CATALOGUE 034 · PAPER 07 / 14 Overview
Lifting sections from the reduced support of an adjoint
From generation on the whole reduced support to finite-order lifting and section growth
The paper seeks ambient sections from generation on the reduced support of an effective adjoint multiple. It retains gluing across the entire support and uses positive frame weight to eliminate obstructions on finite neighborhoods. Supported lifting holds in every dimension; fourfold nonvanishing is neither a hypothesis nor an input to its proof.
Read the manuscript on GitHub1. Main results
Theorem 1.1 — Supported boundary lifting
Let $(V,C)$ be a normal projective $\mathbb Q$-factorial dlt pair over an algebraically closed field of characteristic zero, with effective rational boundary $C$, and put $A=K_V+C$. For a sufficiently divisible positive integer $q$, suppose a nonzero effective Cartier divisor $G$ satisfies $$0\ne G\ge0,\qquad G\sim qA,\qquad\operatorname{Supp}G\subseteq\operatorname{Supp}\lfloor C\rfloor.$$ If $\mathcal O_V(G)|_{G_{\mathrm{red}}}$ is semiample on the whole reduced scheme, then $\kappa(V,A)>0$.
No additional nefness is assumed. The support inclusion can be strict: writing $C=S+H+C_0$, with $S=G_{\mathrm{red}}$, the extra coefficient-one divisor $H$ may meet $S$ and its strata. Semiampleness separately on normalized components cannot replace the whole-support hypothesis.
Theorem 1.2 — Abundance after nonvanishing
For a projective lc pair $(X,\Delta)$ over $\mathbb C$ of dimension at most four with effective rational boundary, suppose $D=K_X+\Delta$ is $\mathbb Q$-Cartier, nef, and $\kappa(X,D)\ge0$. Then $D$ is semiample; if $\kappa(X,D)=0$, then $D\sim_{\mathbb Q}0$. No uniform index is asserted. The extension to arbitrary algebraically closed characteristic-zero fields is separately stated as Corollary 7.2 (p. 29).
Theorem 1.3 — Reduction to smooth canonical nonvanishing
Assume, in every dimension, that every smooth connected projective complex variety $W$ with pseudo-effective $K_W$ has $H^0(W,mK_W)\ne0$ for some $m>0$. Then every normal projective lc pair $(X,\Delta)$ over any algebraically closed characteristic-zero field, with effective rational boundary and nef $\mathbb Q$-Cartier adjoint $D=K_X+\Delta$, has semiample $D$. The premise is not restricted to nef canonical divisors.
Corollary 1.4 — Fourfold log abundance
For a normal projective lc pair $(X,\Delta)$ over an algebraically closed characteristic-zero field, of dimension at most four with effective rational boundary and nef $\mathbb Q$-Cartier adjoint $D=K_X+\Delta$, the divisor $D$ is semiample. If $\kappa(X,D)=0$, then $D\sim_{\mathbb Q}0$. This is where NV Corollary 1.2 adds fourfold lc nonvanishing.
References on the arrows
Unprefixed references denote this manuscript. External inputs use the following keys.
- KS: Popa, Kodaira–Saito vanishing and applications (author PDF).
- Saito: Mixed Hodge Modules (1990).
- FG: Fujino–Gongyo, Log pluricanonical representations and abundance conjecture (final author PDF).
- Bir: Birkar, On existence of log minimal models (arXiv:0706.1792).
- Hash: Hashizume, On the non-vanishing conjecture and existence of log minimal models (arXiv:1609.00121v4).
- NV: Fourfold nonvanishing by minimal metrics and moving jets (September 27, 2026).
2. Proof of Theorem 1.1
Proof route. Lifting in Theorem 1.1 enters abundance after nonvanishing in Theorem 1.2. The last diagram separates the conditional all-dimensional Theorem 1.3 from fourfold Corollary 1.4 and their distinct inputs.
Start from generation on the entire reduced support $S=G_{\mathrm{red}}$ and show that $h^0(V,\mathcal O_V(NG))$ is unbounded. To eliminate finite-neighborhood obstructions, treat the horizontal symbol and the vertical Euler action separately.
