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ABUNDANCE AFTER NONVANISHING FOR COMPACT KAHLER FOURFOLDS
A reader-friendly, mathematically faithful edition
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How to read this edition
------------------------
This file is a complete, self-contained guide to the paper
"Abundance after nonvanishing for compact Kähler fourfolds."
It is written in layers:
Part I. The story in plain language.
Part II. The main theorem and every important word in it.
Part III. The proof at a glance.
Part IV. The technical engine room: what each hard lemma does.
Part V. Formal statements and a dependency chart.
Part VI. Glossary, notation, and questions for first-time readers.
A high-school reader should read Parts I, II, III, and VI first.
Parts IV and V are the reference layers: they tell you exactly which
technical result is being used and why. No hypothesis has been dropped
and no logical step has been replaced by hand-waving. Long coordinate
computations are compressed, but their purpose and their output are
stated clearly.
============================================================
PART I. THE STORY IN PLAIN LANGUAGE
============================================================
1. What is the problem?
------------------------
Imagine a compact curved space X of complex dimension four. "Complex
dimension four" means that near each point it looks like C^4, so it has
real dimension eight. It is allowed to have mild singularities, just as
a cone is a slightly singular version of a smooth surface.
On X we have a "divisor" Delta. A divisor is a formal sum of walls of
complex codimension one. Its coefficients are rational numbers, and
Delta is effective, meaning all coefficients are nonnegative.
The basic object is the "adjoint"
D = K_X + Delta.
Here K_X is the canonical class, a measure of how curved X is. In the
smooth case, K_X is the line bundle of holomorphic 4-forms. The pair
(X,Delta) is assumed klt, a standard "mild singularities" condition.
We ask whether D is semiample. Roughly:
Semiample means that some positive multiple mD has enough global
holomorphic sections to draw X (or its image) inside projective
space.
We assume two things:
(1) D is analytically nef. This means D never bends away from the
space: for every epsilon > 0, some Cartier multiple has a smooth
Hermitian metric whose curvature is bounded below by -epsilon
times a Kähler form.
(2) D has at least one nonzero holomorphic section, written
kappa(X,D) >= 0.
The theorem says: under these assumptions, D really is semiample. In
the special case kappa(X,D)=0, it is even rationally trivial: some
positive multiple mD is isomorphic to the trivial line bundle.
The title "abundance" refers to the philosophy that a nef adjoint should
have as many sections as its numerical behavior predicts. The theorem
proves the last open fourfold case when a section already exists.
2. Why is this hard?
--------------------
If X were projective and K_X + Delta were ample, Kodaira vanishing and
Kodaira embedding would finish quickly. But here:
* X need not be projective.
* X need not be Q-factorial.
* D need not be big.
* D may be nef only in the analytic sense.
* The boundary Delta can have complicated intersections.
So the proof cannot choose a global ample divisor and reduce to ordinary
projective geometry. It must work with the actual holomorphic line
bundle D and with analytic positivity.
The paper splits into two regimes:
kappa >= 1: a known Kähler abundance theorem applies after a dlt
modification. Sections descend back to X.
kappa = 0: this is the hard case. A nonzero section gives an
effective divisor. We build a "supported model" where
the effective divisor sits exactly on the reduced floor
of the boundary. Then two boundary theorems force a
contradiction unless the original section is nowhere
zero. If the section is nowhere zero, D is rationally
trivial, and we are done.
The rest of the paper is the construction of those two boundary
theorems:
A. Floor generation:
The restriction of the adjoint to the entire reduced floor is
semiample.
B. Supported lifting:
If the effective representative is supported exactly on the floor
and the floor restriction is semiample, then the ambient adjoint
has positive Iitaka dimension.
Together they say: in Iitaka dimension zero, a nonzero supported
representative cannot exist unless it is empty. Hence the original
section is nowhere zero.
3. The two boundary theorems in one paragraph each
--------------------------------------------------
Floor generation:
The floor S is a union of lower-dimensional pieces, and different
pieces meet along smaller strata. On each piece and each intersection,
the adjoint restricts to an actual adjoint line. We need sections on
all pieces that agree through every intersection. Adjunction gives the
line identities. Residue maps compare sections. A "link" is the one
birational comparison needed when a floor curve sits horizontally over
the semiample target. Finite residue actions and norm products let us
choose invariant generating sections. They descend to the whole floor.
