CATALOGUE 034 · PAPER 04 / 14 Overview
Log abundance in characteristic zero
Nonvanishing, boundary deformations, and dimension induction
The manuscript derives smooth nonvanishing from incompatible jet and Frobenius estimates, then combines it with boundary geometry to obtain good minimal models and log abundance.
Read the manuscript on GitHub ↗1. Main results
The following are the manuscript’s claims, retaining their original numbering, hypotheses, and scope. Theorem 9.6 is the conditional induction step; Theorem 11.1 records the result after induction closes.
Theorem 1.1（Log abundance）
Let $(X,B)$ be a normal projective lc pair over an algebraically closed field of characteristic zero. Assume that $B$ is an effective rational divisor and $K_X+B$ is $\mathbb Q$-Cartier. If $K_X+B$ is nef, some positive Cartier multiple is globally generated.
Theorem 9.6（The smooth nonvanishing step）
Assume that every projective complex lc pair of dimension less than $n$, with effective real boundary and pseudo-effective adjoint, has a good log minimal model (Assumption 2.1). Then, for a smooth projective complex $n$-fold $X$,
Theorem 11.1（Real-boundary good models over the complex numbers）
Every projective lc pair $(X,B)$ over $\mathbb C$, with effective real boundary and pseudo-effective $\mathbb R$-Cartier adjoint $K_X+B$, has a good log minimal model.
Real semiampleness means $\mathbb R$-linear equivalence to the pullback of an ample real divisor under a contraction. This real-boundary conclusion is the hypothesis passed to the next dimension.
2. Theorems 1.1 / 11.1: the induction
The induction retains good models for pseudo-effective real-boundary lc pairs in all smaller dimensions. This supplies both boundary adjunction and the reductions to lower-dimensional bases.
Statements above: Theorem 11.1·Theorem 1.1·Manuscript proof ↗
1. Establish nonvanishing first
The hypothesis $\mathrm{GLM}_{\leq n-1}$ concerns projective complex lc pairs in dimensions below $n$, with effective real boundary and pseudo-effective adjoint. Starting from a point, first prove $\mathrm{NV}_n$, Theorem 9.6. General good models in dimension $n$ are not yet available. Boundary semiampleness in Proposition 2.3 and special termination in Proposition 6.9 use good models only for smaller-dimensional pairs.
Theorem 3.1 is a geometric criterion assuming semiampleness on the whole boundary; it does not assume $\mathrm{NV}_n$. Sections 6–8 use this criterion and the lower-dimensional hypothesis to construct fixed data from a counterexample. The independent Theorem 9.1 excludes their coexistence. Only its application establishes $\mathrm{NV}_n$.
2. Make the nef model good
After Theorem 9.6, apply Hashizume Theorem 1.4. It supplies real-boundary lc nonvanishing and ordinary log minimal models through dimension $n$, not yet good models. Proposition 2.5 decomposes the boundary on this nef model into a finite convex combination of rational nef boundaries and proves semiampleness for each rational adjoint. Positive Iitaka dimension is handled by [FG17, Lemma 3.5], using lower-dimensional good models.
[Hash] Theorem 1.4 ↗·[FG17] Lemma 3.5 · p. 5 ↗·Proposition 2.5 · pp. 9–12 ↗
3. Resolve the zero-Iitaka case by boundary perturbation
When $\kappa(L)=0$, split according to pseudo-effectivity after perturbing the boundary. The non-pseudo-effective branch descends through an $L$-trivial MMP. The pseudo-effective klt branch constructs an effective representative on the full reduced boundary and applies Theorem 3.1. The remaining non-klt branch reduces to two klt perturbations. The next section separates these arguments.
Once each rational adjoint is semiample, the product of its contractions recovers semiampleness of the original real adjoint. This establishes $\mathrm{GLM}_{\leq n}$ and supplies the hypothesis in dimension $n+1$. Theorem 11.1 records this output in all dimensions; it is not fed back as an input to Theorem 9.6.
