness over a different, contracted intermediate space.

A related construction of Cadorel–Deng–Yamanoi also retains an exceptional curve after deleting a boundary strict transform (Cadorel et al. 2025, Example 4.34). Here the product \(E\times\mathbf P^1\) produces a positive-dimensional nonspecial quasi-Albanese fibre. Complete elliptic slices and orbifold Riemann–Hurwitz verify specialness on every required upper model. A complete section and the absence of nonconstant units establish the quasi-Albanese universal property. Local discs through the retained exceptional curves kill every puncture loop and give the ordinary fundamental group.

Section 2 proves Theorem 2, and Section 3 derives Theorem 1 and its logarithmic-Kodaira-zero consequence. Section 4 sets out the orbifold conventions and surface geometry. Sections 5 and 6 prove Theorem 3; Section 7 explains the composition calculation. Section 8 records the remaining ordinary-group question. The representation results do not assume that the ordinary group embeds in its completion, and no such embedding is supplied by the argument.

The canonical quotients and their weights

Fix a connected smooth complex quasi-projective variety \(V\) and a base point \(v\in V\). Put \(\Gamma=\pi_1(V,v)\) and write \(\mathfrak m_V\) for the complete pronilpotent Lie algebra of its rational unipotent completion. In characteristic zero the exponential and logarithm identify a unipotent algebraic group with its nilpotent Lie algebra, using the Baker–Campbell–Hausdorff multiplication. For \(c\geq1\), define the canonical \(c\)-step quotient by \[ \mathfrak g_c=\mathfrak m_V/\overline{\gamma_{c+1}\mathfrak m_V}, \qquad U_c=\exp(\mathfrak g_c), \tag{2}\] where \(\gamma_1\mathfrak m_V=\mathfrak m_V\), \(\gamma_{r+1}\mathfrak m_V=[\mathfrak m_V,\gamma_r\mathfrak m_V]\), and closure is taken in the inverse-limit topology. The group \(\Gamma\) is finitely generated, so each \(\mathfrak g_c\) is finite-dimensional. The canonical image of \(\Gamma\) is Zariski dense in \(U_c\), and \[ \mathfrak g_c/[\mathfrak g_c,\mathfrak g_c]\simeq H_1(V,\mathbf Q). \tag{3}\] These constructions and their universal properties can be obtained from the completed group algebra (Hain 1987, sec. 2.5, Theorem 2.5.3 and Proposition 2.5.5).

We recall exactly the mixed-Hodge information used below. A rational mixed Hodge structure is a finite-dimensional rational vector space with an increasing weight filtration \(W\) and a decreasing Hodge filtration on its complexification, such that each weight-graded piece is a pure Hodge structure. We say it has weights at most \(r\) if \(W_r\) is the entire space, and is pure of weight \(r\) if also \(W_{r-1}=0\). Morphisms preserve the filtrations and are strict: their images receive the same filtration as subobjects of the target and as quotients of the source. The category is abelian (Deligne 1971, Theorem 2.3.5 and §2.3.8). Tensor weights add.

For smooth quasi-projective \(V\), first cohomology has weights \(1,2\), so \(H_1(V,\mathbf Q)\) has weights \(-1,-2\) (Deligne 1971, Corollary 3.2.15(ii)). The canonical Lie quotients \(\mathfrak g_c\) carry mixed Hodge structures for which brackets and the Hurewicz map (3) are morphisms. Moreover a pointed algebraic map induces a morphism of these structures. We use Hain’s pointed construction (Hain 1987, Theorem 6.3.1, p. 321) for this functoriality. It equips the completed Lie algebra with a pro-mixed Hodge structure compatible with its bracket. The closed lower-central ideals are topologically generated by iterated bracket images and are therefore pro-mixed-Hodge subobjects. The finite-dimensional quotients (2) inherit ordinary mixed Hodge structures; their transition maps and the maps induced by pointed algebraic morphisms are morphisms of these structures. Morgan’s foundational construction (Morgan 1978) is part of the history of the method, but its original functoriality assertion was corrected in (Morgan 1986); the stronger pointed statement needed here is supplied by Hain.

