CONDITIONAL GOOD MINIMAL MODELS FOR COMPACT KAHLER FOURFOLDS
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condensed author's version, October 2026.
Same statements and same proof skeleton as the full manuscript.
Proofs are summarized to one paragraph each; theorem numbers and
cross-references refer to this document. For full technical detail,
bibliographic citations, and the complete proofs of the projective
appendices, see the original file.
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ABSTRACT
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We prove, conditionally, that every globally strongly Q-factorial
compact Kahler klt fourfold pair (X,B) with effective rational
boundary and analytically pseudo-effective actual adjoint D=K_X+B
admits a good minimal model. "Good" means the endpoint adjoint is
analytically nef and a positive Cartier multiple is globally
generated. The three inputs are:
(A) full orbifold Iitaka subadditivity in Fujiki class C,
(B) a finite pseudo-effective klt fourfold MMP,
(C) abundance-after-nonvanishing for nef klt fourfold adjoints.
The body of the proof reduces the problem to canonical nonvanishing
on a smooth non-uniruled fourfold with algebraic dimension zero, and
finishes it there with singular metrics, holomorphic foliations, and
extension from a reduced boundary. The projective case uses a
conditional projective log abundance theorem proved in the appendix
from a single logarithmic Iitaka subadditivity premise.
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CONTENTS
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I. Setup and the three inputs
II. Main theorems
III. Structure of the proof
IV. Reductions to canonical nonvanishing
V. Positive algebraic dimension
VI. Algebraic dimension zero: metric lemma
VII. Algebraic dimension zero: reduced boundary and stratum
VIII.Algebraic dimension zero: extension and completion
IX. Projective appendix: signed-representative theorem
X. Projective appendix: geometric exclusions
XI. Projective appendix: jets and Frobenius
XII. Finite-field descent
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I. SETUP AND THE THREE INPUTS
========================================================================
All spaces are over C and are normal, connected, and complex analytic
of dimension at most four. A Kahler space means a complex analytic
space with a local smooth strictly plurisubharmonic potential on each
local embedding; smooth compact Kahler resolutions and resolutions of
meromorphic maps are obtained by projective modifications.
1. Actual adjoint.
For an effective rational Weil divisor B on X, write D=K_X+B when
for some positive integer r, the reflexive sheaf
omega_X^[r] (x) O_X(rB), double-dualized, is invertible. Positive
tensor powers then define O_X(krD). A Q-linear identity D1~_Q D2
means an isomorphism after a common positive integral multiple of
actual holomorphic line bundles. Numerical equivalence alone never
infers this identity.
2. Analytic positivity.
D is analytically pseudo-effective if c_1(D)/r is represented by a
closed positive (1,1) current with local potentials. D is
analytically nef if this class lies in the closure of the Kahler
cone in real Bott-Chern cohomology. On compact curves this implies
nonnegative degree; the implication is one-way.
3. Iitaka dimension.
kappa(X,D) is -inf if every positive Cartier multiple has zero
sections; otherwise it is the maximum image dimension of the
complete linear systems of positive Cartier multiples. A rational
line is semiample if some positive Cartier multiple is globally
generated. A semiample rational line of Iitaka dimension zero is
torsion as an actual rational line.
4. Global strong Q-factoriality.
A normal compact X is globally strongly Q-factorial when every
coherent rank-one reflexive sheaf F on X has some positive
reflexive power F^[m]=(F^(x)m)^** invertible. This is a global
condition on the whole space, not a hypothesis on every open
subset. Smooth spaces satisfy it, and all constructions below
preserve it when it is needed.
5. Ordinary klt and dlt.
On a smooth model W over X, the log discrepancy of a prime divisor
E is a(E;X,B)=1+coeff_E(K_W-p^*(K_X+B)). The pair is klt if all
these numbers are positive and dlt if they are nonnegative and B
has SNC support with coefficients in [0,1]. In this document,
"klt pair" always means ordinary (not generalized) rational pair
with effective rational boundary. The floor is the sum of the
coefficient-one components.
6. Conventions on cycles.
A prime cycle is an irreducible reduced closed analytic subset of
positive codimension. A "curve" is a compact irreducible reduced
1-cycle. The closed analytic cone NA(X) is the cone of positive
closed (1,1)-currents modulo real Bott-Chern (1,1)-classes.
7. Assumption (A): full orbifold Iitaka subadditivity.
Let X be a smooth compact connected manifold in Fujiki class C,
Delta an effective rational SNC boundary with coefficients in
[0,1], and f:X->Y a surjective holomorphic map with connected
fibers onto a normal compact irreducible complex space. Then for a
very general smooth fiber F with SNC boundary restriction,
kappa(X,K_X+Delta)
>= kappa(F,K_F+Delta|_F) + kappa(f | Delta).
Here the invariant base term kappa(f | Delta) uses the
inf-multiplicity orbifold convention on smooth models: for every
prime P of the base, m(P) is the min over dominating divisors E of
ord_E(f^*P) * (1-coeff_E Delta)^-1, and B(f,Delta) is the sum of
(1-1/m(P))P. Neat models realize the infimum explicitly as
kappa(S,K_S+B). Neither source nor base need be projective. There
is no abundance, positivity, or good-model premise in (A). The
statement is Theorem 1.1 of the orbifold subadditivity preprint.
8. Assumption (B): pseudo-effective klt fourfold MMP.
The closed cone NA(X) is as in (6). Start from an ordinary klt pair
(X,B) on a globally strongly Q-factorial compact Kahler fourfold,
with B effective rational and D=K_X+B Q-Cartier and analytically
pseudo-effective. The program on every non-nef stage (X_i,B_i)
picks a nonzero D_i-negative extremal ray R of NA(X_i) with a nef
supporting class alpha such that alpha - c_1(D_i) is Kahler and
alpha^perp meets NA(X_i) exactly in R. There is a projective
bimeromorphic contraction with connected fibers f_i:X_i->Z_i,
relative Picard number one, and -D_i relatively ample. A
divisorial contraction gives the next pair by pushforward. A small
contraction is flipped by the canonical relative Proj of the log
algebra of multiples of D_i; it is projective, small, and has
connected fibers, and the new adjoint is relatively ample. Every
new pair remains klt, compact Kahler, globally strongly
Q-factorial, with Q-Cartier analytically pseudo-effective actual
adjoint, and every finite prefix extends to a full program. Every
program terminates, no matter which negative rays are chosen. It
ends at an analytically nef pair. The composite map extracts no
prime divisor and satisfies the minimal-model discrepancy
inequalities: weakly for every prime over the models, strictly for
every prime on X contracted by the map.
Assumption (B) follows from Theorem 6.24, Lemma 3.2, and
Propositions 3.4 and 6.23 of the fourfold MMP preprint, in the
intrinsic canonical-sheaf formulation. It gives neither a first
plurisection nor global generation.
9. Assumption (C): effective fourfold abundance.
Let X be a normal compact Kahler fourfold and Delta an effective
rational boundary such that (X,Delta) is klt and J=K_X+Delta is
Q-Cartier. If J is analytically nef and kappa(X,J)>=0, then a
positive Cartier multiple of J is globally generated. This is
Theorem 1.1 of the abundance-after-nonvanishing preprint. The
nonvanishing premise is explicit and is not granted separately.
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II. MAIN THEOREMS
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Theorem A (conditional good minimal models).
Assume (A), (B), (C). Let X be a normal connected compact Kahler
fourfold, B an effective rational Weil divisor, and suppose (X,B)
is klt, D=K_X+B is Q-Cartier and analytically pseudo-effective,
and X is globally strongly Q-factorial. Then there are a normal
connected globally strongly Q-factorial compact Kahler fourfold Y
and a bimeromorphic map phi:X --> Y such that:
(i) phi extracts no prime divisor and B_Y=phi_*B;
(ii) (Y,B_Y) is klt and D_Y=K_Y+B_Y is Q-Cartier and
analytically nef;
(iii) a(E;Y,B_Y)>=a(E;X,B) for every prime divisor over the
models, with strict inequality for every prime on X
contracted by phi;
(iv) some Cartier multiple O_Y(mD_Y) is globally generated.