4. Return to the original pair and change fields
If the original adjoint is nef, the effective exceptional comparison term vanishes by the negativity lemma (Lemma 2.2). Semiampleness on the good model therefore returns to the original pair. Proposition 10.1 spreads the pair, boundary, and a Cartier multiple over a finitely generated field, then descends surjectivity of the actual evaluation map by faithful flatness. This gives Theorem 1.1 over every algebraically closed field of characteristic zero.
3. Proposition 2.5: from nonvanishing to good models
Here $\mathrm{NV}_n$ has already been established. In the remaining zero-Iitaka branch, finish the klt case before returning to the non-klt case.
1. From real to rational boundaries, then to zero Iitaka dimension
Hashizume supplies a $\mathbb Q$-factorial dlt nef model. Fix the support and coefficient-one components; intersect a small rational neighborhood preserving dlt singularities with the rational polytope of nef adjoints. The real boundary lies in a finite simplex of rational dlt boundaries. A positive real combination of their semiample adjoints is the pullback of an ample real divisor from the image of the product of their contractions.
For rational $L=K_Z+\Delta$, fix the finite prime and principal divisors in an effective real representative and solve the rational linear constraints to obtain an effective rational representative. If $\kappa(L)\geq1$, induction supplies the dimension-$n-1$ good-model hypothesis in [FG17, Lemma 3.5]. Lemma 2.2 returns semiampleness from the resulting good model. Thus only nef $L$ with $\kappa(L)=0$ remains.
Proposition 2.5 · pp. 9–10 ↗·[Hash] Theorem 1.4 ↗·[FG17] Lemma 3.5 · p. 5 ↗
2. Non-pseudo-effective perturbation: descend with one Cartier index
Put $P=\Delta$ in the klt case and $P=\lfloor\Delta\rfloor$ otherwise, and fix $m$ with $mL$ Cartier. If all sufficiently small perturbations $L-\epsilon P$ are non-pseudo-effective, choose rational $0<\epsilon<(4nm+2)^{-1}$. Lemma 2.4 gives $0\leq L\cdot C<1/m$ on a curve generating a negative extremal ray. Integrality forces $L\cdot C=0$, and the same line bundle $mL$ descends through contractions and flips. This uniform index prevents the required smallness of $\epsilon$ from changing during the MMP.
The established non-pseudo-effective klt MMP ends in a Mori fiber space $f:Z'\to T$, with $L'\sim_{\mathbb Q}f^*A$, $A$ nef, and $\dim T<n$. For a klt total pair, Ambro Theorem 0.2 supplies an effective klt pair with $A\sim_{\mathbb Q}K_T+\Delta_T$, allowing induction. Otherwise $P'$ is relatively ample and a floor component dominates $T$. Adjunction on its transform $S$ in a crepant dlt blowup makes the lower-dimensional $f_S^*A$ semiample. Equality of sections over the connected-fiber part and norms over the finite part descend this to $A$.
Lemma 2.4 · p. 9 ↗·Proposition 2.5 · p. 10 ↗·[Ambro] Theorem 0.2 · p. 2 ↗
3. The pseudo-effective klt branch: build a full-support representative
If $L-\epsilon\Delta$ is pseudo-effective for a sufficiently small rational $\epsilon>0$, the available nonvanishing gives $L\sim_{\mathbb Q}G_0=H_0+\epsilon\Delta$, with $H_0\geq0$. On a resolution $p:W\to Z$, put every exceptional divisor and the strict support of $H_0+\Delta$ into a reduced SNC divisor $D$. Then $K_W+D=p^*L+R$, $R\geq0$. Positive klt log discrepancies make $G_W=p^*G_0+R$ supported on all of $D$. Pushforward comparison of sections preserves $\kappa(K_W+D)=0$.
