The reverse logarithmic Kodaira inequality and additivity

1. Treat both negative-infinity cases as vanishing statements

If $\kappa_F=-\infty$, the nonvanishing locus of a hypothetical nonzero total-space section has image containing a dense open subset of the base. Restriction and adjunction would give $H^0(F,mL_F)\ne0$ on a very general fiber, a contradiction. If $\kappa_Y=-\infty$, Proposition 1.3 instead contradicts the existence of a total-space section by producing base nonvanishing. Both arguments exclude sections in every positive degree.

§2.2 · p. 5

2. Obtain restriction injectivity in every degree at one fiber

When $\kappa_Y=0$, one has $h^0(Y,aL_Y)\leq1$. Each degree therefore contributes at most one zero divisor of a nonzero section. Choose $y$ outside their countable union, also making all fiber plurigenera attain their very general values. A nonzero $s$ in the restriction kernel would, by Proposition 1.3, produce a nonzero base section vanishing at $y$, contradicting this choice. Hence

The essential point is that Proposition 1.3 applies at the specified point of the original $V$, independently of the auxiliary open set attached to each section. Finite generation of the section rings is unnecessary.

§2.3 and (2.3)–(2.5) · pp. 5–6

3. Reduce positive base dimension to zero on an Iitaka fiber

For $k=\kappa_Y>0$, resolve the logarithmic Iitaka map to $q:Y'\to T$ with $\dim T=k$. Use strict-transform boundaries plus reduced exceptional divisors and make a compatible resolution $f':X'\to Y'$; logarithmic section spaces are unchanged. A very general fiber $G$ of $q$ has $\kappa(G,K_G+D'|_G)=0$. The induced morphism from $H=(q\circ f')^{-1}(t)$ to $G$ retains the required smoothness with its boundary strata, so the preceding case applies:

The last upper bound is the image-dimension estimate for a total-space linear system combined with $q\circ f'$. It includes the case where $G$ is a point. If $H$ has Kodaira dimension $-\infty$, restriction excludes all total-space sections instead.

§2.1 and (2.1) · p. 5; §2.4 · pp. 6–7

Section 5 combines this independent upper bound with OI’s logarithmic lower bound. Its key input is Proposition 1.3, developed next.

RA, §2・§7.6 · pp. 5–7, 50

1. Keep new cyclic-cover discriminants out of the base pole divisor

Taking an $m$th root of $s$ produces a logarithmic mixed Hodge class; projection to a suitable pure weight quotient remains nonzero. The coefficients of the resulting $u:\mathcal O_Y(-L_Y)\dashrightarrow\mathcal E^d$ and its Higgs iterates are measured in the original $\Omega_Y^1(\log D)$ frames. Proposition 3.1 proves their regularity in the unipotent Hodge extension on each prescribed higher model.

In transverse-curve tests, the cotangent tensor frames remain frames of an external pullback bundle. They are not replaced by differentials on the testing curve and then divided back. Thus an extra cyclic-cover discriminant inside the original $V$ introduces no new allowed poles. Regular projection to a pure weight quotient uses the subbundle property of Lemma 3.2, with the semistable logarithmic de Rham/canonical-extension comparison from FF14 and the strictness supplement FF17.

Proposition 3.1 and Lemma 3.2 · pp. 8–13 · FF14, §4.9 and Lemma 4.10, Step 1 · pp. 24–31 · FF17 · pp. 1–2

2. Retain the original source frame when blowing up the specified point

If $s|_{X_y}=0$, its coefficient near the smooth relative SNC fiber belongs to $\mathfrak m_y\mathcal O_X$. Blow up $y$, take a transverse parameter $t$ at the exceptional divisor, and set $t=\tau^e$ with $m\mid e$. In the original $L_X$ frame, the root $r$ satisfies

The right-hand side is regular, and normality makes $r/\tau^{e/m}$ regular. The weight-subbundle property in Lemma 3.2 allows this scalar factor to be divided out before projection. After a homogeneous tensor operation of degree $a$, the vanishing order is still at least $ae/m>0$ in the pulled-back original $\mathcal O_Y(-aL_Y)$ frame. Replacing it by the new logarithmic canonical frame on the blowup would lose the prescribed-point information.

