Logarithmic Kodaira dimension and whole-fiber variation

1. Obtain the candidate field from section systems

OI’s logarithmic lower bound yields $\mathbb C(T_0)\subseteq\mathbb C(Y)$ and $\bar\kappa(U)=d+\dim T_0$. The next section develops this construction.

Proposition 2.10, Remark 2.11 · pp.12–14 · OI, Corollary 6.2 · pp.40–41

2. Turn Hodge-line triviality into birational constancy

Apply OI’s adjoint comparison to the restricted root-cover variation to make its entire top line flat with finite character. Contraction of poles and local flows make $J$ constant; article Section 4 develops this step.

Propositions 3.9, 3.12, Corollary 3.16, Proposition 4.10 · pp.21–25, 42–43 · OI, Theorem 3.1 · pp.17–18

3. Return to the whole fiber while retaining coordinates and actions

Markings fix Iitaka-base coordinates and a finite normalization. Constant cocycles and inertia at moving divisors allow descent of covers and actions and identification of the original function field. Article Section 5 obtains $\operatorname{Var}(f)\leq\dim T_0$.

Proposition 6.2, Propositions 7.2–7.3, §7.4 · pp.47–59

Combine this bound with the section-system dimension identity to obtain Theorem 1.1. Corollary 1.2 sets the boundaries to zero; Corollary 1.3 is a separate consequence using LA and Taji. The diagrams for the parameter field, first constancy, and whole-fibre descent expand these three stages.

1. Combine the two maps while retaining coefficient-one boundary

Choose SNC compactifications $(X,D_X)\to(Y,D_Y)$ with $f^{-1}(V)=U$. Applying OI Corollary 6.2 gives $\bar\kappa(U)\geq d+\bar\kappa(V)$. Resolve the absolute Iitaka map of $K_X+D_X$ as $q:X'\to Z$. The relative algebraic closure of $\mathbb C(Y)\mathbb C(Z)$ in $\mathbb C(X)$ defines $X'\xrightarrow{x}W\xrightarrow{h}Z$ and $g:W\to Y$. The map $W\to Y\times Z$ is generically finite onto its image, and the relevant generic fibers are geometrically integral.

Lemma 2.6, Proposition 2.7 · pp.8–10 · OI, Corollary 6.2 · pp.40–41

2. Contract logarithmic forms to compute the rank of the image family

Fix $0\ne\xi\in H^0(Y,m(K_Y+D_Y))$. Set $r=\dim W_z$ and $p_1=\dim Y-r$. Contracting $g^*\xi$ against $p_1$ directions from $Z$ produces a nonzero logarithmic pluricanonical form on $W_z$. Since $X'_z$ has logarithmic Kodaira dimension zero, subadditivity for $x_z$ gives

Here $J$ is the geometric generic fiber of $x$. The contracted sections in a fixed degree are proportional. Consequently the ratios of maximal minors of the normal deformation map $T_zZ\to N_{g(W_z)/Y,y}$ do not depend on $y$. Its constant kernel gives rank $p_1$ for the differential of the Hilbert parameter map.

Lemmas 2.8–2.9, proof of Proposition 2.10 · pp.10–13

3. Embed the parameter field in the original base field

Take the relative algebraic closure in $\mathbb C(Z)$ of the parameter field of the image family $g(W_z)$, and call its smooth model $T_0$. The resolved universal image family $S$ maps generically finitely to $Y$. In the tower

the first extension is algebraic, while $\mathbb C(Y)$ is relatively algebraically closed in $\mathbb C(X)$. Thus $\mathbb C(S)=\mathbb C(Y)$, an identification stronger than a dimension count. Together with $\dim(W/Y)=d$, it gives

It remains to descend the entire fiber to $b=\overline{\mathbb C(T_0)}\subseteq\Omega$.

Proposition 2.10, Remark 2.11 · pp.12–14

Fix the resulting $b=\overline{\mathbb C(T_0)}$ as the candidate definition field. Constancy of $J$, followed by markings and descent of finite covers, must still recover the original $F$.

WV, §§3–7 · pp. 14–59

For the last conclusion of Proposition 2.10, pull the fixed nonzero base section $\xi$ to $S$ and restrict it nontrivially to very general $S_t$. Adjunction gives $\kappa(S_t,K_{S_t}+D_S|_{S_t})\geq0$, retaining base nonvanishing for the later marked-model comparison.