At finite stages, all lower-central ideals, images, kernels, and abelianizations below are therefore ordinary subobjects or quotients in the abelian category of mixed Hodge structures. Arbitrary unipotent representation quotients need not themselves carry the required Hodge structure. We prove the bound on the canonical quotients first and only then pass to arbitrary representations by universality.

We now assume all the hypotheses of Theorem 2, with \(a:V\to A\) dominant, \(F\) a smooth connected general fibre, and the base point \(v\) chosen in \(F\). We use the ordinary exact sequence (1); properness and specialness of the fibre play no role.

The fibre image is the commutator at every unipotent stage

Let \[N=\mathop{\mathrm{im}}\bigl(\pi_1(F,v)\longrightarrow\Gamma\bigr).\] By (1), \(N\) is normal and equals \(\ker a_*\). The fundamental group of a semiabelian variety is free abelian. The quasi-Albanese map induces an isomorphism \[ H_1(V,\mathbf Z)/\mathop{\mathrm{Tor}}H_1(V,\mathbf Z)\xrightarrow{\sim}H_1(A,\mathbf Z), \tag{4}\] as in (Fujino 2025, Lemmas 3.11–3.12, arXiv v2 PDF, pp. 20–21). The first of those lemmas states integral surjectivity with torsion kernel; the proof of the second gives the rational mixed-Hodge identification. Thus \[ [\Gamma,\Gamma]\subseteq N, \qquad N/[\Gamma,\Gamma]\simeq\mathop{\mathrm{Tor}}H_1(V,\mathbf Z). \tag{5}\] In particular the latter quotient is finite. We do not assert equality of the two discrete subgroups when this torsion is nonzero.

Fix \(c\). Let \(K_c\) be the Zariski closure of the image of \(N\) in \(U_c\), and let \(\mathfrak k_c=\mathop{\mathrm{Lie}}K_c\). The pointed fibre inclusion induces a map of canonical \(c\)-step Lie quotients \[\mathfrak g_c(F)\longrightarrow\mathfrak g_c(V).\] Its image is \(\mathfrak k_c\): images of morphisms of unipotent algebraic groups are closed, and the fibre group is dense in its own completion. This also equips \(\mathfrak k_c\) with its image mixed Hodge structure.

We claim \[ K_c=[U_c,U_c],\qquad\mathfrak k_c=[\mathfrak g_c,\mathfrak g_c]. \tag{6}\] Indeed, every element of \(N\) has a positive power in \([\Gamma,\Gamma]\) by (5). The abelianization \(U_c/[U_c,U_c]\) is an additive vector group, which has no nonidentity torsion. Hence the image of \(N\) in it is zero and \(K_c\subseteq[U_c,U_c]\). Conversely, normality of \(N\) and density of \(\Gamma\) imply normality of \(K_c\) in \(U_c\). The dense image of \(\Gamma\) in \(U_c/K_c\) is abelian, because \([\Gamma,\Gamma]\subseteq N\). The commutator morphism of the algebraic quotient vanishes on this dense subset of its product, and so the quotient is abelian. This gives the opposite inclusion and proves (6). In particular, no left-exactness assertion about unipotent completion has been used.