Lemma 3.3 and (3.15) · pp. 13–14

3. Extract a scalar coordinate for a point quotient; prove positivity otherwise

Proposition 4.1 constructs a homogeneous replacement $u_*:\mathcal O_Y(-aL_Y)\dashrightarrow\mathcal E_0$ and a connected quotient $S$ retaining the full Lie period and compact flags. [BBT, Theorem 1.1] algebraizes the period image of the full Lie variation, in which an integral lattice has been retained. The quotient is not obtained merely from the determinant of the highest Hodge bundle.

If $S$ is a point, the descended variation is constant. In a constant frame, each coordinate of $u_*$ is a rational section of $aL_Y$. Divisorial regularity and normality make it global; choose a nonzero coordinate as $\tau$. The positive exceptional order from the preceding step is the vanishing order at the original $y$, so $s|_{X_y}=0$ implies $\tau(y)=0$. When $\dim S>0$, the next section gives $\kappa_Y>0$. Consequently only the point case can occur for $\kappa_Y=0$, proving both conclusions of Proposition 1.3.

Proposition 4.1 · pp. 14–15; Lemma 4.7 · pp. 19–20; §7.1 and Proposition 7.1 · p. 43; §7.5 · pp. 49–50 · BBT, Theorem 1.1 · p. 1

A base section vanishing at the specified point makes restriction to one fiber injective in every degree when the base has Kodaira dimension zero. The next section handles the positive-dimensional period quotient.

RA, §2.3・proof of Proposition 1.3 · pp. 5–6, 50

1. Construct a divisor representing the complete section spaces

The coefficient system is constant on the generic fiber over $S$, so a nonzero logarithmic section exists there and its Iitaka dimension is nonnegative. Form its relative Iitaka fibration. The unique degree-$a$ generator $\omega$ on the generic $x$-fiber $J$ factors $u_*=\omega x^*u_W$. For a prime $A$ on $W$, define

Here $D_Y$ is the reduced boundary on this model. If a prime on the original source is contracted to codimension at least two on a temporary base, its restricted valuation is first extracted on the base. The minimum rule then yields, in all sufficiently divisible degrees $\ell$, an isomorphism

preserving section ratios. The fixed model $Y_0$ has the same section spaces as the original $(Y,D)$. Moreover, $L_W$ is big over $S$.

Lemmas 6.1–6.2 and (6.5)–(6.6); Proposition 6.3 and (6.8) · pp. 37–41

2. Check the auxiliary subpair's rank condition before applying b-nefness

The manuscript sets $\Delta=-a^{-1}\operatorname{div}_{aK_{Y/W}}(\omega)+x^*\sum t_AA$, obtaining $K_Y+\Delta\sim_{\mathbb Q}x^*L_W$; negative coefficients of $\Delta$ are allowed. The logarithmic zero divisor $Z_J$ on the generic fiber has Iitaka dimension zero. Sections of the discrepancy sheaf are bounded above by sections of $\mathcal O_J(NZ_J)$ and below by the constant section, giving

[FG, Definition 3.2 / Theorem 3.6] now applies. On a suitable model,

The lower bound on $B_W$ comes from testing the discriminant threshold at the same prime attaining the minimum for $t_A$. The external input supplies b-nefness, without requiring semiampleness of the moduli part.

Proposition 6.3 and (6.9)–(6.15) · pp. 41–43 · FG, Definition 3.2 and Theorem 3.6 · PDF pp. 5, 7 (printed pp. 1724, 1726)

3. Account for the period directions in the Higgs-tail numerical dimension

On each component of the boundary $D_0$ with nonidentity Lie monodromy, the period line satisfies $\nu(H|_{D_i})<\nu(H)$. Comparing growth orders of sections removes this boundary and makes $K_S+bH$ big. Let $G_i$ be generated by all contractions of the $i$th Higgs iterate of $u_W$. On a unipotent cover followed by resolution, $\pi:\widehat W\to W$, put

Proposition 5.6 proves that $P$ is nef and $\nu(P+\widetilde H)=\nu(P)$. A curvature-null direction fixes the chosen line and its compact flags; an orbit calculation and the representation-theoretic Lemma 5.7 then show that it fixes the whole highest Hodge space. The chosen vector is not assumed to span that space.

