CATALOGUE 034 · PAPER 01 / 14 Overview
Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
Conditional dimension induction and generation of actual line bundles
Assuming logarithmic Iitaka subadditivity, the manuscript proves semiampleness of nef adjoints for compact Kähler lc pairs in every finite dimension. It constructs good divisorial decompositions by induction, using Hodge lines for fibrations and signed boundaries on simple spaces to reach generation of actual holomorphic lines.
Read the manuscript on GitHub1. Main results
The statements retain the source order: Assumption 1.1, the final target Theorem 1.2, the induction statement Theorem 2.13, and four tools used in the proof. Each retains its own hypotheses and scope.
Assumption 1.1 — Logarithmic Iitaka subadditivity
The paper assumes that for every surjective morphism $f:X\to Y$ with connected fibres between smooth connected complex projective varieties, and reduced effective SNC divisors $D_X,D_Y$, including zero divisors,
(Assumption 1.1, p. 3). Here $F$ is a very general smooth fibre and $D_F=D_X|_F$. The source's convention for $\kappa=-\infty$ is retained.
Theorem 1.2 — Log abundance for compact Kähler spaces
Assume Assumption 1.1. Let $X$ be a normal irreducible compact Kähler complex analytic space, and let $\Delta\ge0$ be a rational boundary such that $(X,\Delta)$ is lc and $J=K_X+\Delta$ is $\mathbb Q$-Cartier. If $J$ is analytically nef, there is a positive integer $m$ such that $mJ$ is Cartier and the evaluation map
is surjective everywhere. This includes every finite dimension and zero boundary, but remains conditional on subadditivity. It asserts generation of an actual holomorphic line bundle, a stronger conclusion than expressing its Chern class as the pullback of a Kähler class.
Theorem 2.13 — Good divisorial decomposition
The inductive assertion $G_n$ concerns a smooth connected compact Kähler $n$-fold, an SNC boundary $B$ with coefficients in $[0,1]\cap\mathbb Q$, and a pseudo-effective adjoint $J=K_X+B$. It asks for a smooth Kähler modification $\mu:Y\to X$ and
(Definition 2.1, p. 6). Here $R$ is a rational divisor, $N$ is the analytic divisorial negative part, and $\sim_{\mathbb Q}$ means an isomorphism of actual rational holomorphic lines. Theorem 2.13, p. 13 asserts all $G_n$ under Assumption 1.1.
Theorem 4.1 — Generation on the entire reduced boundary
Assume $G_j$ for $j<n$. Let $(V,B)$ be a normal irreducible compact Kähler dlt $n$-fold, globally $\mathbb Q$-factorial, with effective rational boundary and analytically nef $J=K_V+B$. Under the lc-strata resolution condition of Definition 3.1, an actual Cartier multiple of $J|_{\lfloor B\rfloor}$ is generated on the whole reduced boundary. The resolution condition asks for a resolution with exceptional crepant coefficients strictly below one, which is an isomorphism at the generic point of every lc centre.
Proposition 5.1 — The rank-one fibration case
Assume Assumption 1.1 and $G_j$ for $j<n$. Let $X$ be a smooth connected compact Kähler $n$-fold, $B$ a rational SNC boundary, and $J=K_X+B$ pseudo-effective. Suppose there are a modification $\pi:\widetilde X\to X$ and a base $W$, both smooth compact Kähler, such that $\widetilde B=\pi_*^{-1}B+\operatorname{Exc}(\pi)_{\mathrm{red}}$ has SNC support. Suppose the proper surjective holomorphic map $g:(\widetilde X,\widetilde B)\to W$ has connected fibres, $0<\dim W<n$, the base is projective or has $a(W)=0$, and $\kappa(F,K_F+\widetilde B_F)=0$ on very general fibres. Proposition 5.1, p. 90 gives $G_n$ in this situation.
Theorem 6.1 — Meromorphic nonvanishing on simple spaces
Theorem 6.1, p. 107 states that a smooth connected simple compact Kähler manifold with $a(X)=0$ has a nonzero meromorphic section of $K_X^{\otimes m}$ for some $m>0$. Its proof is independent of Assumption 1.1. It does not assert a holomorphic section or an effective canonical representative.