Two weight bounds force centrality

We continue with the fixed finite-dimensional nilpotent Lie algebra \(\mathfrak g_c\). Its lower-central graded pieces are generated by iterated brackets of its abelianization. More precisely, the \(r\)-fold iterated bracket induces a surjective mixed-Hodge morphism \[H_1(V,\mathbf Q)^{\otimes r}\longrightarrow \gamma_r\mathfrak g_c/\gamma_{r+1}\mathfrak g_c.\] The target therefore has weights at most \(-r\). Starting with \(\gamma_{c+1}\mathfrak g_c=0\) and taking successive extensions down the lower-central filtration shows that \(\gamma_r\mathfrak g_c\) itself has weights at most \(-r\). In particular, \[ W_{-1}\mathfrak g_c=\mathfrak g_c, \qquad \mathfrak k_c=[\mathfrak g_c,\mathfrak g_c]\subseteq W_{-2}\mathfrak g_c. \tag{7}\] Consider its fibre-image abelianization \[B_c=\mathfrak k_c/[\mathfrak k_c,\mathfrak k_c].\] The surjection \(\mathfrak g_c(F)\to\mathfrak k_c\) induces a surjective mixed-Hodge map \[ H_1(F,\mathbf Q)\longrightarrow B_c. \tag{8}\] The group \(H_1(F,\mathbf Q)\) has only weights \(-1,-2\), because the chosen fibre is smooth and quasi-projective. Strictness of (8) gives \(W_{-3}B_c=0\). On the other hand, strictness for the image \(\mathfrak k_c\subseteq\mathfrak g_c\) and (7) give \(W_{-2}B_c=B_c\). Therefore \[ B_c\text{ is pure of weight }-2. \tag{9}\] This is the point where geometry of the fibre enters: its first homology forbids weights below \(-2\), even though it maps into a commutator ideal.

Because \(\mathfrak k_c\) is an ideal, the bracket followed by its abelianization is a mixed-Hodge morphism \[\mathfrak g_c\otimes\mathfrak k_c\longrightarrow B_c.\] The source has weights at most \(-3\), and the target has \(W_{-3}=0\) by (9). The morphism must vanish. Thus \[ [\mathfrak g_c,\mathfrak k_c]\subseteq[\mathfrak k_c,\mathfrak k_c]. \tag{10}\] Substituting (6) and using the Jacobi lower-central estimate yields \[\gamma_3\mathfrak g_c \subseteq[\gamma_2\mathfrak g_c,\gamma_2\mathfrak g_c] \subseteq\gamma_4\mathfrak g_c \subseteq\gamma_3\mathfrak g_c.\] Hence \(\gamma_3\mathfrak g_c=\gamma_4\mathfrak g_c\). Repeated bracketing with \(\mathfrak g_c\) makes all subsequent lower-central terms equal. Since \(\mathfrak g_c\) is nilpotent, they must be zero. We have proved \[ \gamma_3\mathfrak g_c=0\qquad\hbox{for every }c\geq1. \tag{11}\]

The complete Lie algebra \(\mathfrak m_V\) is the separated inverse limit of these canonical quotients. Every element of \(\gamma_3\mathfrak m_V\) has zero image in every \(\mathfrak g_c\) by (11); separatedness therefore gives \(\gamma_3\mathfrak m_V=0\). Its closure is zero as well. By the definition (2), the map \(\mathfrak m_V\to\mathfrak g_2\) is consequently an isomorphism. Thus the canonical tower actually stabilizes at \(c=2\), and the completion is finite-dimensional. Its abelianization is \(H_1(V,\mathbf Q)\) and its commutator is a quotient of \(\bigwedge^2H_1(V,\mathbf Q)\). The Baker–Campbell–Hausdorff correspondence gives the same class bound for the unipotent algebraic group.

It remains to pass to complex unipotent representations. For the finitely generated group \(\Gamma\), unipotent completion commutes with extension of the characteristic-zero ground field (Hain 2015, sec. 4). More explicitly, the completion over \(\mathbf C\) is the scalar extension of the rational completion and has the universal property for homomorphisms into complex unipotent groups. After extension from \(\mathbf Q\) to \(\mathbf C\), this universal property therefore bounds every complex unipotent representation image. This completes the proof of Theorem 2.