The exponent m may depend on the pair; the conclusion concerns the
actual holomorphic adjoint line on the endpoint Y.
Theorem B (nonvanishing on the nef endpoint).
Assume (A), (B), (C). Let (Y,Delta) be an ordinary klt pair on a
normal connected globally strongly Q-factorial compact Kahler
fourfold. If Delta is effective rational and the Q-Cartier adjoint
J=K_Y+Delta is analytically nef, then kappa(Y,J)>=0.
Theorem A follows from Theorem B in one step: (B) supplies the nef
minimal model with (i)-(iii), (B) proves nonvanishing on it, then
(C) globalizes a Cartier multiple. So the substance of the paper
is Theorem B.
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III. STRUCTURE OF THE PROOF
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Part IV reduces Theorem B to one explicit canonical-nonvanishing
statement. Part V handles positive algebraic dimension via an
actual canonical pullback from a projective base. Parts VI-VIII
handle algebraic dimension zero: a zero-Lelong metric on a nef klt
adjoint, a nonextracting reduced-boundary model, and extension
from the reduced floor. The projective appendices (IX-XI) prove
projective log abundance from part (A) alone; XII transfers the
conclusion to every algebraically closed field of characteristic
zero.
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IV. REDUCTIONS TO CANONICAL NONVANISHING
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The whole body of the paper reduces Theorem B to canonical
nonvanishing in one leftover geometric case.
1. Lemma (A implies smooth projective log Iitaka).
Let f:M->S be a fibration of smooth connected complex projective
varieties, and D_M, D_S reduced effective SNC divisors (possibly
zero) with Supp(f^*D_S) contained in Supp(D_M). Then for a very
general smooth fiber F with D_F=D_M|_F:
kappa(M,K_M+D_M) >= kappa(F,K_F+D_F) + kappa(S,K_S+D_S).
Proof sketch. It is enough to bound the invariant base from below
by kappa(S,K_S+D_S). Flatten f over a base modification q:S'->S,
then resolve. Flatness makes the main strict transform
equidimensional with fibers of dimension dim M - dim S, and the
unchanged smooth connected generic fiber rules out vertical
components. Resolve the source and boundary to SNC, giving a
commuting diagram M'->M over S'->S. Set D_M' equal to the strict
transform of D_M plus the full reduced exceptional divisor. This
is an allowed orbifold modification: any prime over a
coefficient-one prime of D_M is either a strict transform of that
prime or exceptional, and in both cases has coefficient one.
Neatness also holds: a prime E whose image in S' has codimension
c >= 2 has image dimension at most (dim S - c) + (dim M - dim S)
= dim M - c in the equidimensional main space, hence codimension
at least two in M, hence E is exceptional over M. So p is a neat
morphism.
Every pullback component of q^{-1} Supp(D_S) lies in D_M', so it
carries infinite orbifold multiplicity, giving
B(f',Delta) >= (q^{-1} Supp D_S)_red.
The log nonnegativity of discrepancies of the SNC pair (S,D_S)
gives K_{S'}+(q^{-1}Supp D_S)_red = q^*(K_S+D_S)+E_S with
E_S effective and q-exceptional. Monotonicity of kappa under
pushing forward proves
kappa(f|D_M) >= kappa(S,K_S+D_S).
Feeding this into (A) proves the lemma. []
2. Corollary (projective log abundance from (A)).
Under (A), a nef Q-Cartier adjoint of a normal projective lc pair
over C, with effective rational boundary, is semiample.
Proof. Apply the conditional projective log abundance theorem of
the appendix; by Lemma 1 its only subadditivity premise follows
from (A). []
3. Lower-dimensional inputs.
(i) A smooth compact Kahler manifold of dimension at most three
which is not uniruled has kappa(K_X)>=0. In the nonprojective
threefold case take a terminal minimal model (Horing-Peternell)
and apply canonical nonvanishing (Demailly-Peternell) to its nef
canonical class.
(ii) Projective klt pairs of dimension at most three with
effective rational boundary and pseudo-effective adjoint have
nonnegative adjoint Iitaka dimension (klt MMP + log abundance in
dimensions at most three).
(iii) Ou's theorem: a smooth compact Kahler manifold is
non-uniruled iff its canonical class is analytically
pseudo-effective.
4. Lemma (uniruled reduction).
Assume (A). Let (Y,Delta) be a non-Moishezon compact Kahler klt
fourfold with effective rational boundary and pseudo-effective
adjoint J=K_Y+Delta. If a smooth compact Kahler resolution of Y
is uniruled, then kappa(Y,J)>=0.
Proof sketch. Take the rational-chain quotient of a smooth model
W of Y and resolve. The result is a fibration f:W'->T with W',T
smooth compact Kahler, and very general fibers projective
rational-chain-connected. The base T is not uniruled: otherwise
pulling back a covering rational curve on T and using a
Graber-Harris-Starr section produces a rational chain between
two distinct very-general rational-quotient fibers, contradicting
the quotient property. The base dimension is in [1,3], as is the
fiber dimension.
The simultaneous log resolution makes
K_{W'}+Gamma ~_Q p^*J + E,
with 0 <= Gamma < 1 and E >= 0 effective p-exceptional. Very
general fibers, with the SNC boundary restriction, satisfy
kappa(F,K_F+Gamma|_F)>=0 by (3)(ii); the base T satisfies
kappa(T,K_T)>=0 by (3)(i); (A) with smooth base gives
kappa(W',K_{W'}+Gamma)>=0. Pushing forward through E and
extending reflexively across the codimension-two excluded set of
Y produces a section of a positive multiple of J. []
5. Lemma (irregular canonical nonvanishing).
Assume (A). If W is a smooth non-uniruled compact Kahler fourfold
with q(W)>0, then kappa(W,K_W)>=0.
Proof sketch. Take the Albanese Stein-factor base T (positive
dimensional, maps generically finitely into the Albanese torus).
A wedge of invariant 1-forms on the torus pulls back to a nonzero
section of K_T, so kappa(K_T)>=0. Very general fibers have
dimension at most three and are non-uniruled (a rational-curve
fibration would fibre the total space), so their canonical classes
have kappa>=0. Then (A) with zero boundary on the resolved total
space gives kappa(K_W)>=0. []
6. Proposition (reduction to the leftover case).
Assume (A), (B), (C). It suffices, for Theorem B, to prove
canonical nonvanishing for every smooth non-uniruled non-Moishezon
compact Kahler fourfold W with q(W)=0.
Proof sketch. Let (Y,Delta) be the nef klt endpoint of Theorem B
with adjoint J=K_Y+Delta. If Y is Moishezon, rationality of klt
singularities (relative vanishing on a projective log resolution)
makes it projective; a resolution-adapted GAGA and Chow argument
identifies the rational boundary and adjoint with algebraic data,
analytic nefness is stronger than curve nonnegativity, and
Corollary 2 gives semiampleness.
Otherwise let W->Y be a smooth compact Kahler resolution with
Delta_W effective compact, using (B). If W is uniruled, Lemma 4
finishes. Otherwise K_W is pseudo-effective (3)(iii). A canonical
plurisection on W pushes forward reflexively through the resolution
and multiply by the boundary section at a common multiple gives
kappa(Y,J)>=0. Lemma 5 handles q(W)>0, leaving exactly W smooth
non-uniruled non-Moishezon with q(W)=0.
Once nonvanishing on this original endpoint Y holds, (C) makes J
semiample. The remaining properties (i)-(iii) of Theorem A have
already been supplied by (B).
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V. POSITIVE ALGEBRAIC DIMENSION
========================================================================
Reduction proposition IV.6 leaves canonical nonvanishing on W smooth
non-uniruled non-Moishezon q(W)=0. This part handles the subcase
0<a(W)<4. The statement is:
Proposition.
Assume (A), (B), (C). Let W be a smooth non-uniruled compact
Kahler fourfold with q(W)=0 and 0<a(W)<4. Then kappa(W,K_W)>=0.