Supported lifting:
A generated multiple of A|_S gives a map f : S -> T = P^b. Near a
compact fiber, take an r-th root of the defining section. The supported
divisor becomes a reduced Cartier divisor E in a finite cover. We must
lift holomorphic sections from E to every finite neighborhood. The
obstruction to lifting one order is a graded derivation. A second root
and a residue map put that obstruction into a Hodge cohomology group.
A filtered Hodge-module vanishing theorem kills the symbol of the
obstruction on a projective parameter cover. Therefore the obstruction
vanishes. Induction lifts every finite order. Then cyclic invariants
produce many sections of high multiples of A. Hence kappa(V,A) >= 1.
4. The shape of the proof
-------------------------
The proof has five big moves:
Move 1. Reduce to a supported dlt model (V,B).
Move 2. Prove A|_S is semiample on the whole floor S.
Move 3. Prove supported lifting: A|_S semiample forces kappa(V,A)>=1.
Move 4. Apply Moves 2 and 3 to get a contradiction in kappa=0.
Move 5. Conclude M=0, so the original section trivializes mD.
The technical engine room supplies:
* Threefold contraction theorems needed to run the MMP with Kähler
targets.
* Conormal-layer vanishings and analytic thickenings that extend a
floor contraction to an ambient contraction.
* The filtered Hodge-module theorem that kills the lifting
obstruction.
* A special resolution of the final model so all floor strata are
globally simple normal crossings and adjunction is clean.
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PART II. THE MAIN THEOREM AND EVERY IMPORTANT WORD
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5. The main theorem
-------------------
Theorem (Abundance after nonvanishing).
Let X be a normal connected compact Kähler complex space of dimension
four. Let Delta be an effective rational Weil divisor such that
(X,Delta) is Kawamata log terminal and the actual adjoint
D = K_X + Delta is Q-Cartier. Assume:
(i) D is analytically nef;
(ii) kappa(X,D) >= 0, i.e. some positive Cartier multiple of D has
a nonzero holomorphic section.
Then there is an integer m > 0 for which mD is Cartier and the
evaluation map
H^0(X, O_X(mD)) tensor_C O_X --> O_X(mD)
is surjective at every point.
If kappa(X,D) = 0, one may choose m so that
O_X(mD) is isomorphic to O_X.
6. What "actual Q-Cartier adjoint" means
----------------------------------------
A naive definition of K_X + Delta requires K_X and Delta separately to
be Q-Cartier. In many analytic problems they are not. The paper uses
a sheaf-theoretic convention.
Let omega_X be the canonical sheaf, and let omega_X^[r] be its
reflexive r-th power. For a rational divisor Delta, choose an integer
r > 0 clearing denominators and define
L_r = ( omega_X^[r] tensor O_X(r Delta) )^{**}.
We say D = K_X + Delta is an actual Q-Cartier adjoint if L_r is an
invertible holomorphic line bundle for some r. Then positive multiples
of D are tensor powers of L_r, after dividing the exponent by r. This
avoids choosing a global canonical divisor and avoids pretending that
K_X or Delta is separately Q-Cartier.
All statements about sections, nefness, and semiampleness refer to this
actual holomorphic line bundle.
7. What "analytically nef" means
--------------------------------
Fix a Kähler form omega on X. A rational line D is analytically nef
if for every epsilon > 0, a fixed Cartier multiple of D has a smooth
Hermitian metric h such that its normalized curvature satisfies
Theta_h >= - epsilon omega
in the sense of local smooth potentials on X. This is the natural
analytic replacement for "intersects every curve nonnegatively" when X
is not projective and curves may not sweep out enough directions.
8. What "klt" and "dlt" mean
----------------------------
Klt means Kawamata log terminal. On a log resolution, the crepant
boundary has all coefficients strictly below 1. Intuitively, the
singularities are milder than ordinary log canonical singularities.
Dlt means divisorial log terminal. It is a standard weakening that
allows coefficient-one components in a controlled simple-normal-crossing
way. The reduced floor S = floor(B) is the union of coefficient-one
components. Dlt pairs are klt away from that floor.
9. What "semiample" means
-------------------------
A rational line D is semiample if some positive Cartier multiple mD is
generated by global sections. That means for every point x there is a
section s of mD with s(x) != 0. Equivalently, the evaluation map
H^0(X, O_X(mD)) tensor O_X --> O_X(mD)
is surjective. Then the sections define a holomorphic map to projective
space.