The surface and its two projections

We use the notation of Theorem 3 and put \[S=E\times\mathbf P^1,\qquad H_i=E\times\{q_i\},\qquad A_i=b^{-1}(e_i,q_i).\] Write \[\bar a=\operatorname{pr}_E\circ b:Y\to E,\qquad a=\bar a|_X,\qquad j=\operatorname{pr}_{\mathbf P^1}\circ b:Y\to\mathbf P^1.\] The points \(e_i\) are permitted to coincide. The distinctness of the \(q_i\) ensures that all blowup centres are distinct and that the curves \(D_i\) are pairwise disjoint. We have \[ A_i\simeq\mathbf P^1,\qquad D_i\cdot A_j=\delta_{ij},\qquad A_i^\circ:=A_i\cap X\simeq\mathbf A^1. \tag{12}\] Only the single intersection point with \(D_i\) is removed from \(A_i\). These retained affine lines will account both for the orbifold multiplicities and for the disappearance of puncture loops in \(X\).

The infimum convention

A smooth orbifold pair is a smooth variety \(Y\) together with a simple normal crossing \(\mathbf Q\)-divisor \[D=\sum_P\left(1-\frac1{m_P(D)}\right)P, \qquad m_P(D)\in\mathbf Z_{\geq1}\cup\{\infty\}.\] Only finitely many coefficients are nonzero. We use \(1/\infty=0\); thus a component of a reduced boundary has multiplicity \(\infty\), whereas a prime divisor outside its support has multiplicity \(1\).

Here a fibration is a surjective morphism with connected fibres. Let \(f:Y\to Z\) be such a morphism, with \(Y\) smooth projective and \(Z\) smooth. For a prime divisor \(Q\subset Z\) and a prime divisor \(P\subset Y\) dominating \(Q\), write \(r(f,P)\) for the coefficient of \(P\) in \(f^*Q\). The orbifold base divisor is \[ \Delta(f,D)=\sum_Q\left(1-\frac1{m_f(Q)}\right)Q, \qquad m_f(Q)=\inf_{f(P)=Q}r(f,P)m_P(D). \tag{16}\] This is the infimum convention of (Cao et al. 2026, Definition 2.1); it is not a greatest-common-divisor convention. A divisor \(P\) with \(\mathop{\mathrm{codim}}_Zf(P)\geq2\) is \(f\)-exceptional and does not occur in the infimum for the immediate base. It may occur for a later composition.

For the surface above, (14) gives \[ m_j(q_i)=\min\{1\cdot\infty,1\cdot1\}=1, \qquad \Delta(j,D)=0. \tag{17}\] The unchanged fibres give multiplicity one at every other point. Thus \((Y,D)\) has logarithmic Kodaira dimension one, but the orbifold base of \(j\) is \((\mathbf P^1,0)\). The retained component \(A_i\) is what makes its multiplicity at \(q_i\) equal to one.

Upper models and specialness

A dominant morphism of smooth orbifold pairs \(u:(Y',D')\to(Y,D)\) is an orbifold morphism if, for every prime divisor \(P\subset Y\) and every component \(P'\) of \(u^*P\), its coefficient \(r\) in that pullback satisfies \[r\,m_{P'}(D')\geq m_P(D).\] It is an elementary birational orbifold morphism if, in addition, it is birational and \(u_*D'=D\). In particular, when \(D\) is reduced, every component of \(u^{-1}D\) must lie in the coefficient-one part of \(D'\). An equality of pushed-forward boundaries alone is weaker.