1. Retain the obstruction group through cyclic covers and residues
Replace $G$ by a multiple so that $L=\mathcal O_V(G)$ satisfies $L|_S=f^*\mathcal O_{\mathbb P^\ell}(1)$. Pass to the nonzero-frame bundle $P=L^\times$ and construct two cyclic covers, retaining every component when the algebras split. On a resolution there is a logarithmic form $\sigma$ of positive frame weight $a$, with residue $\Omega$. The polar divisor $D_h$ retains poles above intersections with $H$. Lemma 2.3 inserts the obstruction group into $$\iota:R^1g_*\mathcal O_E\hookrightarrow R^1h_*\omega_R(D_h)=F_0\mathcal N\hookrightarrow\mathcal N.$$ Here $g:E\to B$, $h:R\to B$, and $B=\mathcal O_{\mathbb P^\ell}(1)^\times$. The lowest piece $F_0\mathcal N$ of a mixed Hodge module on the boundary graph retains cohomology concentrated on smaller supports.
2. Kill the horizontal part using the identity for every boundary function
With $I=(y)$, divide the obstruction from order $k$ to $k+1$ by $y^k$. This gives a derivation $\delta_k:g_*\mathcal O_E\to R^1g_*\mathcal O_E$: the product of two lifting errors vanishes modulo $y^{k+1}$. The graph-residue computation gives, for every boundary function $F$, $$\iota(\delta_k(F))+\sum_i\iota(F\delta_k(t_i))\partial_{t_i}=0,$$ using the right $\mathscr D$-module action, as in (5.5). Set $F=1$ and take the first symbol. The resulting zero-symbol tensor has weight $a+k>0$. A transverse finite cover and the adjugate of its differential turn a nonzero horizontal part into a nonzero map from an ample line into the lowest symbol kernel. Lemma 3.2, derived from Kodaira–Saito vanishing, excludes it.
3. Kill the vertical part using the actual Euler action
For the vertical part, vanishing of the symbol alone is insufficient. The actual Euler action $s\varepsilon=-(a+k)s$ and the residue identity kill that part; returning to arbitrary $F$ then gives $\delta_k(F)=0$. This separation is the core of the lifting argument.
4. Turn finite-layer lifting into ambient section growth
Finite-group invariants and frame degree zero descend the construction to divisorial layers on $V$. The periodicity $Q_{j-r}=Q_j\otimes\mathcal O(1)$ accumulates positive twists. Higher cohomology remains bounded while sections grow. Passing from finite neighborhoods to the ambient variety loses at most the fixed number $h^1(V,\mathcal O_V)$: $$h^0(V,\mathcal O_V(NG))\ge h^0(V,\mathcal O_V(NG)/\mathcal O_V)-h^1(V,\mathcal O_V)\longrightarrow\infty.$$ When $\ell=0$, the number of layers still grows. No convergent formal neighborhood or extension of the boundary morphism to a whole neighborhood is assumed.
3. Proof of Theorem 1.2
Proposition 6.3 assumes ordinary minimal models for effective lc pairs through dimension $d$, and full lc abundance below $d$. It deduces abundance after nonvanishing in dimension $d$; good minimal models are not assumed at the outset.
1. Exclude a nonzero effective representative in Kodaira dimension zero
For $\kappa(D)=0$, suppose $0\ne M\ge0$ and $M\sim_{\mathbb Q}D$. Raise support coefficients to one on a resolution, then pass to an ordinary minimal model $(V_*,C_*)$. Kodaira dimension zero and an effective representative $M_*$ are retained. Nefness of the original $D$ and negativity ensure $M_*$ does not disappear. Lower-dimensional abundance and FG normalization gluing give semiampleness on the whole floor. Diagram 1 applied to $G=qM_*$ contradicts Kodaira dimension zero, so $D\sim_{\mathbb Q}0$.
2. Use fibre minimal models in positive Kodaira dimension
For $\kappa(D)=k>0$, on a resolution write $L=K_W+\Gamma=\pi^*D+N$, with effective exceptional $N$. The fibre adjoint $L|_F$ need not be nef. Take an ordinary minimal model of that effective fibre pair and apply lower-dimensional abundance there. Comparison (6.6) and nonnegative intersections yield $\pi^*D|_F\equiv0$, and then $\nu(D)=\kappa(D)$. Whole-floor semiampleness also gives abundance on normalized lc centers.