1. A local index bound.
Lemma. Let (Z,Delta) be an ordinary rational klt pair on a normal
n-dimensional compact Kahler space, with Q-Cartier adjoint
J=K_Z+Delta, and pi:Z->T a projective bimeromorphic morphism with
connected fibers to a normal compact Kahler space. Suppose -J is
relatively ample and every contracted curve lies in one J-negative
ray R of NA(Z). Let k>0 satisfy: kJ is Cartier. Let L be a line
with L.R>0. Then every nontrivial fiber F satisfies
r_F := sup{ u : L+uJ nonneg on all curves of F } >= 1/(k(n+1)).
Proof sketch. Every curve in F is in R and has nonzero class, so
r_F=L.C/(-J.C) is independent of C. The restriction of L is
numerically a positive multiple of -J and hence ample on the
projective fiber; openness of fiberwise ampleness makes L
relatively ample over a small Stein neighborhood of the image
point. At r_F, L+r_F J is nef and not ample on the fiber. Fujino's
analytic rationality theorem for projective morphisms over a Stein
base with a property-(P) compactum, applied to this finite positive
threshold, gives denominator at most k(dim F+1). The non-lc locus
condition is empty for klt. Dividing by dim F+1 gives the bound.
The line-bundle form is directly supported by that theorem. No
analytic Q-factoriality on open subsets is used. []
2. An actual canonical pullback.
Fix a=dim W. Resolve a map to PP^d defined by a algebraically
independent meromorphic functions on W. Stein factorization has a
projective base, and after resolving base and graph we may assume
W is a smooth compact Kahler model with a fibration b:W->S_0 to
a smooth connected projective S_0 of dimension d, with connected
fibers. Fix a very ample H on S_0.
Because K_W is pseudo-effective and 4-d<=3, the parameter
observation from III pushes generic nonvanishing on fibers through
to rk b_*O_W(lK_W)>0 for some l>0. Coherence plus ample twisting on
the projective base gives, for some N_0,
kappa(K_W + N b^*H) >= 0 for every integer N >= N_0.
Proposition (the pullback model). Under (A), (B), there are a
canonical globally strongly Q-factorial compact Kahler fourfold V,
a normal projective T of dimension d, a fibration g:V->T, and an
actual rational A_T in Pic(T) (x) Q such that
K_V ~_Q g^*A_T.
The space V is obtained from the current smooth W by a finite
empty-boundary K-program allowed by (B), and K_V remains
pseudo-effective.
Proof sketch. At a stage V_i still mapping to S_0, put
H_i=b_i^*H. A common projective resolution of every negative
canonical step gives
p^*K_before ~_Q q^*K_after + G,
with G effective and exceptional over the after model (analytic
negativity over the contraction base; no extraction). Discrepancies
do not decrease, so canonicity persists from smooth W.
Choose a stage index k_i (Cartier index of K_{V_i}) and a twist
N >= N_0 with N > 5 k_i. The line K_{V_i}+N H_i has an effective
rational klt boundary representative by a general member of a high
free multiple of N H_i. The adjoint is pseudo-effective. If not
nef, (B) supplies a negative extremal ray of K_{V_i}+N H_i; H_i is
the pullback of a semipositive class, so the same ray is
K_{V_i}-negative. Lemma 1, applied with L=H_i and J=K_{V_i},
forbids H_i . R_i > 0; therefore H_i . R_i = 0, and every
connected projective fiber of the contraction maps to a point under
b_i. Proper descent over the normal contraction target factors
b_i; in a flip, composition with the positive-side morphism gives
b_{i+1}. So the actual pullback line H_i, not merely its class,
keeps pulling back from the same H.
The procedure is a single empty-boundary K-program from W. If
infinite, it would contradict (B). Termination gives a stage V
with K_V+N H_i nef for its selected N. Section spaces at that N on
V already contain the original plurisections from W (pushed
reflexively). So kappa(K_V+N H_i)>=0 directly, before invoking
(C). Then (C) makes this actual line semiample.
Let h:V->PP^r be the morphism of a free integral multiple
m(K_V+N H_i), and g:V->T its Stein factorization over (b_i,h).
T is normal projective and dim T=d. Write
A_T = (1/m) c^*O_{PP^r}(1) - N a^*H, in Pic(T) (x) Q,
where a:T->S_0 and c:T->PP^r are the induced maps. The evaluation
isomorphism for h and the actual identity H_i=g^*a^*H give
K_V ~_Q g^*A_T. []
Choose a smooth projective resolution mu:S->T and put A=mu^*A_T.
Resolving the graph gives a smooth compact Kahler M with p:M->V
and fibration f:M->S such that
K_M ~_Q f^*A + R, with R effective and p-exceptional.
Pullback of holomorphic one-forms gives q(S)<=q(M)=q(W)=0.
Push of a positive representative of c_1(K_V) against omega^{4-d}
gives v c_1(A) for a positive constant v; hence A is
pseudo-effective. Since S is projective, A is also numerically
pseudo-effective.
3. Base of dimension 1.
The smooth projective curve S has q(S)=0, so S=PP^1. A
pseudo-effective rational line on PP^1 has nonnegative degree and
a section after a divisible multiple. []
4. Base of dimension 2: fiber powers.
On very general smooth fibers, h^0(F,mK_F)=1 for every m divisible
by a fixed index m_0. At least one section is supplied by the
effective exceptional divisor R|_F. If some m had generic dimension
>=2, two sections would have a nonconstant ratio on a general
fiber; combined with the two-dimensional projective base, this
would give a(M)>=3. Since a(M)=2, that is a contradiction. []
Lemma (base intersection inequality).
There are nonnegative rational c_P indexed by the mu-exceptional
curves P on S, such that
(A-K_S+sum_P c_P P) . H' >= 0
for every ample H' on S.
Proof sketch. Choose a dense open S^0 over which f is smooth and
the canonical generator from K_M~_Q f^*A+R is not identically zero
on any fiber. Discard isolated bad points; include all mu-exceptional
curves among the finitely many complementary base curves.
For each complementary curve P, fix a divisor D in f^*P dominating
P, write e=ord_D(f^*P), h=coeff_D R. If P is not mu-exceptional,
choose D with h=0 (the strict transform of a component over a
general point of its image divisor in the normal T has coefficient
zero in R). For an exceptional P, set c_P=h/e.
Choose a smooth curve C in a high very ample multiple H' with
genus at least one, transverse to all complementary curves, and
through a parameter where all the rank-one conditions hold. The
pullback M_C=f^{-1}(C) is smooth and connected. Very general
fibers of f_C:M_C->C satisfy the h^0=1 condition by Bertini.
Take the r-fold fiber product over C, resolve, apply (A) with zero
boundary in dimension 2r+1 to get a nonzero section of
mK_{Z_r}. On good parameters, the product formula pieces apart
into r relative generators times a base pluriform. The base
coefficient extends meromorphically. Local coordinate analysis at
a point of C cap P, with t=x^e and R_C=h{x=0}, gives order
m (r h - (r-1)(e-1)) >= -m r h (pole allowance)
plus nonnegative terms. Dividing by r and letting r->infty, with
C fixed, proves the intersection inequality. []
Zariski decompose A=P_0+N_0 with P_0 nef and N_0 effective
negative-definite. Every mu-exceptional P is orthogonal to P_0.
If P_0 equiv 0, torsionness of numerically trivial lines on
q(S)=0 makes P_0=0 in Pic(S) (x) Q. If P_0^2>0, P_0 is big. In
the remaining case P_0^2=0 and P_0 not=0: test the inequality
against ample classes approaching P_0 to get K_S . P_0 <= 0;
Riemann-Roch makes chi(S,lP_0)>0 for large l, and
h^2(S,lP_0)=0, so h^0(S,lP_0)>0. Multiplication by a section of
lN_0 gives kappa(S,A)>=0. []
5. Base of dimension 3: elliptic fibration and modular forms.
The very general fiber is a smooth connected elliptic curve. The
Hodge line H of the family is a holomorphic line by periods.
Evaluation f^*H -> omega_{M/U} is an isomorphism (a nonzero
holomorphic 1-form on an elliptic curve has no zero). Therefore
the actual identity
H^{x m} ~ L_m|_U = O_S(m(A-K_S))|_U
holds on a smooth open U of the family; no section of the fibration
is needed.