On an ordinary minimal model $(V,D_V)$, the effective exceptional comparison of Lemma 2.2 preserves a representative $K_V+D_V\sim_{\mathbb Q}G_V=\sum b_iD_i$ with all $b_i>0$. Hence $0<c<\min b_i$ makes $K_V+D_V-cD_V$ pseudo-effective. Proposition 2.3 supplies semiampleness on the entire reduced boundary. Every hypothesis of Theorem 3.1 is now met, giving $\nu(K_V+D_V)=\kappa(K_V+D_V)=0$.
4. Return the vanishing representative to the original nef divisor
Intersection gives $G_V\cdot H_V^{n-1}=0$ for ample $H_V$. Effectivity forces $G_V=0$, and full support gives $D_V=0$. To return to $G_0$, use a common resolution $W\xleftarrow{q}U\xrightarrow{v}V$: one has $0\leq q^*p^*G_0\leq q^*G_W$ and $v_*q^*G_W=0$. Thus $q^*p^*G_0$ is effective and $v$-exceptional, and rationally linearly equivalent to the pullback of the original nef $L$. The negativity lemma forces it to vanish, proving $L\sim_{\mathbb Q}0$.
Good models for klt pairs in dimension $n$ have now been established. The next branch uses this restricted conclusion, not the general lc conclusion still being proved.
5. The remaining non-klt case: two perturbations and numerical dimension
Set $P=\lfloor\Delta\rfloor$ and suppose $L-\delta P$ is pseudo-effective for some $\delta>0$. Choose rational $\epsilon>0$ with $2\epsilon\leq\delta$ small enough for the perturbed boundaries to be effective and klt. Convexity makes $L-\epsilon P$ and $L-2\epsilon P$ pseudo-effective. Nonvanishing and $\kappa(L)=0$ give Iitaka dimension zero for both. The just-established klt case supplies good models, so both have Nakayama numerical dimension $\kappa_\sigma=0$.
An effective rational representative of $L-2\epsilon P$, together with the identity below, gives $\kappa_\sigma(L)=0$ by monotonicity and homogeneity. The perturbations have not been assumed nef. The original $L$ is nef, so $\nu(L)=0$. Available nonvanishing writes $L\sim_{\mathbb Q}E\geq0$; ample intersection forces $E=0$. Return to the rational simplex for the real-boundary conclusion, and use Lemma 2.2 to descend semiampleness to the original rational nef pair.
4. Theorem 3.1: abundance from the boundary
Theorem 3.1 allows negative coefficients in a representative supported on the boundary. Its conclusion is abundance, $\kappa(L)=\nu(L)$, which is then used in Propositions 2.5 and 6.10.
1. Glue sections on the whole boundary
Proposition 2.3 applies lower-dimensional induction after adjunction to each $D_i$. Fujino–Gongyo’s slc gluing theorem descends semiampleness from the normalization across the conductor, giving semiampleness on all of $D$. Componentwise semiampleness alone is not the hypothesis of Theorem 3.1.
Separate the supply of this hypothesis from the proof of Theorem 3.1 itself. If $D=0$, the signed relation gives $L\sim_{\mathbb Q}0$. If $v=\nu(L)=0$, the inequality $(L-cD)H^{n-1}\geq0$ forces $D=0$. If $v=n$, the nef divisor is big. Only the remaining range $0<v<n$ requires the following deformations.
2. What Hodge theory lifts
Put $v=\nu(L)$. Lemma 3.2 finds a positive boundary component with image dimension $v-1$, while its intersections with the negative and zero parts have smaller image. After the root construction and adjunction data of Proposition 3.3, §4 lifts both local coordinates on that image and a transverse coordinate through every infinitesimal neighborhood. The obstruction lies in a negative twist of the lowest Hodge piece; the graph calculation places it in the symbol-map kernel, where Lemma 4.3 kills it.
