Orbifold and logarithmic Iitaka subadditivity

1. Preserve exceptional coefficients and fix the original base

Resolve the base and flatten before resolving the source. Source primes mapping into base codimension at least two become exceptional over a fixed smooth reference. Taking the strict transform plus reduced exceptional boundary makes the difference of log canonical divisors effective and exceptional; pluricanonical sections, products, and ratios are preserved. Fix the resulting smooth base $Y$ and set $C=B(f,\Delta)$. Further changes of $W$ retain $C$ and the coefficient-one condition, together with $\kappa(Y,K_Y+C)\geq b$.

Lemma 2.1 · Corollaries 2.3, 2.5 · Lemmas 2.4, 2.6 · pp. 5–9; §6 · p. 39

2. Replace the Kodaira-zero fiber by an actual Hodge line

The relative Iitaka factorization $X\xrightarrow{g}W\xrightarrow{h}Y$ has $\dim W_y=r$ and logarithmic Iitaka dimension zero on a very general $g$-fiber $G$. Take a root of a relative generating form $s$ in a sufficiently divisible degree and use the highest Hodge part of its distinguished character. This does not assert that the entire highest Hodge space is a line. Coefficient-one boundary requires the mixed Hodge structure of an open fiber and passage to the unique nonzero pure weight grade. The extension theorem FF retains this identification at the boundary.

Lemmas 2.6, 2.8–2.10 · pp. 9–15 · FF, Theorem 1.1(ii), (iv) · p. 2

The boundary $T$ compares the original form's allowed poles with its Hodge-extension order. In the notation of §2.6.3,

Here $a_E=\operatorname{ord}_E(g^*D)$ and $l_E=m^{-1}\operatorname{ord}_E(s\wedge g^*\omega^m)$, where $\omega$ is a local ordinary base volume form. The minima include every component of the prepared SNC source. This gives $B(g,\Delta)\leq T\leq1$ and, in sufficiently divisible degrees $q$,

The untested primes mapping into base codimension at least two are exceptional over the reference. Descend first to $X_0$, then extend across codimension two. Specifying this reference is the bridge from a generic formula to actual global section spaces.

Proposition 2.7 · (2.12), (2.14), (2.16) · pp. 10, 13–17

3. Recover the hypotheses for addition from relative section comparison

Proposition 2.7(iv) also identifies direct images over $Y$ and preserves section ratios. The relative Iitaka image has dimension $r=\dim W_y$, so $K_W+T+M$ is big on very general $h$-fibers. Independently, the multiplication of weighted orders in Lemma 2.12 gives $B(h,T)\geq C$ and coefficient one on every prime of $W$ mapping into codimension at least two in $Y$. Both are hypotheses required for addition.

Proposition 2.7(iv) · Lemma 2.12 · pp. 10, 17; (6.3)–(6.5) · pp. 39–40

4. Supply adjoint positivity on the period base

Theorem 3.1 starts with the highest Hodge line of a complex direct summand of a pure real-polarizable variation. After modification it supplies a smooth projective $S$ with $M=p^*P$, $P$ nef, and $K_S+a_0P$ big. The integral lattice belongs to the ambient variation; the chosen complex summand need not be rational.

The proof constructs a quotient from the full periods of rational adjoint factors. Compactification Vil and the Moishezon criterion BC provide a projective model. If $D_S$ marks nonidentity local monodromy, including finite nonidentity monodromy, then $K_S+D_S$ is big. Lemma 4.6 uses BT and other inputs to assert $\nu(P|_{D_i})<\nu(P)$ for each component. In Lemma 4.7, the available section count has degree $\nu(P)$ in the coefficient $t$, whereas the boundary loss has smaller degree. This makes $K_S+a_0P$ big. The rank of the chosen-line map is distinguished from the rank of the full period map.

Theorem 3.1 · pp. 17–18; Proposition 4.5 · Lemmas 4.6–4.7 · pp. 25–31 · Vil, Theorem 2.19 · p. 16 · BC, Theorem 1.1 / Corollary 1.3 · pp. 1–2 · BT, Theorem 1.1 · p. 2

5. Apply addition and return the sections to the original source

All hypotheses of Proposition 3.4 now hold. The section operation explained below gives

Proposition 2.7 and exceptional descent return these systems to the original $(X,\Delta)$. Ratios and image dimensions are preserved throughout, transferring the lower bound to the theorem's left side. If $S$ is a point, $M$ is rationally trivial and Proposition 3.2 applies directly.

Propositions 3.2, 3.4 · pp. 18–20; §6 · (6.6) · p. 40

Section 4 transfers this lower bound to a logarithmic base. The intermediate Theorem 3.1 also supplies WV’s restricted Hodge comparison.