FG Theorem 4.2 now applies.
3. Insert the fourfold inputs and transfer the ground field
For $d=4$, insert known lower-dimensional abundance and Birkar's ordinary minimal-model theorem for effective lc fourfolds to obtain Theorem 1.2. Transfer to other characteristic-zero fields descends and extends a specified Cartier multiple of the original divisor and its evaluation map; it does not assert preservation of $\mathbb Q$-factoriality under field descent.
4. Proofs of Theorem 1.3 and Corollary 1.4
The conditional all-dimensional assertion of Theorem 1.3 and the fourfold consequence of Corollary 1.4 receive nonvanishing from different sources. Both routes finally transfer generation of a Cartier multiple of the original adjoint between ground fields.
1. Supply ordinary models from the all-dimensional premise
For Theorem 1.3, assume smooth canonical nonvanishing in all dimensions.
Hash Theorem 1.4 supplies lc nonvanishing and ordinary minimal models. The passage from real-linear to rational-linear effectivity uses only the finite rational linear algebra in NV Lemma 8.1, separately from that paper's fourfold nonvanishing theorem. Proposition 6.3 then closes dimension induction.
2. Obtain the first fourfold section from a separate input
For Corollary 1.4, NV Corollary 1.2 supplies a nonzero section in a multiple of a prescribed Cartier multiple on the original normal lc variety; Theorem 1.2 and field transfer finish the argument. That nonvanishing input does not enter the proofs of Theorems 1.1–1.2. Appendix A gives a separate conormal–period method, not an input to diagram 1.
5. Which papers supply which steps
Use in effective Iitaka. Theorem 1.2 makes a nef adjoint semiample after nonvanishing in the qualitative model step of Fourfold Iitaka. Uniformity of the degree is a separate subsequent argument.
| Input | Use | Scope checked |
|---|---|---|
| KS Kodaira–Saito vanishing | Lemma 3.2, p. 13: negative hypercohomology with an inverse ample twist | Compared Popa author's PDF Theorem 8.2, PDF pp. 14–15. The manuscript cites published Theorem 28, a different numbering scheme |
| Saito Projective direct-image strictness for mixed Hodge modules | Lemma 3.1, pp. 12–13: the entire coherent lowest piece | Read the application and local calculation; rechecking Saito 1990 Theorem 2.14 / Proposition 2.15 is pending |
| FG Theorems 4.3 / 4.2, final author PDF p. 18 | Whole-floor gluing in Lemma 6.2 and nef log abundance in Proposition 6.3 | Compared statements, hypotheses, and uses; proofs excluded |
| Bir Corollary 1.6, accessed arXiv PDF p. 3 | §6.3, p. 27: ordinary minimal models of effective lc fourfolds | Statement checked; distinguished from good models |
| Hash Theorem 1.4, arXiv v4 §1 | Theorem 1.3 only: smooth nonvanishing gives lc nonvanishing and ordinary models | Compared Conjectures 1.1–1.3 and the implication |
| NV Corollary 1.2, p. 2; Lemma 8.1, p. 35 | The former feeds Corollary 1.4; the latter rationalizes Theorem 1.3 | Statements and uses on pp. 27, 29 checked; full nonvanishing proof excluded |
Theorem 1.2 also occurs in the final assembly of fourfold log Iitaka in Ufour, §10, pp. 48–49; its detailed use belongs to that article. Classical threefold abundance and its correction, and all external splitting/vanishing theory used for residue insertion, are not collectively certified here.
6. Return to the sources
pp. 8–13 treat extra coefficient-one poles, residue injection, and the entire coherent lowest piece. pp. 14–23 treat transverse covers, Euler action, the graph identity, finite-layer lifting, and growth. pp. 23–29 treat ordinary model comparison, the whole floor, Iitaka fibres, and field transfer. These core passages were read and the main statements and listed direct inputs compared.
Pending work includes all details of the cyclic covers and split residue insertion in §2, independent reconstruction of each Hodge-filtration identification from external theory, proofs of classical lower-dimensional abundance, Appendix A, and full external proofs. Diagram arrows present the manuscript's logical connections; they do not certify those connections independently. Japanese and English text, diagrams, references, and limitations are aligned.