Lemma 3.2 / Proposition 3.3 · pp. 13–18 ↗·Proposition 4.1 / Lemma 4.3 · pp. 19–25 ↗
3. From formal deformations to a big divisor downstairs
Fix the lifted image coordinates and vary the transverse coordinate to move a general boundary fiber off $D$. Proposition 5.1 algebraizes this deformation and produces a dominating family of compact $L$-trivial subvarieties of dimension $n-v$. The nef-reduction base therefore has dimension $b\leq v$. Pseudo-effectivity of $L-cD$ makes $D$ vertical for that reduction, and Lemma 5.2 descends the signed divisor to the base.
The nef divisor $G_B$ has full numerical dimension and is therefore big. Pulling back sections gives $\kappa(L)\geq v$, while the usual inequality for a nef divisor gives the reverse bound. Lemma 3.2 handles $v=0$ and $v=n$ separately.
Proposition 2.5 applies this to a zero-Iitaka model with a positive full-support representative. In the nonvanishing preparations, Proposition 6.10 uses it in a contradiction proving bigness of $K_W+D$ for the reduced support of a signed canonical representative. There Proposition 6.9 supplies a nef endpoint using lower-dimensional special termination, and $c=1$ makes $L-D=K_Z$ pseudo-effective. No general good-model assertion in the current dimension is used in advance.
Proposition 5.1 · pp. 26–28 ↗·Lemma 5.2 / §5.1 · pp. 28–30 ↗·Propositions 6.9–6.10 · pp. 41–43 ↗
The conclusion is abundance, not yet the final semiampleness statement. Use it in the zero-Iitaka branch of the induction and in the signed-representative branch of nonvanishing.
5. Theorem 9.6: construct fixed data from a counterexample
The source handles dimensions zero and one directly. For $n\geq2$, assume a smooth counterexample with pseudo-effective $K_X$ and $\kappa(X)=-\infty$. Use the exclusions in §§6–8 to realize the three kinds of input to Theorem 9.1 in characteristic zero.
Statements above: Theorem 9.6·Manuscript proof ↗
1. Counterexample geometry supplies two estimates
Use $q(X)=0$ from Lemma 6.1, general type from Proposition 6.3, and the exclusion of covering curves of bounded genus and normalized degree in Proposition 6.8. Section 7 tracks moving base components caused by excessive vanishing and bounds their degree and canonical divisor. In Proposition 8.1, general-type components would yield the forbidden curves, establishing a uniform scalar bound.
On the two-slot projective bundle, restrict a component arising from failed jet generation to a slice over one factor. Positive-dimensional branches use curve exclusion and the signed-representative bigness of Proposition 6.10. The branch generically finite over both factors is treated by excluding moving branch divisors, fixing a branch complement, and then fixing finitely many covers. Proposition 6.11 excludes the remaining dominant family of birational correspondences. This gives the large Seshadri constant of Proposition 8.3. The auxiliary semiample models use established big-klt results, not the general abundance theorem being proved.
2. Choose the parameters and fix a polarization and flag
On the late terminal model $Y$, set $K=K_Y$, fix $mK$ Cartier, and put $\mu_t=\operatorname{vol}(K_X+tA_X)^{1/n}\to0$. First choose rational $\epsilon>0$ satisfying both small-constant conditions, next a sufficiently large integer $q$ from Proposition 8.3, and only then small rational $t>0$ and integers $r$ with $r\mu_t\to\epsilon$ and $r>qm+1$.
On a common resolution $W$, set $L=r(K+tA)$ and $K_W=K+E$, $E\geq0$. Pull back the semiample positive part of $L$ on its scaling model to obtain $H_0$, with $H_0^n\to\epsilon^n$, $LH_0^{n-1}=H_0^n$, and $K_WH_0^{n-1}\leq H_0^n/r$. Fix $t,r,W$ with strict margins, then make $H=H_0+\eta H_{\rm amp}$ ample for small rational $\eta>0$, preserving the intersection inequalities of Theorem 9.1. Generic semipositivity for pseudo-effective $K_W$ and restriction give one fixed complete-intersection flag with $\mu_{\min}(\Omega_W^1|_C)\geq0$.