Theorem 7.1 — Signed rigidity
Theorem 7.1, p. 125 assumes a normal irreducible, globally $\mathbb Q$-factorial, simple compact Kähler space $X$ with $a(X)=0$, a dlt pair with reduced boundary $D=\sum D_i$, and the resolution condition of Definition 3.1. Suppose $L=K_X+D$ is analytically nef; $L\sim_{\mathbb Q}\sum a_iD_i$ as actual rational lines, allowing negative $a_i\in\mathbb Q$; the pullback of $\{L-cD\}$ to a projective resolution with smooth compact Kähler source is pseudo-effective for some $c>0$; and $L|_D$ is semiample on the whole reduced boundary. Then $L$ is torsion.
References on the arrows
Unprefixed numbers refer to this manuscript P. Catalog references LA, CGM4, and ANV4 are used only for the specified results. OILS supplies rank-one Hodge lines and periods; the remaining keys are external tools.
2. Proofs of Theorems 2.13 and 1.2: the global induction
Proof route. Theorem 1.2 is the target and Theorem 2.13 the induction statement. Generation on the whole boundary is a shared tool. The positive-dimensional fibration and the simple case via meromorphic nonvanishing and signed rigidity feed back into this diagram.
Assume the lower-dimensional assertions $G_j$ and prove $G_n$. Split by algebraic dimension and simplicity, then descend generation to the original nef lc adjoint.
1. Locate the use of subadditivity
The assumption enters through the projective good-model statement Proposition 2.10, p. 12. That proposition invokes the induction in LA, Proposition 2.5, Theorem 9.6 and §10, and identifies its subadditivity use in LA, Lemma 6.1, pp. 30–31: a resolved projective Albanese fibration with both boundaries zero. This is the stated entry point of the condition in the present paper.
2. Use projective input or algebraic reduction in positive algebraic dimension
If the algebraic dimension is $a(X)=n$, Kähler–Moishezon makes $X$ projective and Proposition 2.10 applies. If $0<a(X)<n$, algebraic reduction gives a nontrivial fibration.
3. Use a covering family and norms in the nonsimple case
When $a(X)=0$ and $X$ is not simple, Cam, Lemma 17 / Corollary 18, p. 10 supplies an incidence space with a nontrivial fibration and a generically finite evaluation onto $X$. Its algebraic dimension is still zero, so a semiample positive part there is torsion. Finite analytic norms and uniqueness of the negative part descend this purely negative decomposition (Proposition 2.9, p. 11). A space is simple when no positive-dimensional proper compact subvariety passes through a very general point.
Proposition 2.9; Theorem 2.13 · pp. 11, 13 · Cam, Lemma 17 / Corollary 18
4. Close the simple case by signed rigidity
For simple $X$ with $a(X)=0$, Theorem 6.1 supplies a signed canonical representative. Add a boundary and pass to a nef model; boundary generation and Theorem 7.1 give torsion. Proposition 2.12 subtracts the added boundary from the negative part. Sections 5–6 explain this route.
5. Remove the exceptional error and descend generation
The final passage to the main theorem is Proposition 2.7, pp. 9–10. Giving exceptional divisors coefficient one on an lc resolution yields $K_Y+B_Y=p^*J+E$, with $E\ge0$ exceptional. Lemma 2.3 shows that $E$ is added unchanged to the negative part, so it can be cancelled in the actual line identity. If the original $J$ is nef, its remaining negative part is zero. Normality, $p_*\mathcal O_Y=\mathcal O_X$, and the projection formula identify sections and descend generation to the original singular space.
3. The interface of §§3–4: models and whole-boundary generation
What is compared on strata. CGM4 Lemmas 7.16–7.17 supply actual adjoint lines and strict comparisons across small dlt steps, entering Theorem 3.13 here. Torsion-freeness from ANV4 Lemma 7.1 enters Lemma 4.3 to pass from residue compatibility to section extension. This is separate from the later use of ANV4 §§10–12 in the lifting argument of §7. Theorem 3.13 · pp. 29–30; Lemma 4.3 · p. 74
Separate contraction of a known negative part from gluing sections on the entire boundary. The first preserves the actual nef line; the second uses induction on proper strata and compatible residues.
1. Contract the known negative part and preserve the nef line
Proposition 3.8, pp. 23–24 starts with an actual presentation $J\sim_{\mathbb Q}P+E$, where $P$ is nef and $E$ remains the entire negative part on a resolution. It contracts $E$ while descending a fixed Cartier multiple of $P$ at every step.