10. What "Iitaka dimension" means
---------------------------------
Kappa(X,D) measures how many independent sections positive multiples of
D have and how large their images are. If every positive multiple has
no nonzero section, kappa = -infinity. Otherwise kappa is the maximum
dimension of the image of the rational map defined by a complete linear
system of a positive multiple.
In this paper we only need two facts:
* kappa >= 0 means some positive multiple has a nonzero section.
* kappa = 0 means no positive multiple has two independent sections.
If X is irreducible, two independent sections of one line would
have a nonconstant meromorphic quotient, giving kappa >= 1.
============================================================
PART III. THE PROOF AT A GLANCE
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11. Step 1: The positive-Iitaka case
------------------------------------
Assume kappa(X,D) >= 1. Das-Hacon-Paun's dlt modification theorem gives
a projective bimeromorphic map
g : (X', Delta') --> (X, Delta)
with X' compact Kähler and ordinary Q-factorial, the pair dlt, and the
adjoint crepant. Crepancy is an actual line identity:
O_X'(m(K_X' + Delta')) is isomorphic to g^* O_X(mD)
for a common index m. The pulled-back line is analytically nef, and its
sections are exactly the sections of mD on X.
Now apply the Horing-Lazic-Lehn theorem: a nef adjoint of positive
Iitaka dimension on an ordinary Q-factorial compact Kähler klt pair is
semiample, unconditionally through dimension four. Thus the pulled-back
line on X' is semiample. A generated multiple descends to X because
g_* O_X' = O_X. So D is semiample on X.
Hence the only remaining case is kappa(X,D) = 0.
12. Step 2: The section and its divisor
---------------------------------------
Choose a nonzero section
s in H^0(X, O_X(mD)).
Normalize its divisor:
M = (1/m) div(s).
If M = 0, then s is nowhere zero. A nowhere-zero section of a line
bundle is an isomorphism O_X -> O_X(mD). So mD is trivial, and we are
done.
Assume for contradiction that M != 0. We will build a supported model
and derive kappa >= 1, contradicting kappa = 0.
13. Step 3: The supported model
-------------------------------
Resolve X so that the support of Delta + M is simple normal crossing.
On the resolution, raise to coefficient one every exceptional prime and
every strict transform of a prime appearing in M. This produces an
effective rational SNC boundary B_Y and a divisor P_Y with
K_Y + B_Y ~_Q P_Y,
Supp P_Y = Supp floor(B_Y),
kappa(Y, K_Y + B_Y) = 0.
Then run the supported minimal model program for fourfolds with Kähler
targets. The paper's technical work supplies the needed contractions
on threefold floors and extends them to the ambient fourfold. At the
end one obtains a normal ordinary Q-factorial compact Kähler dlt
fourfold (V,B) and a nonzero effective rational Q-Cartier divisor P
such that
A = K_V + B is analytically nef,
A ~_Q P,
Supp P = Supp floor(B),
kappa(V,A) = 0.
The model also has a special projective resolution pi : Vhat -> V with:
* globally smooth distinct SNC strict boundary and exceptional
components;
* every exceptional crepant coefficient strictly below 1;
* pi generically an isomorphism on the image of every irreducible
component of an intersection of distinct strict floor components.
Why does P survive? Suppose P = 0. Then the pullback of M to a common
resolution is an effective nonzero exceptional divisor. Its line is
the pullback of the original nef D, so it is relatively nef. But an
effective relatively nef exceptional divisor on a normal compact Kähler
space must be zero. Contradiction. Thus P != 0.
14. Step 4: Floor generation
----------------------------
Let S = floor(B). The first boundary theorem is:
A|_S is semiample.
The proof is an induction over the strata of S. A stratum is an
irreducible component of an intersection of distinct floor components.
On each stratum Z, adjunction gives a normal compact Kähler dlt pair
(Z,B_Z) and an actual line identity
O_Z(m(K_Z + B_Z)) is isomorphic to O_V(mA)|_Z.
The induction builds compatible sections on all strata up to dimension
d. There are two cases when extending from the floor C_Z of a stratum Z
to Z itself.
Case 1: C_Z is vertical over the semiample target Y_Z of Z.
Then a torsion-free obstruction shows that every section of a large
multiple on C_Z extends automatically to Z. No comparison is needed.
Case 2: C_Z is horizontal.
