CATALOGUE 033 · PAPER 04 / 5 Overview
Projective Hodge lines and ordinary Iitaka subadditivity
From rank loss of the highest Hodge line to ordinary Iitaka subadditivity
The manuscript claims ordinary Iitaka subadditivity by combining adjoint positivity for an entire rank-one highest Hodge piece with a canonical bundle formula and weak positivity. Its central mechanism separates the dimension of the full period image from the rank of the highest-line map, removes boundary and finite-quotient ramification, and constructs sections without assuming semiampleness.
Read the manuscript on GitHub1. Main results
The following statements report the manuscript's claims in their original order and with their hypotheses.
Theorem 1.1 — Ordinary Iitaka subadditivity
Let $k$ be an algebraically closed field of characteristic $0$, and let $f:X\to Z$ be a surjective projective morphism with connected fibres between smooth connected projective $k$-varieties. For its geometric generic fibre $F$,
Use $(-\infty)+a=-\infty$ and $\kappa(\mathrm{point})=0$.
Theorem 1.2 — Projective adjoint positivity
Let $Y$ be a smooth connected projective complex variety, $U\subset Y$ a dense Zariski open subset, and $\mathbb V$ a rationally polarized integral pure variation of Hodge structure on $U$. Assume that its entire highest nonzero Hodge filtration step is a line: $F^{p+1}=0$ and $\operatorname{rank}F^p=1$. Let $M\in\operatorname{Pic}(Y)\otimes\mathbb Q$. Assume that, after a smooth projective birational modification if needed, a smooth projective alteration $\tau:\widehat Y\to Y$ and $c\in\mathbb Q_{>0}$ exist such that the pulled-back variation has unipotent monodromy along an SNC boundary and its extended highest Hodge line satisfies
After further smooth projective birational modification, there are a surjective morphism $p:Y\to S$ with connected fibres onto a smooth projective variety and a nef rational line $L$ such that
If $S$ is a point, the conclusion is $M\sim_{\mathbb Q}0$.
References on the arrows
Unprefixed theorem, lemma, section and equation numbers refer to Projective Hodge lines and ordinary Iitaka subadditivity. External inputs use the following keys.
- BBT Benjamin Bakker; Yohan Brunebarbe; Jacob Tsimerman — o-minimal GAGA and a conjecture of Griffiths: Algebraicity of integral period images and positivity of Hodge lines. arXiv:1811.12230v3 (2022-10-23).
- BC Yohan Brunebarbe; Benoît Cadorel — Hyperbolicity of varieties supporting a variation of Hodge structure: Log-general type from an immersive period map. arXiv:1707.01327v1.
- FM Osamu Fujino; Shigefumi Mori — A canonical bundle formula: Residual divisors and all-degree direct images in the canonical bundle formula. JDG 56 (2000), author-hosted journal PDF.
- Amb Florin Ambro — Shokurov’s boundary property: Moduli b-divisors and the extended Hodge eigensheaf. arXiv:math/0210271v1 (2002-10-17).
- Fuj Osamu Fujino — Notes on the weak positivity theorems: Weak positivity for producing sections with a negative ample twist. 2015-06-30, version 0.54.
- Has Kenta Hashizume — Log Iitaka conjecture for abundant log canonical fibrations: Addition for general-type fibers of effective klt pairs. arXiv:1902.10923v2 (2019-05-01).
Each citation gives the result and pages. When printed pagination differs from the PDF position, both are shown. GitHub previews do not automatically jump to the cited page.
2. Theorem 1.2 — Removing boundary and ramification from the period image
Proof target：Theorem 1.2 · p. 2 See the hypotheses and conclusion.
Proof route. Two preparatory arguments meet in Theorem 1.1. This section establishes the adjoint positivity of Theorem 1.2; the canonical bundle formula supplies the section comparison. The final two-base argument and interpolation combine them to prove ordinary Iitaka subadditivity.
Let $T$ be a smooth projective model of the actual full period image at neat level, $J$ its extended highest line, and $L_T=cJ$. Write $D_N$ for the boundary with nonzero monodromy logarithm, $R_{\rm div}$ for divisorial ramification over the finite quotient, and $R_{\rm exc}$ for the Jacobian components exceptional over the quotient base. The number $r=\nu(J)$ measures the highest-line map and need not equal $\dim T$.