A fibration \(f:(Y,D)\to Z\) is neat if there is a commutative birational diagram \[\begin{tikzcd}[column sep=large] (Y,D) \arrow[r,"f"] \arrow[d,"w"'] & Z \arrow[d,"v"]\\ (Y_0,D_0) \arrow[r,"f_0"'] & Z_0 \end{tikzcd}\] with smooth spaces, \(w\) an elementary birational orbifold morphism, and every \(f\)-exceptional divisor also \(w\)-exceptional. The definition is the one in (Cao et al. 2026, Definition 2.3). A fibration is high if its morphism to the base equipped with the induced divisor \(\Delta(f,D)\) is an orbifold morphism. Birational equivalence of fibrations is generated by diagrams in which the source modification is an elementary birational orbifold morphism and the base modification is birational. The canonical dimension of a fibration is the infimum of the Kodaira dimensions of its orbifold bases over these models. Here the Kodaira dimension of a \(\mathbf Q\)-divisor is the exponent measuring growth of its plurisections, with value \(-\infty\) if all positive plurisection spaces vanish. For a neat fibration, \[ \kappa(f,D)=\kappa\bigl(Z,K_Z+\Delta(f,D)\bigr) \tag{18}\] by (Cao et al. 2026, Lemma 2.7), attributed there to (Campana 2011, Corollaire 5.11(3)).

For a smooth projective curve \(C\), a divisor \(K_C+\Delta\) is of general type precisely when its degree is positive. In dimension two we will also use the elementary fact that a birational fibration is equivalent to the identity fibration on its source. Consequently its canonical dimension is no larger than the logarithmic Kodaira dimension of that source. Equation (18) then makes this useful for Definition 7.

A two-dimensional base

Suppose \(\dim Z=2\). A general connected zero-dimensional fibre is a point, so \(g\) is birational. Compare \(g\) with the identity fibration on \((Y',D')\), using the identity on the source and \(g\) on the base. By the definition of canonical dimension, \[ \kappa(g,D')\leq\kappa(Y',K_{Y'}+D'). \tag{19}\]

We claim that the right side is at most one. Choose a positive integer \(m\) divisible enough that \(mD'\) is integral. Off the finite set of exceptional images, \(u\) is an isomorphism and its pushed-forward boundary is \(D\). A section of \(m(K_{Y'}+D')\) therefore restricts to a section of \(m(K_Y+D)\) away from that finite set. A section of a line bundle on a smooth surface extends across a subset of codimension two: after trivializing the line bundle, this is extension of regular functions on a normal variety. We obtain an injection \[ H^0\bigl(Y',m(K_{Y'}+D')\bigr) \lhook\joinrel\longrightarrow H^0\bigl(Y,m(K_Y+D)\bigr). \tag{20}\] The target grows linearly with \(m\) by (15). This proves the claim. Neatness, (18), and (19) now show \[\kappa\bigl(Z,K_Z+\Delta(g,D')\bigr)\leq1<2.\] Notice that this uses a one-sided bound on sections; it does not assume birational invariance for arbitrary changes of the boundary.

A curve base

Suppose \(Z=C\) is a smooth projective curve, and set \(\Delta=\Delta(g,D')\). Assume for contradiction that \[ \deg(K_C+\Delta)>0. \tag{21}\] Choose \(t\in\mathbf P^1\) away from the \(q_i\) and from the images, under \(j\), of the finitely many exceptional images of \(u\). The fibre \[C_t=(j\circ u)^{-1}(t)\] is then an unchanged smooth complete elliptic curve, disjoint from both \(D'\) and the exceptional locus of \(u\).

Suppose \(h=g|_{C_t}:C_t\to C\) is nonconstant. It is finite and surjective. For a point \(c\in C\), write \(m_c\) for its orbifold multiplicity. Any component \(P\) of \(g^*c\) meeting \(C_t\) lies outside \(\mathop{\mathrm{Supp}}D'\), so \(m_P(D')=1\). Formula (16) implies \[r(g,P)\geq m_c.\] For \(x\in C_t\) above \(c\), the ramification index \(e_x\) of \(h\) is the intersection multiplicity of \(C_t\) with \(g^*c\) at \(x\). It is the sum of the positive intersection orders multiplied by the corresponding \(r(g,P)\), so \(e_x\geq m_c\). If \(m_c=\infty\), this is impossible, since surjectivity supplies a point \(x\) above \(c\). Thus the existence of \(h\) first rules out all infinite base multiplicities. For finite \(m_c\), \[e_x-1\geq e_x\left(1-\frac1{m_c}\right).\] Summing this inequality over the fibres above \(\mathop{\mathrm{Supp}}\Delta\) and using Riemann–Hurwitz gives \[\begin{align*} 0=\deg K_{C_t} &=\deg(h)\deg K_C+\sum_{x\in C_t}(e_x-1)\\ &\geq\deg(h)\deg(K_C+\Delta)>0, \end{align*}\] a contradiction. This argument only needs the inequality of multiplicities, so it applies to the infimum convention rather than requiring divisibility of ramification indices.