Then a Mori model reduces the problem to matching two marked points on
a general P^1 fiber. The two markings come from the two branches of the
horizontal floor. In even degree q, the residues of the logarithmic
form are
a(w) xi^{tensor q} and (-1)^q a(w) xi^{tensor q},
which are equal because q is even. This birational residue comparison
is called a link. A section extends if its two pullbacks agree under
the link.
Links generate a groupoid on the surface strata. To impose invariance,
we need finite isotropy actions on each fixed section space. On
projective surfaces this is Fujino-Gongyo's log pluricanonical
representation theorem. On nonprojective surfaces the paper uses the
single ambient Kähler class: a returning link preserves the class of the
ambient Kähler form modulo the Neron-Severi group. The Hodge index
theorem forces the induced scalar on the holomorphic 2-form to be a
root of unity, and only finitely many orders are possible. Thus the
scalar image is finite.
Now use norm products. If G_Z is the finite isotropy image on
H^0(Z,L_Z^k), form
P_Z(s_Z) = product_{g in G_Z} g(s_Z).
Choose a common h divisible by all |G_Z| and use P_Z(s_Z)^{h/|G_Z|}.
This raises the lower restrictions to a common power and makes the
section invariant. Finite hyperplane avoidance then chooses enough
such invariant sections to generate at every point. They agree on all
intersections by construction. Proposition "strata adjunction" descends
them uniquely to the reduced floor S. Therefore A|_S is semiample.
15. Step 5: Supported lifting
-----------------------------
The second boundary theorem is:
If (V,B) is a normal irreducible compact Kähler dlt pair, A=K_V+B
is the actual adjoint, and a nonzero effective rational Q-Cartier
divisor P satisfies P ~_Q A and Supp P = Supp floor(B), then
A|_S semiample implies kappa(V,A) >= 1.
Here is the idea.
Choose a sufficiently divisible even q and a line
L = O_V(qA)
with a section s whose divisor is G = qP. The semiampleness of A|_S
gives a morphism
f : S -> T = P^b
and an actual identity L|_S = f^* O_T(1).
Near a compact fiber of f, take an r-th root of the defining section.
This produces a finite cover pi : Z -> W and a reduced Cartier divisor
E = (y=0) with
O_Z(E) = pi^* P_1.
Let I = O_Z(-E). The layers are A_j = I^j/I^{j+1}. We need to lift
sections from E to all finite neighborhoods:
For every j and k >= 1,
g_* (I^j / I^{j+k+1}) --> g_* (I^j / I^{j+k})
is surjective.
The obstruction to lifting one order is the connecting map
delta_k : g_* A_j --> R^1 g_* A_{j+k}.
It is a graded derivation. If all delta_k vanish, every finite order
lifts.
To kill delta_k, take a second full normalized cover, this time using a
q-th root of s as a meromorphic q-pluricanonical form. Resolve the
resulting cover and principalize the pulled ideal of E equivariantly.
Write H for the reduced SNC divisor above E. The residue insertion
gives a split injection
tau_* : R^j g_* A_{-a} --> R^j h_* omega_H.
Thus the obstruction sits inside a Hodge cohomology group R^1 h_* omega_H.
Filtered Hodge theory on the SNC support H gives a right D-module
M = H^1 p_+ K with
F_0 M = R^1 h_* omega_H.
There is a first-symbol map
sigma : F_0 M tensor T_T --> gr^F_1 M.
The key Hodge vanishing says: for every ample line N on a smooth
projective T,
Hom_{O_T}(N, ker sigma) = 0.
Now fix a parameter point t_* in T. Build a projective parameter cover
T' -> T which is unbranched over t_* and a single compact SNC graph over
the whole cover. On this graph, the local adjugate identity turns the
obstruction into a global map
R^m --> F_0 M' tensor T_{T'}.
Its image lies in ker sigma. But R^m is ample. The vanishing theorem
forces this map to be zero. At the chosen point t_* the cover is
unbranched, so the obstruction itself vanishes there. Since t_* was
arbitrary, delta_k = 0 everywhere. Induction on k proves all finite
lifting orders.
Finally, use cyclic invariants. Define
J_j = O_V(-ceil(jG/r)), j in Z.
These are coherent divisorial sheaves. Their quotients B_j = J_j/J_{j+1}
are coherent O_S-modules. Translation by G gives the periodicity
F_{j-r} = F_j tensor O_T(1),
where F_j = f_* B_j. For N > 0, the finite filtration of
F_{-Nr,0} has quotients F_j with -Nr <= j < 0. Each such j is
uniquely j = l - tr with 0 <= l < r and 1 <= t <= N, so F_j = F_l(t).