2. Use induction on proper strata for signed presentations
With only a signed presentation, Theorem 3.13 / Corollary 3.14, pp. 29–32 apply. Lower-dimensional $G_j$ supplies one model carrying a nef parameter interval; every divisorial negative multiplicity is affine there. A nontrivial wall changes its slope, so such walls cannot remain in that interval. Curves away from the floor have zero intersection with an adjoint represented on the floor, precluding further negative operations after special termination.
3. Match residues on every lower stratum
Generation on individual adjunction components does not immediately descend to their union. The source uses residue comparisons in divisible even degree, compatibility on every lower stratum, and finite images of self-comparisons on vertical strata. Products over those finite images give invariant generating tuples in a common degree. Gluing them produces sections on the whole boundary (Proposition 4.11 and the conclusion of Theorem 4.1, pp. 88–89). The full auxiliary proofs of the restricted model theory in §3 and boundary theory in §4 are outside this draft's independent audit.
Lemma 4.2; Proposition 4.11; Theorem 4.1, conclusion · pp. 71–73, 88–89
4. Proof of Proposition 5.1: descent along a fibration
Algebraic reduction and relative Iitaka construction reduce general nontrivial fibrations to this rank-one case. If $a(X)=0$ and the fibre is of log general type, slightly lowering horizontal coefficients below one allows a stable-family comparison that produces a positive-dimensional projective factor, contradicting algebraic dimension zero (pp. 90–91).
Construct the rank-one base presentation, identify the error with the entire negative part, and then generate the positive part.
1. Prepare the Hodge line and error coefficients
The rank-one Hodge line and period positivity from OILS give a prepared model with
where $T$ is a rational SNC boundary, $p:W\to S$ contracts onto a smooth projective variety, and $P_S$ is a nef rational line. If $S$ is a point, $M\sim_{\mathbb Q}0$; otherwise $K_S+a_0P_S$ is big for some $a_0>0$. Lemma 5.2, pp. 91–94 retains nonnegative coefficients $(A^*)_E$ at primes dominating a base divisor, with minimum normalized coefficient zero. Signs at primes mapping into codimension at least two are not yet determined.
2. Subtract the horizontal negative part and descend pseudo-effectivity
Fibre induction puts the horizontal part $A^{*,\mathrm{hor}}$ inside $N(J)$. Subtracting it from a positive curvature current leaves a metric whose weight is constant on connected smooth fibres and descends to the base. The zero coefficient above each base prime gives local upper boundedness and extension of the descended weight, proving that $H$ is pseudo-effective (Lemma 5.3, pp. 94–95).
3. Lower the base dimension or reach big or torsion data
If $a(W)=0$, then $S$ is a point and lower-dimensional induction gives $H\sim_{\mathbb Q}N(H)$. For projective $W$, a chosen conditional generalized MMP either lowers the base dimension or reaches a nef $H_m$ that is big or torsion. In a second program decreasing the nef data, adding a sufficiently large fixed multiple of $H_{\mathrm{nef}}$ combines the extremal-ray length bound with Cartier integrality to force every contraction to be $H_{\mathrm{nef}}$-trivial (Lemmas 5.6–5.7, pp. 95–101). Semiampleness of the Hodge line itself is not assumed.
4. Identify every vertical error with the negative part
Lemma 5.8, pp. 101–103 proves that the resulting divisor $A$, including the pulled-back base error, is effective and equals $N(J)$. Mixed Hodge index handles images of codimension at least two. For divisorial images, the intersection matrix has only whole-fibre multiples in its kernel; the zero minimum coefficient excludes the residual vertical part.
5. Extend boundary sections and remove the base locus
A torsion positive part finishes the argument. In the big nef case, Proposition 3.8 contracts the negative part and Proposition 5.9, pp. 103–106 applies. It extends the boundary sections of Theorem 4.1 through analytic injectivity FM, then creates a new floor over any remaining base locus using an lc threshold. Boundary generation for this new pair gives a contradiction, proving semiampleness of the actual adjoint.
Proposition 3.8; Theorem 4.1; Proposition 5.9 · pp. 23–24, 71, 103–106
5. Proof of Theorem 6.1: two diagonals and meromorphic nonvanishing
Assume that no canonical power has a meromorphic section. One diagonal produces high rank, a second forces determinant vanishing, and comparison with the point-pole bound yields a contradiction.
