Two-step monodromy of special quasi-projective varieties

1. Choose the cover and obtain exactness there

Put $L=\overline{\rho(\pi_1(V))}^{\mathrm{Zar}}$ and take the connected finite étale cover $V_0\to V$ corresponding to $\Gamma_0=\rho^{-1}(L^0)$. CDY gives $L^0=U\times T$. The cover remains special; its quasi-Albanese map is dominant, has connected general fibre, and satisfies the ordinary exact sequence. The weakly special hypothesis in Lemma 4.4 follows from Theorem 3.1. Generic smoothness supplies a smooth general fibre.

[CDY25, Theorem A, p. 2] [CDY25, Theorem 3.1, p. 5] [CDY25, Lemma 4.4, p. 9] [CDY25, Proposition 4.13, p. 12] [Groth, Exposé XII, Theorem 5.1, print p. 251 / PDF p. 267]

2. Kill triple commutators in both factors

Theorem 1.2 applied to $\pi_1(V_0)\to U$ kills triple commutators. Their projection to the abelian group $T$ also vanishes, hence they vanish in the image in $U\times T$. This image has finite index in the original image. Exactness must be re-established on $V_0$: completion need not be invariant under passage to finite index. The manuscript credits Cao–Deng–Hacon–Păun with the prior class-two linear-image conclusion and presents an independent proof.

[CDHP26, Theorem A, p. 1]

§3 · pp. 7–8

1. Identify the fibre image with the commutator

Exactness gives $N=\operatorname{im}(\pi_1(F)\to\Gamma)=\ker a_*$. The first-homology property of quasi-Albanese yields

Do not identify the two discrete subgroups. Instead take the Zariski closure $K_c$ of $N$ in $U_c$. Since $U_c^{\mathrm{ab}}$ is a torsion-free additive group, $K_c\subseteq[U_c,U_c]$. Conversely the dense image of $\Gamma$ in $U_c/K_c$ is abelian, so the commutator morphism vanishes and the reverse inclusion follows. Thus $\mathfrak k_c:=\operatorname{Lie}K_c=[\mathfrak g_c,\mathfrak g_c]$. No left-exactness of completion is invoked.

Hain's construction applied to the pointed algebraic inclusion $F\hookrightarrow V$ makes $\mathfrak g_c(F)\to\mathfrak g_c(V)$ a mixed-Hodge morphism, and gives its image $\mathfrak k_c$ the induced structure. An arbitrary representation quotient is not assumed to carry such a structure.

§2.2; (2.3)–(2.5) · pp. 5–6 [Fuj24, Lemmas 3.11–3.12, pp. 20–21] [Hain87, Theorem 6.3.1, print p. 321 / PDF p. 51]

2. Combine the fibre's lower weight bound with the commutator's upper bound

Iterated brackets give weights at most $-r$ on $\gamma_r\mathfrak g_c$. Meanwhile

and strictness restrict the weights of $B_c$ to $-1,-2$. Its position in $\mathfrak k_c=\gamma_2\mathfrak g_c$ therefore makes it pure of weight $-2$. The bracket

has source of weights at most $-3$ and consequently vanishes. Hence

All subsequent lower-central terms coincide, and nilpotence forces $\gamma_3\mathfrak g_c=0$. This proves the assertion for each canonical quotient.

§2.3; (2.6)–(2.10) · p. 6 [Del71, Theorem 2.3.5, print p. 30 / PDF p. 27] [Del71, Corollary 3.2.15(ii), print p. 39 / PDF p. 36]

3. Pass to full completion and complex unipotent representations

The Lie algebra $\mathfrak m_V$ is the separated inverse limit of its canonical quotients. Vanishing of $\gamma_3$ in every quotient gives $\gamma_3\mathfrak m_V=0$. Thus $\mathfrak m_V\simeq\mathfrak g_2$: the tower stabilizes and full completion inherits finite-dimensionality of $\mathfrak g_2$, which follows from finite generation of $\Gamma$. A uniform class bound on the finite stages alone would not explain this conclusion.

Completion commutes with characteristic-zero field extension. Extend from $\mathbb Q$ to $\mathbb C$ and use universality to factor each complex unipotent representation. This proves the third conclusion. Injectivity of $\Gamma\to\Gamma^{\mathrm{un}}(\mathbb Q)$ is not supplied; killing $\gamma_3\Gamma$ in completion does not identify its elements inside the discrete group.

§2.3 · p. 7 [Hain15, §4, p. 8] Question 8.1; §8 · p. 18

1. Exclude a surface base by the growth of sections

Fix any allowed $u:(Y',D')\to(Y,D)$ and neat fibration $g:Y'\to Z$. If $\dim Z=2$, then $g$ is birational. Neatness and canonical dimension bound the base Kodaira dimension by $\kappa(Y',K_{Y'}+D')$. For sufficiently divisible $m$, restrict sections away from the finite exceptional image of $u$ and extend by normality to obtain

The target grows linearly, so the base has Kodaira dimension at most one. Birational invariance under arbitrary boundary changes is not assumed.