1. Relate actual relative-form orders to the Hodge line

The connected cyclic cover obtained from the least $p>0$ and $0\ne\omega_J\in H^0(J,pK_J)$ has a resolution $\widetilde J$ with $h^0(\widetilde J,mK_{\widetilde J})=1$ for every $m>0$. Its root form spans the entire top-form space. For a base divisor $P$, distinguish the minimum $t_P$ on actual source components from the threshold $\lambda_P$ on all higher models:

Here $m_Q=\operatorname{ord}_Q(x^*P)$, $r_Q=p^{-1}\operatorname{ord}_Q(\omega_J)$ is measured in the actual relative canonical bundle, and $d_Q=\operatorname{coeff}_Q(D')$. The integrability exponent of the Hodge norm identifies the parabolic order of its line $M$ as $\lambda_P-1$. Thus $T=\sum t_PP\sim_{\mathbb Q}B+M$, $0\leq B\leq1$, and $B\geq D_W^0$.

Lemma 3.1, Propositions 3.3 and 3.5 · pp.14–19

2. Apply adjoint comparison anew to the restricted variation

Multiplication by the relative generator transfers sections to the fixed $X_{*,z}$ while preserving their ratios, giving

Meanwhile $D=K_{W_z}+B|_{W_z}$ is rationally effective. Apply OI Theorem 3.1 to the restricted integral variation to obtain $M|_{W_z}\sim_{\mathbb Q}a^*A$ with $K_R+cA$ big. Weak positivity from Fuj and section transfer give $\kappa(D+cM)\geq\dim R$. For $c\geq1$,

transfers that lower bound to $D+M$, forcing $R$ to be a point. Hence $M|_{W_z}\sim_{\mathbb Q}0$ and $\kappa(W_z,D)=0$. Curvature on test curves and Del's finite-character theorem, applied on the restriction, give a flat entire top line with finite monodromy.

Propositions 3.7, 3.9 and 3.12, Lemma 3.11, Corollaries 3.13–3.16 · pp.20–25 · OI, Theorem 3.1 · pp.17–18 · Fuj, Theorem 1.1 · pp.1–2 · Del, Corollary 4.2.8(iii)(b) · printed p.47 / PDF p.44

3. Contract poles of the lifted vector field

Over a curve, kill the finite character and lift the invariant class to a global form $\Theta$. Its kernel defines a rational vector field with $d\pi(v)=\partial_t$ and $\iota_v\Theta=0$. Poles may remain along the zero divisor of the relative top form. Lemma 4.4 uses DHP extension, while Lemmas 4.6–4.7 use PT relative Bergman weights. These lead to positive limiting fixed order $\sigma_S(K_D)>0$ at every pole divisor $S$. The weight is constructed before the final fiber is selected, and the regularization, degree, and cutoff limits have a prescribed order.

Sections 4.1–4.5 · pp.26–40 · DHP, Theorem 4.1 · pp.20–21 · PT, Theorem 4.2.7 and Remark 4.3.1 · pp.36–37

Choose one small $\varepsilon>0$ for all the finitely many poles. The big klt adjoint model of $K_D+\varepsilon H$, supplied by BCHM, contracts their positive fixed orders. Normality extends the resulting regular vector field across codimension two. Local flows and algebraic spreading of graphs give birational constancy after a finite field extension. Applying this equivariantly to the root cover and taking invariant function fields makes $J$ constant in the $h$-fiber directions. Descending the original whole $F$ still requires the next step.

Proof of Theorem 4.1, Corollary 4.9, Proposition 4.10 · pp.40–43 · BCHM, Theorem 1.2(2) · p.5

This fixes the Kodaira-zero fiber $J$. The next descent retains Iitaka-base coordinates and group actions to fix the entire geometric generic fiber.

WV, §6 introduction・§7.4 · pp. 46, 58–59

1. Recover the finite normalization using coordinate and sum markings

For an embedding of $P_0$ with coordinates $x_0,\ldots,x_N$, label the pullbacks of $x_0=0$, $x_j=0$, and $x_0+x_j=0$ by distinct small coefficients. Slightly decrease the old boundary to make the generic pair klt, and add further general markings to make it big. The added line bundles restrict trivially to $a_0$-fibers, preserving the upper bound $\kappa(H_0,K_{H_0}+\Delta)\leq d$. Proposition 5.3, using KP, gives the lower bound $d+\operatorname{Var}(\text{marked model})$. The marked model is therefore constant after a finite parameter extension. Let $C_*$ be the underlying normal projective variety of this model pair over $b$.

Proposition 5.3, Proposition 6.2 · pp.45–50 · KP, Theorem 9.9 · p.50

Coordinate divisors determine $x_j/x_0=c_jf_j$ with $f_j\in b(C_*)$; the sum divisors force $c_j\in b$. The morphism $C_*\to P_0$ itself descends, fixing the normalization $V'\to P_0$ in its function field. Images of positive old-boundary components become fixed $b$-divisors as well. With $k=b(V')$ and $K=k(I)$, the preceding constancy statement supplies a reference variety $J_0/k$. The markings are auxiliary identifiers and are discarded from the final fiber.