1. Bound slopes under the contrary hypothesis
Assuming no canonical power has a meromorphic section, the Albanese map and simplicity give $H^1(\mathcal O_X)=0$. Uniruledness and foliation criteria Ou make $L=c_1(K_X)$ pseudo-effective and imply $c_1(A)\le kL$ for every line subsheaf $A\to\Omega_X^{\otimes k}$. Countably many line bundles can be handled simultaneously, carrying the estimate to a blowup at a very general point (Lemma 6.2, pp. 107–108).
2. Use the first diagonal to produce a high-rank subsheaf
Write $K_i,P_i$ for pullbacks from the two factors of $X^2$, and $\xi=c_1(\mathcal O_Z(1))$. A big class $P$ normalized to volume one has point-pole threshold $\tau(P,x)\le C_n$. On the other hand, on $Z=\mathbb P_{X^2}(K_1\oplus K_2)$, with $d=2n+1$, the class $M=P_1+P_2+q\xi$ has volume comparable to $q$. Blowing up the diagonal in $Z^2$ and differentiating restricted volumes produces, for $s\asymp q^{1/d}$, a subsheaf occupying a fixed positive proportion of the rank of $\operatorname{Sym}^{js}\Omega_Z$ (Lemmas 6.5–6.8, pp. 112–118).
3. Force determinant vanishing with the second incidence
An incidence arising from the diagonal in $X^2$ then converts the pole bound $b_V\ge s-C_n$ into vanishing of coefficients. A fixed positive proportion of the rows in each high-rank determinant must vanish to high order.
4. Cancel determinant classes and contradict the point-pole bound
After factoring out the common zero, restrict the determinant to a point blowup and apply the cotangent slope estimate again. The normalized determinant classes cancel, giving
(equations (6.40)–(6.44), pp. 124–125). The left side diverges, a contradiction. The order of limits matters: fix the geometric data for each $q$ before taking the divisible integer $j$ large.
6. Proofs of Theorem 7.1 and Proposition 2.12: signed boundary and torsion
Exclude intermediate numerical dimension $0<\nu(L)<n$ through compact deformations leaving the boundary. Treat zero and maximal numerical dimension separately, then recover the simple-case induction from torsion.
1. Separate positive boundary fibres from negative and zero supports
Suppose $0<\nu(L)<n$. Mixed Hodge index and the intersection matrix of the signed coefficients produce a positive-coefficient component whose boundary-map image has dimension $\nu(L)-1$. Its general fibres avoid the negative- and zero-coefficient components (Lemma 7.2, pp. 126–127). Taking roots and separating the positive and negative supports gives local neighborhoods with a reduced Cartier divisor $S$, a residual boundary $T$, and an actual isomorphism $\mathcal O_Z(aS)\simeq\omega_Z(S+T)$.
2. Retain residual poles in the Hodge-theoretic residue
Residue on a resolution yields a split injection retaining the residual poles $A$,
(Lemmas 7.3–7.4, pp. 128–133). The divisors $A$ and $H$ may intersect, so these extra poles must be retained.
Proposition 7.5, pp. 133–136 localizes SNC local cohomology along the residual poles. Decomposition into proper Kähler closed strata supplies the lowest Hodge filtration and symbol-kernel vanishing.
3. Kill obstructions in every integral degree and finite order
Proposition 7.6, pp. 137–141 makes
surjective for $I=\mathcal O_Z(-S)$, in every integral degree and at every finite order. The obstruction is a degree-$k$ derivation. An adjugate identity on a coordinate-power cover sends it into the symbol kernel without dividing by the Jacobian. Maps from the ample root line to that kernel vanish, killing even obstructions supported at special parameters. This extends the lifting method of ANV4 to residual poles.
4. Contradict simplicity through deformations leaving the boundary
Lift the boundary-fibre equations and normal direction through all finite orders. Douady space and analytic Artin approximation then produce compact deformations leaving the boundary. Scheme-theoretic images under finite maps and cycles of bounded volume allow proper incidence images that include limits. Countability and Baire's theorem give a covering family of positive-dimensional proper subvarieties, contradicting simplicity (Lemma 7.7 and the conclusion of Theorem 7.1, pp. 141–143). The case $\nu=0$ gives torsion through the signed identity; $\nu=n>0$ gives bigness and $a(X)=n$, which is excluded.