Let E_4 and Delta be the classical scalar modular forms of weights
4 and 12 for SL_2(Z). In a normalized symplectic period frame, they
define sections
e_1 = E_4^{m/4}, e_2 = Delta^{m/12}
of L_m|_U. Growth at a complementary prime P, in local parameter
t, is bounded by C (1+|log|t||)^b on either side of a fixed positive
lower bound, using Schwarz-Pick on the period map on a punctured
disk and the boundedness of E_4, Delta on the standard fundamental
domain.
The divisorial order identity is:
Lemma. e_1,e_2 extend meromorphically to S. If
ell_P=min_i ord_P(e_i/s(x)), omitting an identically zero
section, for a base frame s of L_m, then
ell_P / m = 1 - t_P,
where
t_P = min_{E mapping to P} (1+coeff_E R)/ord_E(f^*P).
The same identity holds on every further smooth base model.
Proof sketch. The weighted integrability threshold of \|s\|^{2/m}
across P, computed on the total space by Fubini in SNC coordinates,
equals t_P. A polynomial lower bound \|s\|^{2/m} >= c \|t\|^{2(h-e+1)/e}
holds on the appropriate local model. Combining gives a pole
allowance for the meromorphic coefficients a_i=e_i/s, extend by
Hartogs, and read off the order identity from the growth of
\|e_1\|+\|e_2\|. []
For non-exceptional primes P one also gets t_P<=1, because a
component of the pullback of its image divisor on T is a strict
transform with zero R-coefficient and positive integral pullback
multiplicity. No upper bound is needed for base-exceptional
primes.
Resolve the pencil [e_1:e_2] by projective blowups, then resolve
divisorial supports. On the smooth model S, the transformed line
is
L'_m = rho^*L_m (x) O_{S'}(-mK_{S'/S})
so both section divisors transform by the same relative Jacobian
term. The fixed divisor is sum_P ell_P P with SNC support. A
general complex linear combination e gives a crepant boundary
B_S = sum_P (1-t_P)P + (1/m) Q,
Q a general free-moving member (possibly zero), of total SNC with
every coefficient < 1. Regarding e downstairs as a rational
section of mA_T-mK_T, the corresponding
Xi = (1/m) div_T(e)
is effective, and (T,Xi) is klt: (T,Xi) has an SNC crepant
boundary with every coefficient < 1. The actual-linear identity
K_T + Xi ~_Q A_T
and the crepancy K_S+B_S = mu^*(K_T+Xi) hold on compatible
rational representatives.
Finally put Gamma_S=(B_S)_+. Then
K_S+Gamma_S ~_Q A + E_S,
with Gamma_S effective rational klt SNC, E_S>=0 mu-exceptional.
Since mu_*Xi>=0 makes B_S effective off the codim-two discrepancy
locus, E_S is mu-exceptional. This adjoint is numerically
pseudo-effective on the smooth projective threefold S. Projective
klt nonvanishing in dimension three (a special case of IX below)
gives a section of a positive multiple of K_S+Gamma_S. Push it to
T. The exceptional pole allowance drops in codimension one,
normality extends across codim two, and the result is
kappa(T,A_T)>=0. []
========================================================================
VI. ALGEBRAIC DIMENSION ZERO: METRIC LEMMA
========================================================================
Part V took care of 0<a(W)<4. This part and VII-VIII treat a(W)=0.
Now the central analytic statement is:
Lemma (minimal metric of a nef klt adjoint).
Let p:M->Z be a compact Kahler log resolution of a normal compact
Kahler klt pair (Z,Delta), with Delta>=0 rational and K_Z+Delta
a nef rational line. Then any semipositive metric with minimal
singularities on L=p^*(K_Z+Delta) has zero Lelong numbers
everywhere.
The proof is by contradiction from a positive Lelong number, uses
a normalized Monge-Ampere family, transports its tail, cancels a
concentrating residual measure, and takes a fixed-t limit.
Setup. Write
L = K_M + sum_i g_i D_i, g_i < 1,
and negative g_i only for p-exceptional D_i. Fix a smooth weight ell
of L pulled back from downstairs, with curvature theta. Fix Kahler
omega on M, omega_Z on Z, a smooth probability volume dV. For
0<t<=1, put beta_t^0=theta+t p^*omega_Z; analytic nefness supplies
q_t<=0 with beta_t^0+ddc q_t>=0 (picking up half the base twist
only), so int (beta_t^0)^n > 0 and int_{D_i}(beta_t^0)^{n-1} = 0
when g_i<0.
Compute the minimal-singularity envelope
Phi = ( sup{ v in PSH(M,theta) : v <= 0 } )^*
and suppose, for contradiction, its Lelong number is positive at
some point x. In local coordinates centered at x, pick lambda>0 with
Phi(x) <= lambda log|x|^2 + O(1).
Regularized density. For canonical sections s_i of D_i, an
auxiliary smooth weight d_i^0, small eps>0, put
d_{i,eps} = log(|s_i|^2 + eps e^{d_i^0}),
mu_eps = e^{ell - sum_i g_i d_{i,eps}},
mu_b = e^{ell - sum_i g_i d_i} (the actual adjoint density).
Then mu_eps -> mu_b a.e., mu_eps is smooth positive, and there are
p>1, C such that || mu_eps/dV ||_p <= C and
int (mu_b/dV)^{-s/(1-s)} dV < infinity for some fixed 0<s<1. Fix
C_0 > n/(s lambda).
Family. For t->0 and k=1,2,..., choose delta=delta_{t,k}<=min(t,1/k)
and beta=beta_t^0+delta omega, V=int beta^n (Kahler class,
nonpositive representative allowed), a smooth beta-psh psi with
q_t-k-1 <= psi <= 1 and ||psi-max(Phi,q_t-k)||_1 <= min(t,1/k), and
eps=eps_{t,k}<=min(t,1/k). For 0<=c<=C_0 solve
T = beta + ddc u > 0,
rho = T^n/V = e^{(1+c)u - c psi} mu_eps = e^{u+cH} mu_eps,
where H=u-psi. The Aubin-Yau negative-lambda theorem applies and
gives unique smooth u=u(c).
Uniform capacity. With the normalized capacity cap_beta(A) defined
by sup of int_A(beta+ddc v)^n/V over PSH competitors E-1<=v<=E,
and mu_eps, the exponential capacity bound
mu_eps(A) <= C exp(-a cap(A)^{-1/n})
holds with a,C independent of the class volume V. The standard
Skoda-Demailly-Kollar compactness gives the uniform integrability
used throughout.
First comparison. sup u is bounded above independent of the extra
smallness. Taking v_*=r psi+(1-r) E with C_0/(1+C_0)<r<1 and E the
minimal envelope for beta, Bedford-Taylor comparison on the same
sublevel set B, mixed-product positivity, and iteration give
u >= r psi + (1-r) E - C.
Since E>=psi-1 and psi has a uniform L^1 bound, we get
-C <= H <= C.
A shell estimate follows: {R<=H<=R+1} has o(1) mu_b/mu_eps/rho mass
for a fixed large R.
Differentiating MA. Set h=-d_c u, Delta_T=tr_T ddc. Then
d_c rho = (H - (1+c)h) rho = -Delta_T h rho, Delta_T h=(1+c)h-H.
Delta_T H = n - tr_T(beta+ddc psi) <= n. Maximum principle around
a minimum of h gives -C<=h, and (1+c)h<=H+n/(1+c). Energy estimate:
there is C_* with h+C_*>0. Transport form
G = (h+C_*) T^{n-1}/V, ddc G = -d_c rho.
Order the smallness. Set M_i=max(0,ceil(-g_i)), N=sum_i M_i D_i,
d_{N,eps}=sum M_i d_{i,eps}. Choose delta, then psi, then eps, so
that
sup (h+C_*) int |tr_T(ddc d_{N,eps})| rho = o(1),
int (1-|s_N|^2_{d_{N,eps}}) |d_c rho| = o(1).
Each choice is valid: the signed trace integral at delta=0 vanishes
by int_{D_i}(beta_t^0)^{n-1}=0 for negative-g_i exceptionals; the
negative part of the trace integral goes to zero in the tubules of
a small eps; uniform integrability from mu_eps L^p bounds handles
the second one. Every error is O(t), uniformly in c.