Proposition 5.3 · pp. 30–33; Lemma 5.4 and Proposition 5.5 · pp. 33–34; Proposition 5.6 and (5.11)–(5.18) · pp. 34–36

4. Obtain a fixed comparison from estimates uniform in the boundary

Lemma 7.4 uses the minimizing prime for $t_A$ and the discriminant estimate together to transfer Higgs determinants into orbifold cotangent tensors. Represent $M_W+\epsilon A$ by a general effective $\Lambda_\epsilon$ and apply [CP, Theorem 1.3] whenever $K_W+B_W+\Lambda_\epsilon\sim_{\mathbb Q}L_W+\epsilon A$ is pseudoeffective. [BDPP, Theorem 2.2] converts movable-curve degree inequalities into

with $C\geq0$ and $R>0$ independent of $\epsilon$. A positive pseudoeffective threshold $t_0$ would satisfy $t_0\leq Ct_0/(C+R)<t_0$, so $t_0=0$. Taking limits and using $\pi^*P_W\geq_{\mathrm{pe}}P$ gives $Q\pi^*L_W\geq_{\mathrm{pe}}P$ for the fixed $Q=C+R>0$. Applying [CP, Theorem 3.4] to the same perturbed pairs and pushing down from good models also gives $L_W-h^*K_S\geq_{\mathrm{pe}}0$.

Lemmas 7.4–7.6 and (7.5)–(7.11) · pp. 46–49 · CP, Theorems 1.3, 3.4 and Remarks 3.3, 3.6 · pp. 3, 12–14 · BDPP, Theorem 2.2 · p. 6

5. Subtract the numerical twist and return to the original base

Relative bigness, $L_W-h^*K_S\geq_{\mathrm{pe}}0$, and bigness of $K_S+bH$ first make $L_W+bh^*H$ big. After pullback, choose an ample lower bound $A'$ and add the fixed comparison from the preceding step:

In Lemma 7.3, the equality $\nu(P+\widetilde H)=\nu(P)$ makes the loss term in the nef Morse inequality have smaller degree in $t$ than the supply term. The right-hand side is therefore big for $t\gg0$, and Proposition 7.2 makes $L_W$ itself big. The complete section-space isomorphism from Step 1 returns these systems to the original base and gives $\kappa_Y\geq\dim W\geq\dim S>0$. No equality between numerical and Iitaka dimensions of an arbitrary pseudoeffective divisor is assumed.

Proposition 7.2 and Lemma 7.3 · pp. 44–45; Proposition 7.7 · p. 49; proof of Proposition 1.3 · p. 50

Positive base Kodaira dimension excludes this branch when the base has Kodaira dimension zero. This completes Proposition 1.3 and returns to the upper-bound argument in Section 2.

RA, Propositions 7.7, 1.3 · pp. 49–50

1. Match the boundary condition and the very general fiber

Apply [OI, Corollary 6.2] with $D_X=E$ and $D_Y=D$. Smooth connected projective complex varieties, connected fibers, reduced SNC boundaries, and support inclusion are all supplied by Theorem 1.1. The lower bound does not require stratum smoothness, and its very general pair is exactly $(F,E_F)$. The manuscript restates it as Theorem 7.8.

Theorem 7.8 and §7.6 · p. 50 · OI, Corollary 6.2 · pp. 40–41

2. Keep the vanishing supplied by the upper bound in the negative-infinity case

When both summands are finite, the two inequalities give equality. If either is $-\infty$, the lower bound is automatic; the content of equality is the vanishing $H^0(X,mL_X)=0$ supplied by Theorem 1.1. This vanishing is not extracted from the proof in OI.

Theorem 1.1 · p. 2; §2.2 · p. 5; §7.6 · p. 50 · OI, proof of Corollary 6.2 · p. 41

For smooth families with nonnegative fiber and base Kodaira dimensions, comparison with WV Theorem 1.1 gives an alternative upper bound on variation.

WV, comparison after Corollary 1.3 · p. 5