It follows that \(g\) is constant on all sufficiently general fibres of \(j\circ u\). Those fibres are connected. Descent on function fields gives a rational map \(r:\mathbf P^1\dashrightarrow C\) with \[g=r\circ j\circ u.\] A rational map between smooth projective curves extends everywhere. If \(\deg r>1\), a general fibre of \(g\) is the disjoint union of the fibres of \(j\circ u\) over the distinct preimages of a general point of \(C\); this contradicts connectedness of the general fibre of \(g\). Therefore \(\deg r=1\), and \(C\simeq\mathbf P^1\). Up to this isomorphism, \(g=j\circ u\).

It remains to compute the boundary 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, since \(u_*D'=D\) and \(A_i\) does not occur in \(D\). At its generic point \(u\) is an isomorphism. Thus \(A_i'\) occurs with multiplicity one in \((j\circ u)^*q_i\), and \[m_{j\circ u}(q_i)=1.\] For every \(t\notin\{q_i\}\) the strict transform of the original elliptic fibre similarly has boundary coefficient zero and ordinary multiplicity one. Additional exceptional components cannot increase an infimum already equal to one. Consequently \[\Delta(j\circ u,D')=0.\] The degree in (21) is therefore \(\deg K_{\mathbf P^1}=-2\), the final contradiction. Both possible base dimensions have been excluded, proving specialness on every upper model in Definition 7.

Units and the universal property

Let \(G\) be any semiabelian variety and let \(f:X\to G\) be an algebraic morphism. Write \[1\longrightarrow T\longrightarrow G\xrightarrow{\rho}B \longrightarrow1, \qquad T\simeq\mathbf G_m^r,\] with \(B\) abelian. For \(e\) outside the finite set \(\{e_i\}\), the fibre \(a^{-1}(e)\) is the punctured line \(\mathbf P^1\setminus\{q_i\}\). The restriction of \(\rho\circ f\) to this fibre extends to \(\mathbf P^1\) because \(B\) is proper. Every morphism \(\mathbf P^1\to B\) is constant: invariant one-forms span the cotangent space of \(B\) everywhere, and their pullbacks to \(\mathbf P^1\) vanish. Its differential is zero, which implies constancy in characteristic zero.

Comparing with the point supplied by the section \(s\) gives \[\rho\circ f=(\rho\circ f\circ s)\circ a\] on a dense open subset of \(X\) and hence everywhere. The morphism \[h(x)=f(x)-f(s(a(x)))\] therefore has image in \(T\). Each coordinate of \(h:X\to\mathbf G_m^r\) is a unit, so Lemma 9 makes \(h\) constant. It vanishes on \(s(E)\), and hence vanishes identically. We have proved the unique factorization \[ f=(f\circ s)\circ a. \tag{23}\] Uniqueness follows by composing with \(s\).

For the based universal property, choose the base point \(s(0)\) and suppose \(f(s(0))=0\). Then \(g=f\circ s:E\to G\) is a group homomorphism. Indeed, its projection to \(B\) is a homomorphism by the rigidity theorem for abelian varieties. Its group-law defect \[E\times E\longrightarrow G, \qquad(e,e')\longmapsto g(e+e')-g(e)-g(e')\] therefore takes values in the affine group \(T\). A morphism from a connected projective variety to an affine torus is constant, and the defect is zero at \((0,0)\). This proves the claim. Together with (23), it establishes \[\mathop{\mathrm{Alb}}(X)=E,\qquad a_X=a.\] The general fibre in (13) has logarithmic canonical degree \(n-2>0\). Its identity fibration is of general type, so it is not special. Together with connectedness of all fibres, this proves part (ii) of Theorem 3.