Bochner-Kodaira with cutoff. On M^o=M\setminus union D_i, use
P = -ell - u + b + sum_i M_i d_i + f(-H)
as the weight on -K_M+N, with chi a smooth cutoff jumping from 0 to
1 across the shell [R,R+1], f a smooth convex auxiliary function.
The curvature obeys
ddc P >= -d T + f''(-H) ii dH ^ dH-bar.
Pushing through complete-metric Bochner-Kodaira for (n,1) forms and
Riesz representation on (d_zeta^*, sqrt(d) zeta) gives a solution to
dbar U + sqrt(d) V_1 = g = dbar(s_N chi(H)), dbar V_1=0,
with ||U||^2 + ||V_1||^2 = o(1). The section w=s_N chi(H)-U on M^o
satisfies
int_{M^o} ( |U|^2_{d_{N,eps}} + |dbar w|^2_{d_{N,eps},T} ) W_0 rho = o(1).
Sobolev extension across the divisor gives w in W^{1,2}(M,N).
A non-holomorphic logarithm. With z=1+|w|^2_{d_N} and
A_w=<dbar w,w>/z,
ddc log z = ii d A_w - ii dbar A_w-bar - (|w|^2/z) ddc d_N
+ (ii/z^2) ( dw ^ dw-bar - dw-bar ^ dw ).
Integrate against G, use h transport, err<o(1), and the residual
measure is handled by the shell. Then
int ddc log z ^ G >= -o(1), int log z d_c rho <= o(1).
Replacing log z=log(1+|w|^2) by log(1+chi(H)^2) and setting
eta(H)=log(1+chi(H)^2)/log 2 gives
d_c int eta(H) rho <= o(1).
Integrating from c=0 to c=C_0:
int_{H>R+1} rho = o(1) at c=C_0.
Cancel residual on the same set. At c=C_0 choose a smooth xi(H)
which is 0 for H<=R+1 and 1 for H>=R+2. The measure xi(H)rho+t dV
on M has positive mass a_{t,k}<=m_t->0 uniformly in k. Solve the
prescribed-volume equation with that target; the complementary part
of rho is supported in {H<=R+2}=> rho <= C_R mu_eps + m_t (MA mass).
Choose r_t->0 with r_t^n>=m_t; assume r_t<=1/2. For each capacity
test phi, comparison on the same sublevel set B gives
tau^n cap(A_{l+tau}) <= C_R mu_eps(A_l),
i.e. the residual terms cancel on the same set. (Iterating, without a
cross-set comparison, gives u >= r_t v + (1-r_t) E - C.)
Fixed-t limit. Choose a subsequence k->infty propagating L^1 and
a.e. limits u_t,v_t. Then u_t >= r_t v_t - C_t, so
nu(u_t,x) <= r_t nu(v_t,x) <= C r_t
with C independent of t (normalized bounded-curvature potentials have
uniformly bounded Lelong numbers).
On the other hand, psi_{t,k}->Phi in L^1, eps_{t,k}->0, so Fatou
on the MA equation at c=C_0 gives
int e^{(1+C_0)u_t - C_0 Phi} mu_b <= 1.
The negative-moment bound on mu_b/dV, Holder, and the assumed pole
bound on Phi give
int e^{a u_t}|x|^{-2b_0} dV < infinity,
with a=s(1+C_0), b_0=sC_0 lambda > n after adding a local smooth
potential. The submean inequality on B(x,|x|/2) now yields
u_t(x) <= ((s C_0 lambda - n)/(s(1+C_0))) log|x|^2 + O_t(1),
so u_t has a positive Lelong lower bound independent of t, in
contradiction with nu(u_t,x) <= C r_t as t->0.
Therefore the envelope Phi has zero Lelong numbers everywhere. Any
two minimal-singularity weights on L differ by a bounded amount, so
the conclusion holds for every such metric. []
========================================================================
VII. ALGEBRAIC DIMENSION ZERO: REDUCED BOUNDARY AND STRATUM
========================================================================
The floor-extension argument of VIII needs dlt machinery in dimension
four. This part builds it. In everything here, "elementary step" is
the standard dlt-adapted one: a nontrivial projective bimeromorphic
divisorial contraction, or a small flip, whose adjoint is
relatively antiample on the negative side and relatively ample on
the positive side. All working spaces are normal compact Kahler
and globally strongly Q-factorial.
1. Two lemmas on steps.
(a) Comparison lemma. On a common smooth projective resolution of
an elementary step, the natural adjoint comparison is
p^*J = q^*J' + F, F >= 0, q_*F = 0.
Log discrepancies do not decrease. They increase strictly at every
place whose center on either side lies over the nonisomorphism
locus. Consequently elementary steps preserve klt and dlt.
(b) Exceptional sheaf comparison. If h:Y->T is a projective
modification of normal spaces and P a global rank-one reflexive
sheaf on Y with (h_*P)^** rationally invertible, then for some l,
some line M on T and some integral h-exceptional Weil divisor E:
P^[l] = ( h^*M (x) O_Y(E) )^**.
This lets the global strong Q-factoriality hypothesis be transported
through steps.
2. Cycle count.
A sequence of bimeromorphic transformations extracting no divisors
between normal compact Kahler n-folds contracts divisors only
finitely often. Proof: count the dimension of the span of prime
(n-1)-cycle classes in H_{2n-2}; each divisorial contraction
strictly drops that count.
3. Low-discrepancy counts and low extraction.
(c) Only finitely many exceptional places have log discrepancy
strictly below 1+eps, for some fixed eps>0. They can all be
realized on a single projective SNC resolution.
(d) Low extraction. From any ordinary rational klt pair on a
globally strongly Q-factorial compact Kahler fourfold, any
specified finite set of exceptional places of discrepancy at most
one can be extracted by a projective crepant morphism Y->T with
effective klt boundary. Y is again normal compact Kahler and
globally strongly Q-factorial, and no other exceptional prime is
extracted. Proof uses a small polytope of formal coefficient
adjoints, finite-generation of the multigraded ring, and a
near-zero stabilization of projective models.
4. Terminal threefold termination.
Any sequence of elementary birational steps for an effective
rational terminal pair of dimension at most three is finite,
without pseudo-effectivity. Proof: reduce to small terminal steps;
test each flip whose positive side has a flipped curve contained
in a maximal-coefficient boundary component by counting
low-discrepancy places; then the flip base surfaces form a
zero-boundary Kähler surface program (cycle count suffices);
finally induct on the number of distinct positive boundary
coefficients by removing the maximal-coefficient components.
5. Uniform global reflexive index.
Fix eps>0 and n. For klt threefold pairs on globally strongly
Q-factorial compact Kahler spaces with all discrepancies >= eps,
at most n exceptional places of discrepancy <= 1, and every
exceptional discrepancy > 1 at least 1+eps, there is a bound
I = I(n,eps)>0 making every global rank-one reflexive sheaf
F^[I] invertible. Proof: for n=0 use the terminal-point
canonical-index bound (Kawamata); for n>0 extract one low place,
reduce the count by one, and use a surface degree estimate to
control the rational correction factor by a bounded-denominator
argument on the surface projection.
6. Klt threefold termination.
Every sequence of elementary steps for effective rational klt
pairs of dimension at most three is finite, without
pseudo-effectivity. Proof: after the cycle count, all steps are
small. The set of discrepancy <= 1 places stabilizes, a uniform
epsilon exists, all discrepancies have a common denominator by
the index bound, so the discrepancies stabilize. Extract them
crepantly to obtain a small terminal pair with effective boundary;
run relative flips over the flip base with the terminal threefold
termination on the lift. Each flip below produces a negative
step in its lift; infinitely many flips would give an infinite
small terminal threefold sequence, contradiction.
7. Dlt adjunction and DCC.
Let (V,A) be dlt, T the normalization of an lc center. A chain
of coefficient-one primes through its generic SNC stratum gives a
dlt adjunction pair (T,A_T) and an actual rational identity
K_T + A_T = nu^*(K_V + A).
The different coefficients are
1 - 1/r + (sum k_j d_j)/r
with r the cyclic order of the transverse quotient and k_j the
multiplicities at the coefficients d_j. If the ambient
coefficients are DCC, the adjunction coefficients are DCC in a
set depending only on the ambient set and the chain length.