Serre vanishing bounds the higher cohomology of all F_l(t) independently
of N. At least one section of F_0(t) exists for every t >= 1, so the
zeroth cohomology grows at least linearly. Therefore
h^0(T, F_{-Nr,0}) >= N - C
for a constant C independent of N. Lifting to V loses at most the fixed
finite dimension h^1(V,O_V). Hence h^0(V, O_V(NG)) is unbounded. For
some N, O_V(NG) has two independent sections. Their quotient is a
nonconstant meromorphic function on V, so kappa(V,A) >= 1.
16. Step 6: Completion
----------------------
Return to the original X with kappa(X,D)=0 and M != 0. The supported
model gives (V,B,P) as above. Floor generation gives A|_S semiample.
Supported lifting then gives kappa(V,A) >= 1. But the model was built
with kappa(V,A)=0. Contradiction.
Therefore M = 0. The original section s is nowhere zero, so
O_X(mD) is isomorphic to O_X.
This proves the theorem in Iitaka dimension zero. Together with the
positive-Iitaka case, the theorem is proved.
============================================================
PART IV. THE TECHNICAL ENGINE ROOM
============================================================
17. What the contraction machinery is for
-----------------------------------------
The fourfold program needs to contract negative extremal rays on
threefold floors and then extend those contractions to the ambient
fourfold. The paper does not simply cite a black box for this. It
builds the needed threefold contraction theorem from two analytic
constructions.
17.1 Finite null locus contraction.
Let X be a normal connected ordinary Q-factorial compact Kähler
threefold with klt singularities. Let alpha be a nef and big real
(1,1)-class. Suppose its null locus
Null(alpha) = union of positive-dimensional V with alpha^{dim V}.V = 0
is a finite union of curves. Then there is a proper bimeromorphic
morphism phi : X -> X' to a normal compact analytic space which is an
isomorphism away from C = Null(alpha), and whose nontrivial fibers are
exactly the connected components of C.
The proof resolves X, writes the pullback class as a Kähler class plus
an effective exceptional divisor, and uses Grauert's contraction
criterion for the conormal of C. The key point is that a suitable
actual positive line on the resolution makes the conormal weakly
negative.
17.2 Conormal layers of a prime floor.
Let (X,S+B) be an ordinary plt pair, S an irreducible Q-Cartier prime,
dim S <= 3, and phi : S -> W a projective surjection with connected
fibers. Suppose the actual lines -(K_S+B_S) and -S|_S are phi-ample.
Put I_k = O_X(-kS) and E_k = I_k/I_{k+1}. Then each E_k is a rank-one
reflexive sheaf on S, and for a global Cartier index q there is an
effective rational divisor Xi_k on S with
E_k^[q] = (O_X(-kqS)|_S) tensor O_S(-q Xi_k),
0 <= Xi_k <= Theta <= B_S,
and
R^i phi_* E_k = 0 for i > 0, k >= 1.
The sheaf Xi_k measures the defect between the k-th conormal layer and
the restriction of the actual line. The vanishing is what makes the
analytic thickenings exist.
17.3 Analytic thickenings and prime-floor extension.
Using the conormal vanishings, the paper builds analytic thickenings
S_k = V(I_k)
and proves that the pushforwards B_k = |phi|_* O_{S_k} are genuine
coherent analytic sheaves on W. This lets one extend a contraction of
a prime floor S to the whole ambient space X. The result is a proper
bimeromorphic morphism F : X -> Z with exactly the same fibers on S
and an isomorphism away from S. This is Proposition "prime-floor
extension."
17.4 The threefold contraction theorem.
Combining finite-null contraction, prime-floor extension, the relative
projective MMP, and the specified positivity and rationality inputs
gives:
Let (Z,Gamma) be a normal connected compact Kähler klt threefold
with effective rational boundary and actual Q-Cartier adjoint
A = K_Z + Gamma. Suppose alpha = c_1(A) + [omega] is nef, where
omega is Kähler. Then there is a projective surjection f : Z -> Y
with connected fibers, Y normal compact Kähler with rational
singularities, and a Kähler class [omega_Y] such that
alpha = f^*[omega_Y]. The line -A is f-ample. A compact curve is
contracted exactly when its alpha-degree is zero.