The meridians die inside the given open surface

Remove the retained exceptional curves temporarily and write \[U=X\setminus\bigcup_iA_i^\circ.\] The blowup is an isomorphism away from its centres, and their containing horizontal fibres have already been deleted. Thus \[ U\simeq E\times(\mathbf P^1\setminus Q), \qquad Q=\{q_1,\ldots,q_n\}. \tag{24}\] Let \(m_i\) denote positively oriented puncture loops in \(\mathbf P^1\setminus Q\), with paths to a common base point. They generate its fundamental group with the single relation \(m_1\cdots m_n=1\).

Each \(A_i^\circ\) is a closed smooth complex divisor in \(X\). The inclusion \(U\hookrightarrow X\) induces a surjection on fundamental groups, whose kernel is normally generated by meridians around these divisors. For completeness, loops can be perturbed off a real codimension-two submanifold. A null-homotopy can be made transverse to it, relative to the boundary of its parameter disc. It then meets the divisors in finitely many points. Removing small discs about those points expresses the boundary loop as a product of conjugates of meridians. This argument uses compactness of the parameter disc, not compactness of the divisors.

The meridians can be identified explicitly. Near \((e_i,q_i)\) choose local coordinates \((x,t)\) on \(S\) so the deleted horizontal curve is \(t=0\). The blowup chart \[ b(u,t)=(ut,t) \tag{25}\] contains the whole retained affine line \(A_i^\circ=\{t=0\}\), with coordinate \(u\). The strict transform of the deleted horizontal curve does not meet this chart. At \(u=0\) the loop \[t=\varepsilon e^{2\pi i\theta},\qquad0\leq\theta\leq1,\] is a meridian of \(A_i^\circ\). Under (24), it has fixed elliptic coordinate \(e_i\) and winds once about \(q_i\). It represents a conjugate of \(m_i\). Moreover, it bounds the actual disc \(\{u=0,\ |t|\leq\varepsilon\}\) inside \(X\), whose centre lies on \(A_i^\circ\).

It follows that \[\begin{align*} \pi_1(X) &\simeq \frac{\pi_1(E)\times\pi_1(\mathbf P^1\setminus Q)} {\langle\!\langle m_1,\ldots,m_n\rangle\!\rangle}\\ &\simeq\pi_1(E)\simeq\mathbf Z^2. \end{align*}\] Under the product identification (24), \(a\) is the projection to \(E\), so \(a_*\) is precisely the displayed isomorphism. The section (22) also shows directly that the elliptic generators survive. For a general fibre, all generators of its free fundamental group are puncture loops, so its image in \(\pi_1(X)\) is trivial. This proves part (iii) and completes Theorem 3. Every relation used here is realized by a homotopy in \(X\); no additional orbifold relation has been imposed on its ordinary group.

The models occurring before Claim 7.17

We next check that this discrepancy occurs on the models used in the cited claim. The relative core used there has special general fibres, and its base pair has general-type general fibres over \(E\). In our case it can be represented by the identity of \(Y\): the identity has point fibres, while the general fibres of \((Y,D)\to E\) are the log-general-type punctured projective lines. The identity is a neat and high model in the terminology of (Cao et al. 2026, Definitions 2.3 and Theorem 2.10). The log-canonical calculation (15) gives the Kodaira dimension one required in the subsequent reduction. General Iitaka fibres are complete elliptic curves mapping isomorphically to \(E\).