8. Dlt special termination and modifications.
(i) In dimension <= 4, any dlt elementary sequence has a tail
whose exceptional loci on both sides are disjoint from every
lc center.
(ii) Any ordinary lc pair of dimension <= 4 on a normal compact
Kahler space, with rationally invertible adjoint, has a
projective crepant dlt modification with globally strongly
Q-factorial compact Kahler total space, effective boundary,
and coefficient one on every exceptional prime.
(iii) In dimension <= 3 every dlt elementary sequence is finite.
Proof: simultaneously. Use the stratum diagram of a normalized
lc center; apply the dlt comparison lemma to bound the total
discrepancy increase on strata of smaller dimension; finiteness of
discrepancy-<=1 places at the start of each stage plus DCC of the
adjunction coefficients bounds the number of stratum extractions;
then (ii) is descent from a projective SNC resolution; (iii) is
immediate from the cycle count and the zero-boundary klt
termination theorem after deleting the floor.
9. Reduced-boundary model.
Take (T_0,G_0) ordinary rational dlt fourfold with reduced
boundary, globally strongly Q-factorial compact Kahler, and K_{T_0}
pseudo-effective with a semipositive singular metric on a Cartier
power. The boundary may be empty. Assume (B) for ordinary klt
pairs. Then there is a nonextracting bimeromorphic T_0 --> T
such that:
(i) T is normal compact Kahler globally strongly Q-factorial
and klt for zero boundary; K_T is still pseudo-effective
with such a metric.
(ii) The pushforward G is reduced, (T,G) is lc, and A=K_T+G is
analytically nef. There is t_0 in [0,1) with (T,tG) klt
and K_T+tG analytically nef for every rational t in
[t_0,1).
(iii) There is a projective crepant dlt modification
h:(V,D)->(T,G) with V normal compact Kahler globally
strongly Q-factorial, D reduced, J=K_V+D analytically nef,
h^*A=J. If G!=0 then D!=0.
(iv) On any simultaneous smooth compact Kahler resolution
M->V of all these and boundaries, mu^*J has a semipositive
metric with minimal singularities and zero Lelong numbers.
Proof sketch. Phase 1 runs a dlt program on the full adjoint;
each step is a step for some small klt perturbation K+tG with
t<1 near one, so (B) supplies the contraction; dlt comparison
keeps the transformed pair dlt; the same assumption also gives
termination. Phase 2 makes A_i-trivial steps of a zero-boundary
program, keeping A_i nonnegative on curves; if some rational
K_i+t_0G_i is nef, monotonicity of the nef cone in t and
closedness of the analytic nef cone make K_i+tG_i nef for every
rational t in [t_0,1); if none is, a rational approximation
theorem forces A_i not= nef and (B) supplies a negative ray on
which A_i.R = 0, so the whole step is crepant for A_i. Arbitrary
termination of the zero-boundary program under (B) finishes.
Apply special termination and low extraction to get (iii), and
apply VI (zero-Lelong metric) in the nef klt interval to get (iv).
========================================================================
VIII. ALGEBRAIC DIMENSION ZERO: EXTENSION AND COMPLETION
========================================================================
This part does the algebraic-dimension-zero case, and finishes
Theorem B.
1. Whole-boundary nonvanishing.
Proposition. Let (V,D) be the compact Kahler dlt fourfold from
VII.9.iii, with V globally strongly Q-factorial and D!=0 reduced.
Put J=K_V+D, and suppose J is analytically nef. Then for some
positive integer m with mJ Cartier on V,
H^0(D, O_D(mJ)) != 0.
The section is on the entire reduced scheme D, not just on a
component.
Proof sketch. Use clusters of maximal system dimension on the
normalizations of the components, together with iterated
chain-adjunction. Each node's normalized adjoint is a dlt
log-trivial pair of dimension <= 3, so the internal comparison
comes from VII.7 and the DCC. Equal-degree sections across a
conductor surface are matched by the B-cocycle equality of
canonical even residues. The projective cluster cases are handled
by finite-image projective representation theory: for r=0 an
internal matching; for r=2 a two-mark calculation or a
sheet-exchange involution over the base curve; for r=1 a
flag-curve connectivity argument over a Stein system curve
produced by a small relative program. The nonprojective cluster
case uses a single ambient Kahler class on V together with
conductor glue data to make the cycle count finite. In every case
a nonzero tuple of matching sections on the chosen cluster
extends by zero to the rest and descends to D by
codimension-one matching.
2. Eventual vanishing.
Lemma. Let M be a smooth compact Kahler manifold with
a(M)=q(M)=0. Let L be a rational holomorphic line bundle with
kappa(M,L)=-inf. Then for every fixed holomorphic vector bundle E,
H^0(M, E (x) O_M(mL)) = 0
for all sufficiently large m in any fixed divisible sequence.
Proof: from a maximal-rank tuple of sections, take determinants
and use finite support of prime classes on M (a finite-primes
lemma for a=q=0 spaces) to get an infinite componentwise
nondecreasing sequence, which yields an effective divisor of a
positive multiple of L, contradicting kappa=-inf.
3. Extension across the reduced boundary.
Proposition. In the situation of VII.9, if G!=0 then
kappa(T,A)>=0. Proof: if not, on a simultaneous resolution
p:M->V with L=p^*J and K_M+F ~_Q L (SNC, coefficients <= 1),
the ideal pushforward
p_*O_M(mL - floor F) = I_D (x) O_V(mJ)
turns into a low-degree Leray injection
H^1(V, I_D (x) O_V(mJ)) -> H^1(M, O_M(mL - floor F)).
Hard Lefschetz with multiplier ideals converts the latter into
H^0(M, Omega^3_M (x) L_m) -> H^1(M, K_M + L_m),
where L_m = O_M(-K_M - floor F + mL) has curvature semipositive
with trivial multiplier ideal (zero-Lelong metric on L, plus the
SNC divisor metric of {F} with coefficient < 1). The source
vanishes for large m by the eventual-vanishing lemma applied to
the fixed bundle Omega^3_M (x) O_M(-K_M - floor F). Hence the H^1
above is zero, restriction sections of O_V(mJ)|_D lift, floor
sections from 1 extend to V and push to T. Contradiction to
kappa(T,A)=-inf.
4. Canonical nonvanishing in algebraic dimension zero.
Theorem. Assume (A), (B), (C). If W is a smooth compact Kahler
non-uniruled fourfold with a(W)=0, then kappa(W,K_W)>=0.
Proof sketch. Suppose kappa(K_W)=-inf. Then q(W)=0 by the
irregular reduction, and K_W is pseudo-effective. Split into two
cases.
Case one. No positive power of K_W has a nonzero meromorphic
section. Resolve W to a smooth T_0 and take G_0 the full reduced
prime support; G_0 may be empty. Apply VII.9. A section of a
positive multiple of A would, after pullback and division by
meromorphic divisor factors, give a signed meromorphic
pluricanonical tensor, which we assumed absent. So G=0. Then T
has no prime divisors, is klt, and K_T=A is nef. Section 5
below excludes this case.
Case two. Some positive power of K_W has a nonzero meromorphic
section. Resolve the signed divisor: K_{T_0} ~_Q P-N with
P,N>=0 SNC, disjoint supports. Set G_0 = (Supp P)_red. Apply
VII.9. Nonextraction preserves K_T ~_Q P_T-N_T with
P_T=|Supp G|. If G=0 then P_T=0, and pseudo-effectivity of K_T
forces N_T=0. If G!=0, section 3 gives kappa(T,A)>=0; because
P_T+G-N_T is the unique representative of A in klt(T,A), it
forces N_T=0 and K_T is effective. Run (B) on (T,0) to its nef
endpoint T'. (C) makes K_{T'} semiample, and a(T')=0 forces
K_{T'} torsion. Passing to a smooth resolution M' of T', K_{M'}
is a sum of exceptional pieces, but pseudo-effectivity plus
linear independence of prime classes on M' (a=q=0) forces the
discrepancy coefficients to be nonnegative, so K_{M'} is
effective after all. Contradiction.