1. Retain the finite action on the actual period image
Replace the variation by its smallest rational subvariation containing the highest line. A generic Hodge structure is then rationally generated by that line, allowing one to eliminate rational normal monodromy subgroups fixing it. Algebraize the period image at neat level and retain the function field of its normalization when taking the finite quotient. Tensor powers kill the finite stabilizer characters on the line, permitting rational line-bundle descent. The identity with the original $M$ is obtained by comparing extensions on a common alteration and applying a norm; it is not inferred solely from the open locus.
Lemmas 2.1–2.3 · pp. 5–7; §5, (5.1) · pp. 13–15 · BBT, Theorem 1.1 · p. 1
If the period image is a point, the extended line is constant. Descending its identification on a common alteration by the norm gives $M\sim_{\mathbb Q}0$, which is the point-case conclusion of Theorem 1.2.
2. Treat the nilpotent and finite obstructions separately
The nef line satisfies $\nu(J)=\operatorname{rank}_{\rm gen}d\ell$. If its numerical dimension did not drop on a divisor $E$, the tangential limiting map would have rank $r$. Lemma 3.4 would then put the nearby line image inside that limiting image. For $N\ne0$, it lies in $\mathbb P(\ker N)$, contradicting rational minimality. For finite inertia it lies in a single eigenspace; Lemma 2.3 then forces every full period to be fixed. The final contradiction uses the fact that $\mathbb C(T)$ is the function field of the actual period image. Passing first to an arbitrary finite cover or Stein factor would lose this argument.
Proposition 3.2 · pp. 7–8; Lemmas 3.4–3.5 · pp. 9–10; Theorem 4.1 · pp. 11–12 · BBT, Lemma 6.17 · pp. 43–44
3. Remove both divisors by comparing degrees of section growth
After extending the variation across components with $N=0$, generic immersivity of the full period map makes $K_T+D_N$ big. Apply Lemma 5.1 with
An ample perturbation yields interior section growth of degree $r$ in $j$, whereas restrictions to the removed divisors have degree at most $r-1$. Thus $A+jL_T$ is big. Removing the exceptional part and descending through the generically finite map makes $K_{S_0}+jL_0$ big. Connected-fibre factorization is performed only afterwards; its effective Jacobian preserves bigness of $K_S+jL$.
Lemmas 5.1–5.2 · pp. 12–13; (5.2)–(5.5) · pp. 15–16 · BC, Theorem 1.1 · p. 1
The identity $M=p^*L$ and adjoint bigness on the base enter the weak-positivity and interpolation argument of Section 4. Semiampleness of $M$ is not concluded here.
3. Proposition 6.4 — Canonical bundle data and section comparison
Proof target：Proposition 6.4 · pp. 20–21
Put $d=\kappa(F)\geq0$ and write the relative Iitaka factorization as $\widetilde X\xrightarrow{g}Y\xrightarrow{q}Z$. Then $\dim(Y/Z)=d$, and the geometric generic fibre $C$ of $g$ has Kodaira dimension $0$. The construction must retain both the line required by Theorem 1.2 and its comparison with the original pluricanonical sections.
Proposition 6.4
Work over $\mathbb C$. Let $f:X\to Z$ be a surjective morphism with connected fibers between smooth connected projective varieties, and let its geometric generic fiber $F$ satisfy $d=\kappa(F)\geq0$. There exist smooth projective $\widetilde X,Y$, a birational morphism $\mu:\widetilde X\to X$, and surjective maps with connected fibers $\widetilde X\xrightarrow{g}Y\xrightarrow{q}Z$ such that $qg=f\mu$, $\dim Y-\dim Z=d$, and the geometric generic fiber of $g$ has Kodaira dimension zero.
They carry the data $B,M,R$ of Theorem 6.1 (pp. 16–18): $B\geq0$ has SNC support, $(Y,B)$ is klt, and $M$ is the nef trace of the moduli b-divisor descending to $Y$, with
The image of $R^-$ in $Y$ has codimension at least two, and $R^-$ is exceptional over the original $X$. After a generically finite semistable base change, $M$ is realized as a positive rational multiple of the extended Hodge line of a polarizable rational variation with entire top piece of rank one, as in (6.4). For $D=K_Y+B+M$,
where $\bar\eta$ is a geometric generic point of $Z$. The same bigness holds on very general smooth $q$-fibers.