An isogeny \(E'\to E\) pulls the construction back to the blowup of \(E'\times\mathbf P^1\) at all preimages of the centres. It is finite étale on the entire pair. The boundary is the pullback of \(D\), the exceptional curves still have coefficient zero, and the log-canonical divisor is still pulled back from \(\mathcal O_{\mathbf P^1}(n-2)\). Thus the minimizing-isogeny step in (Cao et al. 2026, Equation (7.15.2)) leaves the relative-core canonical dimension equal to one and retains the exceptional divisors.

For the birational contraction in (Cao et al. 2026, Proposition 7.6), we can take \(b:Y\to S\). Indeed, the fixed semiabelian subvariety in that construction is the image in \(E\) of a general Iitaka fibre, hence \(E\) itself. The quotient is a point, and the resulting contracted family is \(\mathbf P^1\times E\simeq S\). This product is smooth over all of \(J\), so the open set \(J^0\) used for the smooth contracted family can be taken to be \(J\); its complement \(\partial J\) is empty. On the contracted space \(S\), the boundary \(\sum_iH_i\) has the general-type orbifold base shown in (27). This is consistent with (Cao et al. 2026, Claim 7.7); the difficulty is transferring that conclusion back to \((Y,D)\).

The proof next makes a birational change \(J'\to J\) so that the main component of \(Y\times_JJ'\) is flat over \(J'\). Here \(j:Y\to\mathbf P^1\) is already flat. One way to see this is that at each point of the smooth curve base its local ring is a discrete valuation ring, and the local rings of the dominant smooth integral source are torsion-free modules over it. Torsion-free modules over a discrete valuation ring are flat. A smooth birational model of the complete curve \(J=\mathbf P^1\) is isomorphic to \(J\). The choices in the proof can therefore be instantiated with \[J'=J,\qquad\bar X'=\bar X\times_JJ'=S,\qquad \widetilde X=S,\qquad\widetilde Y=Y.\] Here \(\nu:\widetilde Y\to Y\) is the source modification and \(\mu:\widetilde X\to\bar X'\) is the resolution of the intermediate target. Both are identities, and the maps to that target and to \(J'\) are \[\nu=\operatorname{id}_Y,\quad\mu=\operatorname{id}_S,\quad \widetilde c=b,\quad\widetilde j=\bar j.\] The boundary in (Cao et al. 2026, Equation (7.16.7)) is then exactly \(D\), because there are no \(\nu\)-exceptional divisors to add.

The change of flatness target

Claim 7.17 asserts equality of the two divisors in (26) on the resulting models. Its proof treats every \(\widetilde c\)-exceptional divisor as having infinite boundary multiplicity. To justify this, it uses flatness of \[(Y\times_JJ')_{\mathrm{main}}\longrightarrow\bar X'\] although the preceding construction arranged flatness over \(J'\). These are different morphisms. In the displayed instance the former map is the blowdown \(b:Y\to S\). Its fibres are points away from the centres and curves over the centres, so it is not flat. The divisors \(A_i\) are exceptional for \(\widetilde c=b\) but are not exceptional for \(\nu=\operatorname{id}_Y\); they have boundary multiplicity \(1\), not \(\infty\). Omitting them from the direct infimum is precisely what turns (28) into the incorrect answer (27).

This is a numerical failure of the asserted equality, as well as a failure of the stated justification. Adding the \(A_i\) to the boundary would turn the blowdown into an orbifold morphism, but it would replace \(X\) by \[E\times(\mathbf P^1\setminus Q),\] which is not special and has group \(\mathbf Z^2\times F_{n-1}\), where \(F_{n-1}\) is the free group of rank \(n-1\). Alternatively, making the intermediate target remember the blowups retains the reducible fibres \(D_i+A_i\) and restores the zero orbifold base; the general-type base computed on the contracted model is then lost. Neither modification establishes the original assertion for the original open variety.

The counterexample and this calculation concern the fixed version (Cao et al. 2026). Its Theorem A is a separate assertion about linear representation images. Our proof of that conclusion in Section 3 uses independent inputs and does not use the fibre-specialness assertion.