5. Excluded case. Smooth compact Kahler M with no prime divisors,
a(M)=0, K_M analytically pseudo-effective, and no positive
power of K_M having a nonzero meromorphic section: impossible.
Sketch of the empty-divisor argument. Let L=p^*K_Y with
K_M=L+C and alpha=2 pi c_1(L). Since a=q=0 and no signed
canonical frame exists, alphaneq0 in real cohomology. A
transverse spherical-structure argument from holomorphic forms,
plus a fixed point on the compact convex set of positive closed
(1,1)-currents in alpha (Touzet's method), yields a contradiction
with a(M)=0. Concretely, the multiplier-ideal Hard Lefschetz
surjection gives H^{2,0}(M)>0 and H^{3,0}(M)>0; wedging the
exterior square sigma of a nonzero section of Omega^2_M and a
second 3-form beta_2 produces a saturated integrable conormal E
with c_1(E)=-c_1(L) modulo exceptionals; the fixed-point construction
propagates the transverse current; the developing map to PP^1 has
image a compact Kahler manifold, whose fundamental group
representation on PU(2) has finite image by a Shafarevich-type
argument on the standard PCU(2) algebraic group; compactifying
leaves a nonconstant meromorphic function on M, contradicting
a(M)=0.
6. Assembly of Theorem B.
Combine: VII.9 provides a reduced-boundary model; VIII.1 gives
whole-boundary nonvanishing; VIII.3 gives kappa(T,A)>=0 if the
boundary is nonempty; VIII.4 handles the boundary-empty case via
the empty-divisor lemma. Pushing back through the nonextracting
bimeromorphic map gives kappa(Y,J)>=0 for the original nef
endpoint Y, which is exactly Theorem B. Apply (C) to make J
semiample and Theorem A follows.
========================================================================
IX. PROJECTIVE APPENDIX: SIGNED-REPRESENTATIVE THEOREM
========================================================================
Theorem (signed representative).
Let (X,D) be a projective Q-factorial dlt pair with D=sum D_i
reduced. Suppose L=K_X+D is nef, L-cD is pseudo-effective for some
c>0, and L ~_Q G = sum a_i D_i for a signed rational divisor G. If
L|_D is semiample as a rational line bundle on the reduced scheme
D, then L is abundant: kappa(X,L)=nu(X,L).
No subadditivity is used in this theorem.
1. Numerical boundary.
Write G=G_+ - G_-, D_0=sum_{a_i=0}D_i, with G_+,G_- effective and
disjoint prime supports. Fix a positive integer m clearing
denominators of mG_+, mG_-, mD_0 and mL, and set
N_0=O_X(mG), a line scheme on the reduced scheme D with a
morphism phi:D->P=PP^s. Put n=dim X, v=nu(X,L).
If v=n, L is big. If v=0, then D=0 and L~_Q 0. In the remaining
case put r=v-1. Then every component of D has phi-image of
dimension at most r; some component of G_+ attains r; and
dim phi( Supp G_+ cap (Supp G_- cup D_0) ) < r.
Proof: the mixed Hodge index applied to Q_{ij}=D_iD_jL^rH^{n-r-2}
has at most one positive direction, and the signed Q-relation
forces Q on the positive part to vanish.
2. Root gerbe and a signed-scheme package.
Pick a positive integer a and a line O_X(a) so that the normalized
simultaneous root of mG_+, mG_-, mD_0 (of order ell divisible by
the required integers) produces a normal tame DM stack X near a
reduced Cartier divisor S of pure dimension n-1, with reduced
boundary T disjoint from S and:
(A) (X,S+T) is log canonical and
omega_X(S+T) ~ O_X(aS).
(B) (S, T|_S) is lc; setting J=(S cap T)_red, both S and J are
Du Bois on scheme charts, I_{J/S} is maximal CM, and
B := Ext_S(I_{J/S}, omega_S) ~ O_S(aS).
(C) There is a finite representable S->D x_P P (the root gerbe)
whose image covers Supp G_+, with root line O_S(S) the
pullback of the tautological line C.
(D) There is a smooth projective bundle p:Y->P and a
representable projective g:S->Y factoring through a closed
embedding T=Spec_P h_*O_S into Y, with
g_*O_S=O_T, O_S(S)=g^*C, B=g^*C^a.
3. Hodge-theoretic formal lifting.
Let A^0 be the dual of j_! Q_H[N] on S (j:S cap T -> S), let
A_0 = H^0 A^0 (perverse), let M=H^1 g_* A_0; M is supported on
the r-dimensional base P. Filtered-duality identifies
F_{<0} A_0=0, F_0 A_0 = B|_S, F_{<0} M = 0, F_0 M = R^1g_* B|_S.
Negative-ample Kodaira-Saito vanishing on the projectivized
base P (via a finite cover realizing all the roots) kills
H^j of the twisted graded de Rham complex for all j<0.
Formal lifting proposition. For every scheme etale chart U->Y,
every j>=0, every k>=1, the transition
g_*(I^j/I^{j+k+1}) -> g_*(I^j/I^{j+k})
is surjective as a sheaf of abelian groups with kernel
O_T (x) C^{-j-k}, and on affine charts it is surjective on global
sections. Proof by induction on k: the connecting class factors
through a C-linear derivation D_k of the graded length-one algebra
into R^1 g_* O_S (x) C^{-k}; globalizing D_k gives a section of
T_Y (x) F_0 M (x) C^{-(a+k)}; this is killed by the symbol map
T_Y (x) F_0 M -> Gr^F_1 M after twisting, using the negative-ample
vanishing.
4. Null family and nef-reduction descent.
Assume the formal lifting. Then there is a dominating algebraic
family of integral projective L-trivial subvarieties of X of
dimension d=n-v, avoiding D on general members.
Proof: pick regular parameters on the support of T, lift them
compatibly and lift a conormal frame; cut out closed subschemes
Z_q; algebraize via projective Grothendieck existence; the
geometrically integral generic fiber gives an L-trivial compact
variety; the algebraicity, flatness and irreducible-family count
argument (an uncountable irreducible variety cannot be covered by
countably many proper closed subsets) shows the family sweeps X.
Apply the nef-reduction theorem to L: base dimension b<=v (from the
L-trivial d-folds). After graph flattening and base modification
to smooth B, the pullback pi^*L descends to f^*G_B for a Q-divisor
G_B on B. Then nu(X,L)<=b, kappa(X,L)>=b, and b=v; hence
kappa(L)=nu(L).
5. Induction and remaining good-model step.
The signed-representative theorem is exactly what remains, after
(1)-(4), to prove: "lower-dimensional good models + smooth
nonvanishing in dimension n => all projective lc pairs in
dimension n have good log minimal models". The induction uses the
Hashizume reduction, rational-polytope perturbation of the
boundary, the trivial-perturbation lemma (every step of an LMMP
for L-eps P is L-trivial for eps<1/(4nm+2)), and either the
signed-representative theorem (5)/(2)/(3) or the klt
pseudo-effective perturbation argument, whose boundary-reduced
good model is supplied by projective 3-fold nonvanishing.
========================================================================
X. PROJECTIVE APPENDIX: GEOMETRIC EXCLUSIONS
========================================================================
Setup. X smooth projective, K_X pseudo-effective,
kappa(X,K_X)=-inf.
1. Irregularity zero. Every smooth projective birational model of
X has q=0. Proof: if q>0, take the Albanese Stein-factor
f:W->B with positive-dimensional B; wedges of invariant 1-forms on
the abelian variety give kappa(K_B)>=0; very general fiber has
kappa(K)>=0 by lower-dim; (A) with zero boundary gives
kappa(K_W)>=0, contradiction. Consequence: with q=0, a numerically
trivial rational divisor is Q-linearly trivial.
2. Algebraic web quotient. A covering family of proper subvarieties
with dominant evaluation produces a rational map whose very general
fibers are the leaves of the involutive distribution generated by
fiber tangents. Proof: saturation plus Frobenius-chart invariance
plus Chevalley plus Hilbert-scheme counting.