This theorem is used on floors of the fourfold program and on lower
strata in the boundary comparison.
18. What the Hodge-module theorem is for
----------------------------------------
The lifting obstruction lives in R^1 h_* omega_H on a simple
normal-crossing support H. The paper proves two facts.
18.1 Filtered SNC direct image.
Let p : A -> T be a holomorphic map of complex manifolds, and let
i : H -> A be a reduced SNC divisor in a smooth complete intersection,
with h = p|_H proper. Put
K = H^c_[H](omega_A),
F_0 K = image of Ext^c_A(O_H, omega_A) in K.
Then (K,F) is a good filtered regular holonomic right D-module. For
M^j = H^j p_+ K with the induced filtration, every level cohomology map
H^j F_p p_+(K,F) --> M^j
is injective with image F_p M^j. The module has a finite filtration by
polarizable real pure Hodge modules. In particular,
F_0 M^j = R^j h_* omega_H,
F_p M^j = 0 for p < 0.
This is the technical source of the residue injection and the lowest
piece R^1 h_* omega_H.
18.2 No ample line in the symbol kernel.
For the first-symbol map
sigma : F_0 M^1 tensor T_T --> gr^F_1 M^1,
if T is smooth projective and N is ample, then
Hom_{O_T}(N, ker sigma) = 0.
This is the vanishing that kills the global symbol map on the
projective parameter cover. It is proved by comparing the real filtered
module with the Sabbah-Schnell pure Hodge module attached to the same
generic polarized variation, then applying negative-ample graded de
Rham vanishing.
19. What the special resolution is for
--------------------------------------
The final supported model (V,B) must have a projective log resolution
pi : Vhat -> V such that:
* the strict boundary and exceptional support have globally smooth
distinct SNC components;
* every exceptional crepant coefficient is strictly below 1;
* pi is generically an isomorphism on the image of every irreducible
component of an intersection of distinct strict floor components.
This is Lemma "special resolution." It is built from the normal common
graph of the finitely many MMP steps. The key discrepancy calculation
is that every negative step strictly increases discrepancies over all
nontrivial fibers. Hence a final log canonical center cannot be
swallowed by the bad locus. The resolution is an isomorphism over the
open U_V where all steps are isomorphisms. The complement of U_V has
codimension at least two, so the final exceptional support and floor
intersections are controlled.
20. Why the paper needs all this
--------------------------------
The fourfold theorem is not a formal consequence of threefold abundance.
The obstruction is genuinely four-dimensional: a boundary divisor can
wrap horizontally over the semiample target, and sections on its
components need not agree on intersections. The paper solves this by:
* proving the floor restriction is semiample;
* proving that semiampleness of the floor forces positive Iitaka
dimension when an effective supported representative exists;
* doing both in the analytic compact Kähler category.
The technical engine room is the price of working without projectivity
or Q-factoriality.
============================================================
PART V. FORMAL STATEMENTS AND DEPENDENCY CHART
============================================================
21. Main theorem
----------------
Theorem. Let X be a normal connected compact Kähler complex space of
dimension four. Let Delta be an effective rational Weil divisor such
that (X,Delta) is klt and D = K_X + Delta is an actual Q-Cartier
adjoint. If D is analytically nef and kappa(X,D) >= 0, then some
positive Cartier multiple mD is generated by global sections. If
kappa(X,D)=0, then some positive multiple mD is isomorphic to O_X.
22. Supported model
-------------------
Proposition. Under the hypotheses of the main theorem, suppose
kappa(X,D)=0 and the normalized divisor M of a nonzero plurisection of
D is nonzero. Then there are a normal ordinary Q-factorial compact
Kähler dlt fourfold (V,B) with effective rational boundary and a
nonzero effective rational Q-Cartier divisor P such that
A = K_V + B is analytically nef,
A ~_Q P,
Supp P = Supp floor(B),
kappa(V,A) = 0.
The pair has the special projective resolution described above.
23. Floor generation
--------------------
Theorem. Let (V,B) be a normal ordinary Q-factorial compact Kähler
dlt fourfold with effective rational boundary and analytically nef
actual adjoint A = K_V + B. Suppose V has a projective log resolution
with globally smooth distinct SNC components, exceptional crepant
coefficients below 1, and generic isomorphism on every floor
intersection. Then A|_S is semiample on the entire reduced floor
S = floor(B).