1. Use the least index to obtain the entire rank-one top piece
Let $b=\min\{m>0:H^0(C,mK_C)\ne0\}$. Its unique section $s$ defines the root cover by $\xi^b=s/\omega^b$. Lemma 6.3 proves $\kappa(W)=0$ and $h^0(W,K_W)=1$ for a resolution $W$. Spreading out the construction makes the entire top piece of $R^nh_*\mathbb Q$ rank one. Even when the cyclic character is not rational, the full rational variation is retained.
2. Separate the negative auxiliary boundary from the residual divisor
With $B\geq0$, $(Y,B)$ klt, and $M$ nef, the formula gives
The image of $R^-$ has codimension at least two on the base, and $R^-$ is also exceptional over the original $X$. The latter property requires flattening; it does not follow just from the image dimension. On the generic fibre the auxiliary sub-pair has boundary $-\operatorname{div}(s)/b$, which need not be effective. The argument uses nefness and the extended Hodge-line identification, without importing moduli abundance that requires an effective generic boundary.
Theorem 6.1, Remark 6.2 · pp. 16–19 · FM, Theorem 4.5 · PDF pp. 11–12 · Amb, Lemma 5.2(4)–(5), Proposition 5.5 · pp. 15, 17–18
3. Compare sections on the total space and on the generic fibre
Set $D=K_Y+B+M$. Exceptionality of $R^-$ identifies the sections of $K_{\widetilde X}+R^-$ with the original pluricanonical sections. Multiplication by the section of $R^+$ injects $H^0(Y,mD)$ into that space. On the geometric generic $q$-fibre, flat base change and the all-degree direct-image condition give
Thus $D|_{Y_{\bar\eta}}$ is big. Subsequent birational modifications use the positive part of the crepant boundary and an effective exceptional correction to preserve these comparisons.
Proposition 6.4, Lemmas 6.5–6.6 · pp. 20–22
Section 4 uses both $q$ to the original base and $p$ to the period base. The section injection is retained to transfer the final lower bound back to $X$.
4. Theorem 1.1 — Two bases and an interpolation identity
Proof target：Theorem 1.1 · p. 2 See the hypotheses and conclusion.
If $\kappa(F)=-\infty$ or $\kappa(Z)=-\infty$, the inequality in Theorem 1.1 is automatic under its convention. The argument below treats the case where both are nonnegative.
Assume $\kappa(Z)\geq0$. The identity $M=p^*L$ from Theorem 1.2 does not supply a factorization between $p:Y\to S$ and $q:Y\to Z$. If $S$ is a point, apply log-general-type-fibre subadditivity directly to $q$. The remaining argument treats $\dim S>0$.
1. Obtain nonvanishing on a period fibre through the product map
Resolve the Stein factorization of the image of $(p,q)$ as $Y\to W\to S\times Z$. A general $W_s$ is generically finite onto its image in $Z$, while $W\to Z$ is dominant. Contracting top forms with the exterior base-direction factor turns a pluricanonical section on $Z$ into a nonzero section on $W_s$. On the general fibre of $Y_s\to W_s$, the line $M$ becomes trivial, so bigness of $D$ becomes log-canonical bigness. A first use of log-general-type-fibre subadditivity gives log nonvanishing on the geometric generic $p$-fibre.
2. Use weak positivity to construct an effective divisor with a negative ample twist
Choose $j>1$ and an ample rational divisor $H$ with $K_S+jL-H$ big. The preceding nonvanishing makes a relative log pluricanonical direct image have positive rank, allowing weak positivity. Multiplying sections of symmetric powers and pushing down exceptional poles after flattening yields
The extension and pushdown in Lemma 7.4 connect generic generation to effectivity on the whole total space.
Theorem 7.2, Lemma 7.4, (7.6) · pp. 23–26 · Fuj, Theorem 1.1 · pp. 1–2
3. Balance coefficients and reduce to an effective klt pair
Openness of the big cone allows $0<v<1$ with $(K_Y+B+vM)|_{Y_{\bar\eta}}$ big. Set
These choices simultaneously balance the coefficient of $M$ and cancel the ample twist. Since $A_S$ is ample, a sufficiently divisible general section gives $\Gamma\sim_{\mathbb Q}p^*A_S$ with $(Y,B+\Gamma)$ klt. The divisor $E_2$ is its log canonical class and is big on the generic $q$-fibre. Apply subadditivity a second time and multiply by sections of $E_1$ to obtain
Finally descend the data to a finitely generated field $K/\mathbb Q$ and choose $K\hookrightarrow\mathbb C$. Flat base change for every pluricanonical space, including those of the geometric generic fibre, transfers the result to any algebraically closed characteristic-zero field.