3. General-type exclusion. Every proper positive-dimensional
subvariety through a very general point of X is of general type on
resolution. Proof: suppose not; take a covering family of
nongeneral-type subvarieties with smooth domains; restrict K_X,
get K_V pseudo-effective; by lower-dim K_V has a good model and
kappa(V)>=0; non-general-type means its Iitaka fibers G have
kappa(G)=0 and dimension >0, and spreading them gives another
covering family. The canonical current T of X then must be
supported on the fixed canonical divisor of every G, so T
annihilates the G-web. If the web has full rank, T is supported on
a proper algebraic set. Support theorem + rationality of the
canonical class of a q=0 projective space contradict
kappa(X,K_X)=-inf. If not, the web defines a lower-dim rational
quotient; applying the numerical-dimension-zero reduction of
Gongyo-Lehmann gives lower-dim nonvanishing, another
contradiction.
4. Supporting functional. If P is a rational divisor with
kappa(P)>=1, then K_Y+cP is big for every rational c>0.
Consequence: any nonzero supporting functional of the
pseudo-effective cone, vanishing on K_Y, is strictly positive on
P.
5. Valuation ACC. For fixed positive current T and fixed M, the
set { nu_E(T) : A_V(E)<=M } satisfies ACC. Proof: Skoda gives
nu_E(T)<=A_V(E) nu_x(T) at a general point of the center; all
centers lie in the Siu locus, which is algebraic; induct on the
codimension of the centers inside the Siu locus; further prime
divisors inside the locus contribute at most a bounded order.
6. Multiplier-ideal approximation. For every real algebraic class
T on a smooth projective V there exist integral L_k with
c_1(L_k)-k{T} in a fixed ample band, L_k(x)J(kT) globally
generated, and ord_E J(kT) <= k nu_E(T) for every divisorial
valuation E.
7. No-pullback. Let Y be a projective Q-factorial terminal
birational model of X with K_Y non-big, and let B be a smooth
projective base of a connected-fiber rational fibration h:W->B
with 1<=b<n. Then a positive current in the pullback of [K_Y] on a
smooth resolution cannot, on a dense Zariski open, annihilate the
tangents of the general fibers of h.
Proof: strict pullback T_B=p_*h^*; nef-reduction descent; the
current descends to a positive current S on B; K_Y-T_B{S} is
pseudo-effective. A supporting functional eta=T_B^*alpha yields
eta.{S}=0 and finitely many base-prime coefficients; finitely many
exceptional pullbacks; otherwise a rational relation gives a
moving pencil killed by alpha, contradicting (4). Apply
Kovacs-Patakfalvi relative positivity to get kappa(W,K_W-h^*K_B)
>=n-b, hence K_B.eta<0. Multiplier-image of the current S, plus
bend-and-break, gives uniformly bounded-ample, K-negative covering
families of rational curves with arbitrarily small
polarization/canonical ratio, contradiction.
8. Two auxiliary propositions. (Used in XI.)
(a) Moving-curves exclusion. Given a smooth projective variety
Y with K_Y pseudo-effective and a nef divisor P with positive
normalized volume ratio, curves of uniformly bounded geometric
genus and uniformly bounded (K+sP)-normalized degree do not cover
the total spaces Y_s of the K+sP scaling models for arbitrarily
small s.
(b) Signed alternative. On a log resolution of Y with reduced
strict transforms of the signed support of any such P, at least
one of the two log canonical divisors (K+P)_log, (K+cP)_log is
big for some uniform c>0 of dimension type.
========================================================================
XI. PROJECTIVE APPENDIX: JETS AND FROBENIUS
========================================================================
1. The two-slot bundle. On Y x Y set
Z_0=PP(mK_1 + mK_2), O(1)=xi, M=L_1+L_2+q xi.
Fixing one slot gives the slice PP(O_Y+mK) with class L+q xi.
Sections of k xi are symmetric powers of the direct sum.
2. Model, volume, weights. Both classes have big semiample
positive parts P on Q-factorial klt models Z with
K_Z+Delta ~_Q c_t P, Delta>=0 effective, c_t bounded independent
of small t. Asymptotic Riemann-Roch gives
vol(M)=((2n+1)!/(n!)^2) int_0^q vol(L_b) vol(L_{q-b}) db,
vol(L+q xi)=(n+1) int_0^q vol(L_b) db.
Both lie between positive constants times q. On common resolutions
the positive parts decompose as P=N_{b,1}+N_{q-b,2}+E_b, with E_b
effective, and the full system has no fixed subtraction at general
base-divisor points or on general fibers.
3. Scalar jet bound. Fix r=r(t) with r mu_t -> eps and
L=r(K+tA). Let P be the positive part of 2r(K+2tA),
rho=(P^n)^{1/n}. Then at a very general point, every nonzero
section of every divisible multiple kP has order <= 4k rho, and
2r mu_t <= rho <= 4r mu_t.
4. Two-slot Seshadri. For every fixed small eps>0 there is
q>0 such that, for every small rational t>0 and matching r, the
positive part P of M satisfies
epsilon(P;x) > 2n+2
at a very general smooth point x in the torus open. Consequently
some divisible kP separates jets of order strictly greater than
(2n+2)k at x.
Proof sketch. Choose q large enough to fit d+1 levels above
2n+3. If Seshadri failed, the base-locus tracking lemma supplies
a moving proper positive-dimensional component V of the base locus
that is a component of two successive base loci, with polynomial
degree bound and subadjunction degree bound. Case-split by fiber
dimension of V over each slot:
- restricted to a slice: bounded-degree covering curves arise via
the general-type exclusion and the curve construction.
- small adjoint on the base: signed-alternative gives log-general
type on the ruled saturation, followed by the curve
construction.
- multisection over a slice: a symmetric signed pair D_1,D_2 with
D_1-D_2 ~_Q emK, and mixed-adjunction bounds reduce to the
curve construction on the base.
- reduction to a correspondence: if a correspondence between two
copies of Y sweeps with bounded degree, purity plus psef of
K_Y plus Hanamura (birational automorphism group of a
non-uniruled variety is algebraic, and the identity component is
an abelian variety) reduces to a scheme of rational selfmaps of
a fixed cover, giving a birational-to-abelian-variety quotient.
Ramification against psef K contradicts kappa=-inf.
The surviving branch is Seshadri >2n+2 at a very general point.
5. Frobenius side. Spread X, m, A, and the relevant bundles over
a large finitely generated Z-algebra, then reduce mod a large prime
p. The Frobenius pullback produces a diagonal filtration on every
slope-semistable bundle on the two-slot projective bundle. The
uniform curve-slope bound (Langer-Sun, quoted from the appendix)
makes the slopes of that filtration too small to support the two-
slot Seshadri separation of (4) in a degree-m determinant bundle.
The two contradictory intersection numbers are the obstruction to
a hypothetical counterexample, eliminating the very general point
range.
========================================================================
XII. FINITE-FIELD DESCENT
========================================================================
The complex-case conclusion of the projective log abundance theorem
implies it over every algebraically closed field of characteristic
zero. Reduce to a finitely generated subfield k_1, then to its
algebraic closure k_0 inside the target field; k_0 embeds in C.
Discrepancies, nefness, and a chosen generated Cartier multiple are
preserved by base change and specialization on rational curves.
Faithfully flat base change of sections of the semiample multiple
from C to k_0 to k gives the conclusion.
========================================================================
PROVENANCE AND SCOPE
========================================================================
This file is a condensed companion to the full manuscript, not a
replacement for it. It was produced in the halp session of
Oct 29, 2026 from the original file as it stood then. The
statements of theorems, propositions, lemmas and corollaries here
faithfully paraphrase the original ones, but the one-paragraph
"proof sketches" compress and abbreviate the actual arguments and
cannot in general stand in for them. If you intend to cite, teach,
or check any specific result, read the corresponding passage of
the original file first.
Several of the standing inputs (Campana's orbifold
subadditivity, the finite klt fourfold MMP, effective fourfold
abundance, and the projective conditional log abundance theorem
of the appendices) are themselves partial or conditional results
and are stated here as explicit hypotheses. The conclusions are
conditional on those inputs.
This file is not, and cannot be, a high-school-level account of
the minimal model program. Motivated high-schoolers can follow the
big picture (klt, nef, adjoint, abundance) but the interior of
the proof requires substantial graduate-level background.