24. Supported lifting
---------------------
Theorem. Let (V,B) be a normal irreducible compact Kähler dlt pair of
positive dimension, with effective rational boundary and actual
Q-Cartier adjoint A = K_V + B. Suppose a nonzero effective rational
Q-Cartier divisor P satisfies
P ~_Q A, Supp P = Supp floor(B).
If A|_S is semiample on S = floor(B), then kappa(V,A) >= 1.
25. Threefold contraction
-------------------------
Proposition. Let (Z,Gamma) be a normal connected compact Kähler klt
threefold with effective rational boundary and actual Q-Cartier adjoint
A = K_Z + Gamma. Suppose
alpha = c_1(A) + [omega]
is nef for some Kähler class [omega]. Then there is a projective
surjection f : Z -> Y with connected fibers, Y normal compact Kähler
with rational singularities, and a Kähler class [omega_Y] such that
alpha = f^*[omega_Y]. The line -A is f-ample. A compact curve is
contracted exactly when its alpha-degree is zero.
26. Finite null locus contraction
---------------------------------
Proposition. Let X be a normal connected ordinary Q-factorial compact
Kähler threefold with klt singularities. Let alpha be a nef and big
real (1,1)-class whose null locus is a finite union of curves. Then
there is a proper bimeromorphic morphism phi : X -> X' to a normal
compact analytic space such that phi is an isomorphism away from
Null(alpha), and the nontrivial fibers are exactly the connected
components of Null(alpha).
27. Conormal layers
-------------------
Proposition. Let (X,S+B) be an ordinary plt pair with X a normal
connected compact Kähler space, S an irreducible effective Q-Cartier
prime, B >= 0 not containing S, and K_X + S + B an actual Q-Cartier
adjoint. Suppose dim S <= 3 and phi : S -> W is a projective surjection
with connected fibers. Suppose -(K_S+B_S) and -S|_S are phi-ample.
Put I_k = O_X(-kS) and E_k = I_k/I_{k+1}. Then each E_k is a rank-one
reflexive sheaf on S. For any positive global Cartier index q, there
is an effective rational divisor Xi_k on S such that
E_k^[q] = (O_X(-kqS)|_S) tensor O_S(-q Xi_k),
0 <= Xi_k <= Theta <= B_S,
and R^i phi_* E_k = 0 for i > 0, k >= 1.
28. Filtered SNC direct image
-----------------------------
Proposition. For a reduced SNC support H in a smooth ambient A over T
as above, the filtered module K = H^c_[H](omega_A) is good, regular, and
holonomic. The cohomology modules M^j = H^j p_+ K carry a finite
filtration by polarizable real pure Hodge modules. Every level
cohomology map is injective, and
F_0 M^j = R^j h_* omega_H,
F_p M^j = 0 for p < 0.
29. No ample line in the symbol kernel
-------------------------------------
Corollary. In the filtered SNC situation, if T is smooth projective
and N is ample, then
Hom_{O_T}(N, ker sigma) = 0,
where sigma is the first-symbol map of M^1.
30. Dependency chart
--------------------
Here is the logical order of the proof.
Main theorem
|
+-- kappa >= 1: Das-Hacon-Paun dlt modification
| + Horing-Lazic-Lehn abundance
|
+-- kappa = 0:
|
+-- Supported model (V,B,P)
| + dlt resolution and SNC preparation
| + supported MMP with Kähler targets
| + threefold contraction theorem
| + finite null locus contraction
| + conormal layers
| + prime-floor extension
| + discrepancy increase
| + special resolution
|
+-- Floor generation: A|_S semiample
| + strata adjunction
| + restriction criterion / links
| + finite isotropy images
| + norm products and gluing
|
+-- Supported lifting: A|_S semiample => kappa >= 1
+ root covers and finite thickenings
+ obstruction as graded derivation
+ split residue insertion
+ filtered Hodge modules
+ symbol-kernel vanishing
+ cyclic invariants and section growth
|
+-- Contradiction: kappa(V,A)=0 and kappa(V,A)>=1.
|
+-- Therefore M=0 and mD is trivial.
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PART VI. GLOSSARY, NOTATION, AND READER QUESTIONS
============================================================
31. Glossary
------------
Adjunction:
The rule that restricts a canonical or adjoint line to a divisor and
accounts for the normal bundle by adding a boundary term.
Analytically nef:
A line bundle whose curvature is almost nonnegative in the analytic
sense: bounded below by -epsilon times a Kähler form for every
epsilon > 0.