OI, §6 · pp. 39–41 · WV, Theorem 3.8 / Proposition 3.9 · p. 21

1. Construct the positively twisted systems by general-type addition

Openness of the big cone gives $0<c<1$ with $D_W+cM$ still big on very general $h$-fibers. For an ample divisor $A$ on $S$ and any rational $\eta>0$, the divisor $cP+\eta A$ is ample. A general rational member $Q_\eta\sim_{\mathbb Q}cM+\eta p^*A$ leaves $T+Q_\eta$ an SNC boundary, preserves old coefficient-one components, and retains the orbifold base lower bound. Proposition 3.2 gives

To prove Proposition 3.2 over a nonprojective base, §5 slightly decreases only the horizontal boundary, obtaining klt general fibers. Lemma 5.2 compares these with a projective stable family of maximal variation. Theorem 8.1 and Corollary 8.3 of KP supply bigness of its relative log canonical divisor. Forms obtained on a finite cover descend through finite-group invariants; Lemma 5.4 retains their fiberwise image dimension. For the products with base sections, $C\leq B(h,T)$ controls poles above base divisors and $T_E=1$ handles higher-codimension images. The original projective positivity theorem is thus applied to an auxiliary stable family.

Proposition 3.2 · p. 18; (3.5) · p. 19; Lemmas 5.2–5.5 · pp. 32–38 · KP, Theorem 8.1 / Corollary 8.3 · p. 40

2. Obtain one negatively twisted section by weak positivity

Restrict a nonzero section from the previous step to a very general $p$-fiber. The pulled-back twists disappear, giving relative nonvanishing for $D_W$. Choose $a>\max\{1,a_0\}$ and a small positive rational $\lambda$ so that $H=K_S+aP-\lambda A$ is big. Lemma 3.3 uses Fuj to give a nonzero section of a positive multiple of

After flattening, the possible remaining poles are exceptional over the original smooth $W$, allowing descent to that fixed space.

Lemma 3.3 · pp. 18–19; (3.6)–(3.7) · p. 20 · Fuj, Theorem 1.1 · pp. 1–2

3. Cancel exactly and preserve ratios

Set

Then $0<\theta<1$ and

Fix $0\ne e\in H^0(W,qE_-)$. For sufficiently divisible $n$, multiplication by $e^{n\theta/q}$ gives

Away from the zeros of $e$, the common factor cancels from every ratio. The image dimension of the earlier systems is retained. Semiampleness of $P$ is not required for this step.

Proposition 3.4 · (3.8) and the following multiplication · p. 20

Multiplication by a fixed section preserves image dimension, allowing the two positivity inputs in Section 2 to produce the main lower bound.

OI, Proposition 3.4・§6 · pp. 19–20, 40

1. Pull back logarithmic systems on a neat model

In Lemma 6.1 take a neat model $f':(X',\Delta')\to Y'$, with $q:Y'\to Y$, and put $E_Y=(q^{-1}\operatorname{Supp}D_Y)_{\mathrm{red}}$. Every source prime dominating a component of $E_Y$ has coefficient one, so $B(f',\Delta')\geq E_Y$. Pullback of logarithmic volume forms preserves section ratios. The neat-model identity from Cam gives

Lemma 6.1 · p. 40; (2.2) · p. 5 · Cam, Corollary 4.11 · p. 49

2. Substitute into the main theorem and pass to open varieties

Apply Theorem 1.1 with $\Delta=D_X$. For a dominant morphism $U\to V$ of smooth quasi-projective varieties with geometrically connected general fiber, resolution of the graph and boundaries yields compatible compactifications satisfying Corollary 6.2. Thus $\bar\kappa(U)\geq\bar\kappa(U_v)+\bar\kappa(V)$ for very general $v$. Zero boundaries give ordinary subadditivity over $\mathbb C$.

Proof of Corollary 6.2 and the following discussion · pp. 40–41

This lower bound is used in WV’s parameter-field construction and in the corollary turning RA’s independent upper bound into equality.

WV, Theorem 2.1 · p. 6 · RA, Corollary 1.2 / §7.6 · pp. 2, 50

For compact class-C manifolds, the same neat-model boundary comparison and substitution into Theorem 1.1 apply without adding projectivity of the base or total space. This gives the compact class-C conclusion of Corollary 6.2.

1. Compare sections and image dimensions degree by degree

For a complex projective family, apply cohomology and base change in every divisible degree, and choose opens on which the relative rational maps have constant fiberwise image dimension. Outside a countable union of exceptional sets, the very general and geometric generic fibers have the same Iitaka dimension. For a field extension $K\subseteq K'$,

preserves nonvanishing and the image dimension of the complete linear system.

Proof of Corollary 6.3 · p. 41

2. Descend to a finitely generated field and apply the complex inequality

Descend $f$, its projective embeddings, and boundaries to a subfield $K\subset k$ finitely generated over $\mathbb Q$, and choose $K\hookrightarrow\mathbb C$. Smoothness, geometric integrality, and boundary conditions descend, as does $f_*\mathcal O_X=\mathcal O_Y$ by faithful flatness. Apply the complex logarithmic inequality and identify all three Iitaka dimensions with those over $k$ by field-extension invariance.

Proof of Corollary 6.3 · p. 41

Ordinary and logarithmic subadditivity are now available over algebraically closed fields of characteristic zero. PH’s ordinary subadditivity is a separate proof.

OI, Corollary 6.3 · p. 41 · PH, Theorem 1.1 · p. 2