§7.3, (7.5)–(7.7) · pp. 25–26; Lemma 7.5, §7.4 · p. 27
This yields ordinary subadditivity, Theorem 1.1. WV’s alternative adjoint comparison concerns the separate results of §8; this main theorem is not substituted as a required WV input.
PH, Theorem 1.1 · p. 2; Theorem 8.1 · p. 28 · WV, Remark 3.17 · p. 25
5. Which papers supply which steps
| Input | Use and role | Checking scope |
|---|---|---|
| BBT arXiv:1811.12230v3, Theorem 1.1 / Lemma 6.17 (pp. 1, 43–44) | Period image in §5; nefness and numerical rank in §3 | Statements compared with uses |
| BC arXiv:1707.01327v1, Theorem 1.1 (p. 1) | Bigness of $K_T+D_N$ in §5 p.15 | Statement, immersivity, and choice of boundary compared |
| FM JDG 56 (2000), Theorem 4.5 (PDF pp. 11–12; journal pp. 177–178) | Residual divisor and sections in Theorem 6.1 | Statement and boundary-zero use compared; complete model construction unchecked |
| Amb arXiv:math/0210271v1, Theorem 4.4, Lemma 5.2(4)–(5), Proposition 5.5 (pp. 12–15, 17–18) | Hodge eigensheaf and pullback in Theorem 6.1 and (6.4) | Relevant statements compared; full cited package and base-change proofs unchecked |
| Fuj 2015-06-30, v0.54, Theorem 1.1 (pp. 1–2) | Theorem 7.2 → effectivity in Lemma 7.4 | Statement and application to the projective klt pair compared |
| Has arXiv:1902.10923v2, Theorem 2.11 (p. 5) | Addition in Lemma 7.3 and §7.3 | Statement compared; no abundance of the base canonical class required |
These are direct inputs to the main route. The uses of the fixed-part theorem, monodromy normality, and nilpotent-orbit and extension theory were read in PH, but their external source statements have not all been checked. Within catalogue 033, WV, Remark 3.17 · p. 25 cites Lemma 8.4 / Theorem 8.1 as an alternative adjoint comparison. WV explicitly uses its direct comparison in its own proof, so this relationship is alternative. The ordinary-subadditivity comparison with OI is likewise treated as a separate proof route; no direct PH→OI edge is asserted. Other relationships remain unexamined.
6. Return to the sources
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Theorem 1.2 · p. 2 — Distinguish the highest-line hypothesis, nef descent, and adjoint bigness
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Proposition 6.4 · pp. 20–21 — Read the canonical bundle reduction preserving exceptionality of the residual divisor and the Hodge identification
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Theorem 1.1 · p. 2 — Read the main hypotheses and conclusion alongside the proof passages below
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§§2–5 · pp. 5–16: rational minimality, tangential limits, the two rank losses, quotient descent, and volume estimates.
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§6 · pp. 16–22: the least-index root cover, the Hodge identification across the boundary, and residual section comparisons.
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§7 · pp. 22–27: the product map, effectivity from weak positivity, interpolation, and scalar extension.
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§§8–9 · pp. 27–39: alternative boundary estimates by integral tensor replacement and Thom forms, separate from the main route.
The reading covers Theorems 1.1–1.2, Lemmas 2.1–2.3 / 3.4, Theorem 4.1, descent and volume estimates in §5, and the central proof passages of §§6–7. Section 5 specifies which input statements and applications were compared. Only the statements of Theorem 8.1 / Lemma 8.4 and the corresponding WV comparison were read. The boundary mixed-Hodge identification, multivariable extensions, every auxiliary proof, the alternative proofs in §§8–9, and full external proofs have not been independently verified. This is an AI-generated account of the authors' argument, not certification of the manuscript or expert peer review. The source is the 40-page version of 2026-09-27, pinned to commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.