§8.1 / (30)–(32) · p. 52 ↗·§9.2.1 / (61)–(64) · pp. 69–70 ↗·[CP] Theorem 2.1 · print p. 844 / PDF p. 11 ↗
3. One movable curve controls every determinant weight
Put $D^+=2r(K+2tA)+2E$. Proposition 8.1 bounds the order of every nonzero section in every sufficiently divisible degree $k$ at a very general $x$ by $32\epsilon k$. Hence $\pi_x^*D^+-40\epsilon J$ is not pseudo-effective on the blowup: otherwise a convex combination with big $\pi_x^*D^+$ would produce excessive vanishing for some $32\epsilon<c<40\epsilon$.
[BDPP] duality detects this by a strongly movable class. Strictness permits one actual complete intersection $\gamma=f_*(A_1\cdots A_{n-1})$, on a fixed smooth birational model $f:V\to\widehat W$ with ample integral Cartier divisors $A_i$. One has $J\cdot\gamma>0$. Put $S=mK$ and $Q_\lambda=L+K_W/2+\lambda S$. The difference below is pseudo-effective for every $0\leq\lambda\leq q$. Thus the same $\gamma$ bounds all weights, and only the two endpoint inequalities at $0,q$ need to be retained for reduction.
§9.2.2 / (65)–(66) · pp. 70–71 ↗·[BDPP] Theorem 2.2 · p. 6 ↗
4. Fix finite jets and invoke incompatibility
Proposition 8.3 gives a Seshadri constant greater than $2n+2$. Choose $z_*$ in the torus open where the semiample contraction is an isomorphism. Fix one denominator-clearing degree $k_0$ and finitely many sections generating jets through order $(2n+2)k_0$. Its two base coordinates need not equal the testing point $x$ of the previous step.
All polarization, flag, movable-curve, and jet inputs of Theorem 9.1 now exist in characteristic zero. Inside that theorem, also fix $T_0$ and finitely many remainder sections before passing to large characteristic. The next section contradicts two bounds for the same determinant. This excludes the counterexample $X$ and proves Theorem 9.6. Only afterwards may Proposition 2.5 be applied.
6. Theorem 9.1: incompatibility of fixed data
This theorem assumes neither nonvanishing, abundance, nor minimal models. The preceding counterexample construction supplies its explicit intersection conditions, flag, jets, and movable curve.
1. Reduce finitely many sections and one flag
Choose one sufficiently ample $T_0$ and finitely many sections of $T_0+aS$, $0\leq a\leq(k_0-1)q$, nonzero at the two base coordinates of $z_*$. Spread all varieties, maps, divisors, flags, points, the complete-intersection witness, and jet and filler sections over one finitely generated $\mathbb Z$-algebra. Shrinking preserves smoothness, the required ampleness, finite jet surjectivity, intersection numbers, and the nonnegative minimum slope on the flag curve.
Only finite data are preserved. Neither a characteristic-zero pseudo-effective cone nor an MMP is transferred to characteristic $p$. Nonnegative pairing of $\gamma$ with effective divisors on the reduction follows there by pullback to the fixed birational model and intersection with its ample complete intersection.
2. Amplify the fixed jets to obtain a nonzero determinant
Put $N_p=\lfloor p/k_0\rfloor$ and $B_p=k_0N_pL+T_0$. Products of $N_p$ copies of the fixed system generate jets through order $(2n+2)k_0N_p$. Previously fixed filler sections handle the remainder $(p-k_0N_p)q$, giving jets of $pq\xi+B_{p,1}+B_{p,2}$. Since $\dim Z=2n+1$, for large $p$ these surject onto the local quotient by the $p$th powers of all regular parameters. Triangularization by leading monomials uses no division by factorials.