§4.1; (4.1)–(4.4) · pp. 8–9 §5 · pp. 11–13 [CDHP26, Definitions 2.1–2.9; Lemma 2.7; Proposition 2.8, pp. 7–9]

2. Exclude a curve base using elliptic slices and retained components

Suppose $Z=C$ with $\deg(K_C+\Delta(g,D'))>0$. A sufficiently general fibre $C_t$ of $j\circ u$ is an unchanged complete elliptic curve avoiding the boundary and exceptional locus. If $h=g|_{C_t}$ is nonconstant, its ramification index satisfies $e_x\ge m_c$ for the infimum multiplicity $m_c$. Infinite $m_c$ is impossible, and Riemann–Hurwitz gives

Thus $g$ factors through $j\circ u$. Connected general fibres force the intervening map of curves to have degree one, hence $C\simeq\mathbb P^1$. The strict transform of each $A_i$ still has coefficient zero in $D'$ and ordinary multiplicity one above $q_i$. Strict transforms of the other elliptic fibres do likewise. Therefore $\Delta(j\circ u,D')=0$: additional exceptional components cannot increase an infimum already equal to one. This proves specialness on every upper model.

§5 · pp. 11–13

1. The section and units determine quasi-Albanese universality

Choose a complete section $s:E\to X$ using $q_*\notin\{q_i\}$. The divisor of a unit $v\in\mathcal O(X)^*$ is $\sum b_iD_i$. Intersecting with $A_j$, using $D_i\cdot A_j=\delta_{ij}$ and zero intersection for a principal divisor, gives $b_j=0$. Consequently every unit is constant.

For an algebraic morphism $f:X\to G$ into a semiabelian extension $1\to T\to G\to B\to1$, its projection to $B$ is constant on the general punctured rational fibre. The difference $f-(f\circ s)\circ a$ therefore takes values in $T$. It is constant by the units calculation and zero on the section, hence zero everywhere. After fixing origins, $f\circ s:E\to G$ is a homomorphism: its projection to $B$ is a homomorphism, and its group-law defect is a map from the projective variety $E\times E$ to an affine torus, hence constant and zero. The section proves uniqueness. This establishes the algebraic universal property, beyond a dimension count of logarithmic one-forms.

Every special fibre consists of the strict transform of a vertical curve with retained affine exceptional lines attached at individual points, and is connected. The general fibre is the $n$-punctured projective line; its log canonical degree $n-2>0$ proves it is nonspecial.

Lemma 6.1; §6.1 · pp. 13–14 §4.1; (4.1)–(4.4) · pp. 8–9

2. Discs inside X kill every puncture meridian

We have $U=X\setminus\bigcup A_i^\circ\simeq E\times(\mathbb P^1\setminus\{q_i\})$. The inclusion induces a surjection on fundamental groups whose kernel is normally generated by meridians of the $A_i^\circ$. In the local blowup chart $b(u,t)=(ut,t)$, a meridian of $A_i^\circ=\{t=0\}$ corresponds to the puncture loop $m_i$ about $q_i$ and bounds the actual disc $\{u=0,|t|\le\varepsilon\}\subset X$. Hence

The map $a_*$ is this isomorphism; the section also shows that the elliptic generators survive. Since puncture loops generate the free fibre group, its image in the total group is zero. No additional orbifold relations have been imposed.

§6.2; (6.3)–(6.4) · p. 15

1. Compare specialness conventions and apply both theorems

Choose a smooth projective SNC compactification $(Y,D)$ with $\kappa(Y,K_Y+D)=0$. Apply Campana's specialness theorem and the calculation on neat models. The comparison after Definition 4.1 resolves an open fibration so that $D_{\mathrm{open}}\le D_{\mathrm{high}}$. Infimum multiplicities then give $\Delta(g,D_{\mathrm{open}})\le\Delta(g,D_{\mathrm{high}})$. This supplies the required implication to open-variety specialness.

Theorem 1.1 yields the finite-index linear-image assertion. For completion, apply ordinary quasi-Albanese exactness to $V$ itself and then Theorem 1.2. This gives finite-dimensionality and class two for full rational completion, together with class two for complex unipotent images. The last two conclusions do not follow merely from the finite-index assertion for linear images.

[Cam, Theorem 6.7, p. 92] [Cam, Corollary 4.11, p. 49] §§4.2–4.3; Definition 4.1 · pp. 10–11 Corollary 3.1 · p. 8