5. Subtract the added boundary and return to the original adjoint
In the proof of Proposition 2.12, pp. 144–145, enlarge to a reduced SNC boundary $D$ containing the signed canonical divisor from Theorem 6.1, then use Corollary 3.14 to reach a nef dlt model. Theorem 4.1 supplies boundary generation; Ou and exceptional negative-part subtraction supply pseudo-effectivity with $c=1$. After Theorem 7.1 gives torsion, uniqueness of the positive current in Lemma 2.6, p. 9 allows subtraction of the added boundary, recovering a purely negative decomposition of the original $J$. This closes the simple case, then Theorem 2.13 and finally Theorem 1.2.
7. Which papers supply which steps
| Input | Direct use | Editorial comparison |
|---|---|---|
| LA Log abundance in characteristic zero, 2026-09-24 | Proposition 2.10 here; input Proposition 2.5 p. 9, Theorem 9.6 p. 69, §10 p. 72, Lemma 6.1 pp. 30–31 | Statements, completion of induction, and subadditivity use compared; entire input proof outside scope |
| OILS Orbifold and logarithmic Iitaka subadditivity, 2026-09-26 | Lemma 5.2 and §§5.1 / 5.4 here; input Lemma 2.6 p. 9, Proposition 2.7 p. 10, Theorem 3.1 pp. 17–18, Lemma 3.3 p. 18, Lemma 5.2 p. 32 | Hypotheses, conclusions and uses compared; proofs of Hodge-line construction, positivity and stable-family comparison outside scope |
| CGM4 Conditional good minimal models…, 2026-10-05 | Adjunction and strict comparison on strata in §3; input Lemmas 7.16–7.17 pp. 58–60 | Dimension-free lemma statements compared; its four-dimensional main theorem is not an all-dimensional input |
| ANV4 Abundance after nonvanishing…, 2026-09-27 | Root neighborhoods, residue, filtered direct image and obstruction derivations in §7; input Lemma 10.1 p. 66, Proposition 10.4 p. 71, Proposition 11.1 p. 75, Lemma 11.2 p. 80, Lemmas 12.1–12.2 pp. 82–84 | Statements compared; extension to residual poles is made here in Lemma 7.4 / Proposition 7.5 |
| HX Hacon–Xie, arXiv:2607.24986v1 | Theorem 1.3 p. 2: analytic cone and length bound, used here on p. 16 | Statement compared; the entire restricted model theory here is not independently audited |
| TX arXiv:2301.09186v3 / Xie arXiv:2211.10800v1 | Proposition 5.5, p. 97. TX Theorems A (=4.2), B (=5.4), D (=5.2), p. 2; Xie Theorem 1.5, pp. 2–3 | Compared relative smooth-model induction, NQC decompositions, scaling hypotheses and descent of actual lines; input proofs outside scope |
| Ou arXiv:2501.18088v1 | Theorems 1.1 / 1.4 pp. 1, 3; Lemma 6.2 and canonical pseudo-effectivity on p. 144 | Statements and applications compared; not all slope-related auxiliary inputs compared |
| FM Fujino–Matsumura, author version 2021-07-15 | Theorem A p. 3; section extension in Proposition 5.9, p. 105 | Curvature lower bound, multiplier ideal and multiplying-section hypotheses compared |
| Sai arXiv:2204.09026v5 | Theorem 1 p. 1; applied to smooth closed strata in Proposition 7.5, p. 135 | Constant-source proper Kähler direct-image statement compared; independent verification of mixed/localized transitions remains open |
| Cam arXiv:2605.19713v2 | Lemma 17 / Corollary 18 p. 10; Theorem 2.13, p. 13 here | Generically finite incidence statement and use compared |
8. Return to the sources
The editorial reading covers the main theorem and the selected routes in §§2, 5 and 7; the two contraction/termination connections in §3; the generating-tuple conclusion in §4; and the slope, two-diagonal and determinant-cancellation passages in §6. This is not a record of verifying all 148 pages. The cited passages above provide entry points into the source.
Further editorial checks concern the restricted model induction and finite geography in §3.8; the full residue-comparison and finite-image proofs in §§4.2–4.4; the direct-image estimate in §6.3 and all local calculations of Lemma 6.10; and strictness of the residual-pole filtered direct image in §7.3 and its connection to Sabbah–Schnell vanishing. Uses of Boucksom's negative-part theory, mixed Hodge index and Douady/Artin/Fujiki results were read here, but independent comparison of all their original hypotheses is incomplete. Uninvestigated catalog relations remain uninvestigated, rather than being recorded as absent.