Proposition 6.2, especially (6.12) · pp.47–51

2. Remove the nonconstant cocycle and the inertia at moving divisors

Lemma 7.1 uses Han's structure theorem for birational automorphism groups. The abelian identity component and the unique divisibility of $A_1(\overline K)/A_1(\overline k)$ allow averaging over a finite quotient to remove the nonconstant cocycle as a coboundary. The resulting finite Galois splitting has a constant cocycle with faithful pairs $(z_g,g|_{k'})$.

Lemma 7.1 · pp.52–54

Comparison with the original Han statements remains pending.

At a moving divisor $P$, the old boundary coefficient is $B_P=0$. Equality in the order inequalities produces an actual source component $Q$ with $m_Q=1$ attaining $\lambda_P$. On the split constant family the special fiber is the unique minimizing divisor. The identities

give $e_Q=e$. Total and base inertia coincide, and faithfulness of the constant cocycle makes inertia trivial. Repeating the argument after every finite parameter extension is needed to control all moving divisors over the algebraic closure.

Corollary 3.4 · p.18, Lemma 6.4 · pp.51–52, Proposition 7.2 · pp.55–56

3. Descend the cover, its action, and its coefficient field together

Remove the fixed branch divisors and singular locus to obtain $O\subset V'$. The manuscript invokes purity to make the splitting cover finite étale over $O$. In characteristic zero, Lan's base-change invariance descends covers and their morphisms, retaining both the group action and the map carrying $k'$. For the field $E_1$ of a connected component and its stabilizer $G_1$, set

Commutation of finite-group invariants with field extension and identification of the original field tower give

The right side is the function field of the entire original geometric generic fiber. Hence $\operatorname{Var}(f)\leq\operatorname{trdeg}_{\mathbb C}b=\dim T_0$. Combine this with the dimension identity and subadditivity from article Section 3 to obtain Theorem 1.1.

Proposition 7.3 and Section 7.4 · pp.57–59 · Lan, Theorem 1.1 and Remark 1.5 · pp.2–3

For purity, only the receiving argument and its stated hypotheses were checked; the original SGA1 statement was not rechecked.

This completes the lower bound involving variation in Theorem 1.1. Corollary 1.2 uses this output with empty boundaries.

WV, Theorem 1.1 / Corollary 1.2 · p. 2; §7.4 · p. 59

1. Replace logarithmic by ordinary Kodaira dimensions

With $D_X=D_Y=0$, one has $\bar\kappa(X)=\kappa(X)$ and $\bar\kappa(Y)=\kappa(Y)$. The hypothesis $\kappa(Y)\geq0$ is exactly the base hypothesis of Theorem 1.1, so its inequality gives Corollary 1.2 immediately.

Proof of Corollary 1.2 · p.2, Section 7.4 · p.59

The ordinary Iitaka–Viehweg inequality applies to projective fibrations over a base of nonnegative Kodaira dimension. The additional relative-Iitaka route is not a premise of this proof.

WV, Corollary 1.2 · p. 2; §9 introduction · p. 69

1. Obtain canonical good models from LA's comparison

For every closed fiber $F_v$, BDPP makes $K_{F_v}$ pseudo-effective. LA Corollary 11.2 supplies a semiample $K_{M_v}$ and a common-resolution identity

For every prime $P$ exceptional over $M_v$,

Thus $M_v$ is canonical and gives a good minimal model in the sense used by Taji.

Proof of Corollary 1.3 · p.5 · BDPP, Corollary 0.3 · p.2 · LA, Corollary 11.2 · pp.73–74

2. Apply Taji with whole-fiber variation

For smooth projective families whose fibers have good minimal models, Taj Theorems 1.1–1.2 supply isotriviality over special bases and the two variation bounds according to the sign of the base's logarithmic Kodaira dimension. Taji's definition on p.2 also uses a birational field of definition of the entire fiber. When $\bar\kappa(V)=0$, combine the bound with $\operatorname{Var}(f)\geq0$.

Corollary 1.3 · p.5 · Taj, Theorems 1.1–1.2 and definition of variation · p.2

For a smooth family with $\kappa(F)\geq0$ and $\bar\kappa(V)\geq0$, comparing RA Corollary 1.2's additivity with Theorem 1.1 gives the same upper bound. This alternative does not cover the negative-base branch or all special bases.

Comparison paragraph · p.5 · RA, Corollary 1.2 · p.2

A base of logarithmic Kodaira dimension zero gives birational isotriviality. For the negative-base and special-base conclusions, retain the hypotheses of the LA–Taji route.

WV, Corollary 1.3 · p. 5