For relative Frobenius $F:W\to W'$, set $E=F_*\mathcal O_W(B_p)$, $s=p^n$, and $R=s^2$. Taking the constant coefficient in the torus-coordinate basis $1,z,\ldots,z^{p-1}$ retains weights $pa$. With $U_a=H^0(W,B_p+paS)$, the evaluation below therefore has generic rank $R$ and a nonzero maximal minor $\sigma$. The full-rank point need not be the point used to test orders on the diagonal.
3. Diagonal rank loss forces high vanishing
On the diagonal, every weight $a+b=q$ becomes the same line bundle $\mathcal L_p=\mathcal O_{\Delta_F}(B_{p,1}+B_{p,2}+pqS_1)$ on $\Delta_F=W\times_{W'}W$. The diagonal evaluation rank is therefore at most $h^0(\Delta_F,\mathcal L_p)$. Filtering by the ideal of the reduced diagonal gives $G_j=\mathcal O_W(2B_p+pqS)\otimes A_j(\Omega_W^1)$ for $0\leq j\leq n(p-1)$. Here $A_j$ is the degree-$j$ part of the symmetric algebra modulo $p$th powers, and its ranks satisfy $\sum_j e_j=p^n$.
The fixed-curve hypothesis $\mu_{\min}\geq0$ and [Langer, Corollary 2.5] give the slope bound of Lemma 9.4, uniform in $j,p$. Successive restrictions along the fixed flag bound the sections of every $G_j$ by its rank times the same scalar polynomial. Summing ranks, without an extra factor for the number of pieces, gives $h^0(\Delta_F,\mathcal L_p)\leq R(7^nH^n+O(p^{-1}))<R/4$. The last inequality uses $H^n\leq(2\epsilon)^n$ and $(14\epsilon)^n<1/4$.
The matrix rank at a diagonal point is less than $R/4$. After constant row operations, more than $3R/4$ rows vanish there, forcing every determinant term into the corresponding power of the maximal ideal.
Lemmas 9.2–9.5 / equations (46)–(56) · pp. 63–67 ↗·Equation (60) · p. 69 ↗·[Langer] Corollary 2.5 · print p. 258 / PDF p. 8 ↗
4. Bound the same determinant from above
The determinant is a section of the actual external product $Q_1\boxtimes Q_2$. The normalized first Chern class of each factor is $L+K_W/2+\lambda_iS+O(p^{-1})$, with $0\leq\lambda_i\leq q$. The fixed movable witness bounds the order of every nonzero section of each factor by $R(40\epsilon+O(p^{-1}))$.
Bases adapted to vanishing orders have independent leading terms, so the two factor bounds add for the tensor product.
More precisely, if $m_i$ is the sum of the selected column weights in factor $i$, then $Q_i=(\det E)^{\otimes p^n}\otimes\mathcal O(m_iS')$ and $\lambda_i=m_i/R\in[0,q]$. Thus $c_1(Q_i)/R=B_p/p+(p-1)K_W/(2p)+\lambda_iS$. The numerator of the error, $B_p-pL-K_W/2$, ranges over a fixed finite list of divisor classes, making $O(p^{-1})$ uniform in the selected columns. The movable curve tests the effective divisor of every nonzero section on the reduction itself; that section need not lift to characteristic zero.
Since $80\epsilon<3/4$ was chosen first and all other data are fixed, large $p$ gives a contradiction. This proves Theorem 9.1. Applying it to the data constructed from a counterexample in the previous section gives Theorem 9.6, returning to the nonvanishing step of the overview. Proposition 2.5 then establishes real-boundary good models in the same dimension.
Return Theorem 9.6 to step 1 of the main diagram. Hashizume and Proposition 2.5 then produce good models in the same dimension; induction gives Theorem 11.1.
7. External inputs and consequences
Inputs to the main proof
Hashizume is used only after smooth nonvanishing has been proved. Fujino–Gongyo gluing supplies the whole-boundary condition before that step, using smaller-dimensional models. The positive-Iitaka lemma and Ambro’s descent enter the subsequent good-model argument. BDPP, Campana–Păun and the restriction theorem construct the fixed witnesses of Theorem 9.6; Langer’s slope bound is used inside the independent Theorem 9.1. Their hypotheses and uses are specified in the reference guide and the corresponding proof stages.
Proposition 2.5·Theorem 3.1·Theorem 9.6·Theorem 9.1
Catalogue 033 → LA: subadditivity in one lemma
[OI], Corollary 6.2 is stated as LA’s Theorem 1.2. Its only use is Lemma 6.1, with both boundaries zero. If the Albanese image had positive dimension, its resolved Stein base would have nonnegative Kodaira dimension, and the smaller-dimensional fiber would have nonvanishing by induction. Subadditivity would contradict the counterexample.
For this specific base of maximal Albanese dimension, the established Hacon–Popa–Schnell Theorem 1.1 supplies the same required inequality, as the LA introduction also records. This alternative covers the application in Lemma 6.1, not the full logarithmic theorem of [OI].
[OI] Corollary 6.2 · pp. 40–41 ↗·Hacon–Popa–Schnell, Theorem 1.1 · p. 2 ↗·LA §1 / Lemma 6.1 · pp. 6, 30–31 ↗
Within 034: the independent nonvanishing tools
Lemmas 7.1–7.5 control centres and moving components; Theorem 9.1 compares determinant orders for fixed witnesses. Fourfold nonvanishing imports these tools in its §§3, 6–7. This use does not invoke the abundance main theorem. The receiving argument →
Within 034: the good-model output
Theorem 11.1 supplies the qualitative model before effective degrees are studied. Its uses differ: model existence in Uniform Iitaka, the semiample fibration in Log Iitaka, perturbed klt models in the Hodge-rank argument for slc indices, and nonvanishing in Fourfold Iitaka.
- Uniform Iitaka · Theorem 2.1
- Log Iitaka · Theorem 2.2 / §8
- slc indices · Theorem 2.3 / Lemma 5.6
- Fourfold Iitaka · §10
Kähler abundance instead invokes the full projective induction in its Proposition 2.10, while retaining its own subadditivity assumption. The projective input →
Within 034: LA → Schnell
For a smooth projective complex variety $X$ with pseudo-effective $K_X$, Theorem 11.1 applied to $(X,0)$, together with the comparison lemma, gives Corollary 11.2: a Q-factorial klt good model V and an effective q-exceptional comparison on a common resolution. This is the input used by Theorem 3.1 of the Schnell manuscript.
Continue to the mixed-intersection argument in Schnell →
Theorem 11.1 / Corollary 11.2 · pp. 73–74 ↗·Catalogue 034 dependency table →
Consequences after the main proof
Sections 11–14 derive rational-linear triviality in Iitaka dimension zero, slc abundance and boundary extension, finite generation of rational lc rings, geometric consequences for rational curves and cotangent tensors, and relative analytic abundance near compact base subsets. These consequences follow after the induction. Section 12 additionally cites the ordinary-pair case of the generalized-minimal-model manuscript in catalogue 036; that companion is not an input to the good-model induction above.
§§11–14 · pp. 73–81 ↗·Catalogue 036: generalized minimal models · Corollary 1.2 ↗
8. Source entry points and reading scope
- §2 / Proposition 2.5 · pp. 7–12 — The two induction outputs and the zero-Iitaka cases.
- Theorem 3.1 / §§3–5 · pp. 12–30 — Signed representatives, formal lifting, compact families, and descent.
- Proposition 2.5 / §10 / Theorem 11.1 · pp. 9–12, 72–73 — Close each dimension only after smooth nonvanishing; return from rational to real boundaries.
- Propositions 8.1, 8.3 · pp. 53–61 — The scalar bound and the two-factor jet construction.
- Theorem 9.1 / Theorem 9.6 · pp. 62–72 — The determinant contradiction and construction of its witnesses.
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