Canonical class K_X:
The line bundle of holomorphic top-degree forms. Its first Chern
class measures curvature.
Dlt:
Divisorial log terminal. A mild singularity condition allowing
coefficient-one boundary components with simple normal crossings.
Effective:
A divisor with nonnegative coefficients.
Floor:
The reduced union of coefficient-one components of a boundary.
Iitaka dimension kappa:
A measure of how many independent sections positive multiples of a
line have and how large their images are.
Kähler:
A complex manifold with a positive closed (1,1)-form. It has a
Riemannian metric compatible with the complex structure.
Klt:
Kawamata log terminal. A stronger mildness condition: exceptional
crepant coefficients are strictly below 1.
Line bundle:
A family of complex lines attached to points of a space. Sections
are holomorphic ways to choose a point in each line.
Nef:
Intuitively, "not bending outward." Curve-theoretically, nonnegative
degree on every curve. Analytically, almost nonnegative curvature.
Q-Cartier:
Some positive integer multiple is Cartier, i.e. locally defined by
one holomorphic equation.
Semiample:
Some positive multiple is generated by global sections.
SNC:
Simple normal crossing. Local equations look like products of
coordinate functions.
32. Notation
------------
X, V, Z:
Complex spaces. Usually compact Kähler, normal, connected.
Delta, B, Gamma:
Effective rational Weil divisors, usually boundaries of pairs.
K_X:
Canonical class.
A = K_X + Delta or K_V + B:
The actual adjoint line.
S = floor(B):
The reduced floor, the union of coefficient-one boundary components.
P:
A nonzero effective rational Q-Cartier divisor representing the
adjoint on a supported model.
L = O_V(qA):
A sufficiently divisible actual Cartier multiple.
G = qP:
The divisor of the chosen section of L.
E:
The reduced Cartier divisor produced by the root cover.
I = O_Z(-E):
The ideal sheaf of E.
A_j = I^j/I^{j+1}:
The j-th conormal layer on E.
T_{j,k} = I^j/I^{j+k}:
Finite truncation of the layer filtration.
delta_k:
The obstruction derivation to lifting one finite order.
R^1 h_* omega_H:
The Hodge cohomology group where the obstruction is inserted.
M = H^1 p_+ K:
The filtered Hodge module whose lowest piece is R^1 h_* omega_H.
sigma:
The first-symbol map of the filtered module.
33. Questions a first-time reader often asks
--------------------------------------------
Q: Why can we assume X is connected?
A: The theorem is local over connected components. A common positive
multiple works over finitely many components.
Q: Why does a nowhere-zero section trivialize the line?
A: A section of O_X(mD) is a holomorphic function locally. Nowhere zero
means it is a unit locally. Multiplication by that unit is an
isomorphism O_X -> O_X(mD).
Q: Why does kappa=0 imply at most one section?
A: If one line has two independent sections, their quotient is a
nonconstant meromorphic function. That gives a positive-dimensional
image for some complete linear system, so kappa >= 1.
Q: Why is the floor so important?
A: In Iitaka dimension zero, a section cannot vanish on a divisor and
still have room for another independent section. The supported model
puts the entire vanishing locus exactly on the floor. Then the floor
restriction and the lifting theorem show that the section must have
been nowhere zero.
Q: Why are residue maps needed?
A: The floor is a union of divisors meeting along smaller strata.
Sections on different components must agree on intersections. Residue
is the natural operation that restricts a logarithmic form to a
divisor. It remembers not just the line bundle but the actual
meromorphic frame.
Q: Why even degree q?
A: In even degree, swapping two marked points on a P^1 fiber has sign
(-1)^q = +1. That makes the two residue comparisons equal. Odd degree
would introduce a sign and spoil the gluing.
Q: What is the main new difficulty compared with threefolds?
A: A fourfold boundary is three-dimensional. It can carry complicated
nonprojective surfaces. A link between surface strata need not act by
a finite group. The paper uses the single ambient Kähler class to
control the induced scalars on holomorphic 2-forms. This is the
"finite residue actions from one Kähler class" section.
Q: Is the theorem projective?
A: No. X is compact Kähler, not assumed projective. The proof is
analytic throughout. Projective statements are used only after
passing to a projective resolution or a projective parameter cover.
Q: What happens if Delta is empty?
A: The theorem still applies. It says a nef klt adjoint on a compact
Kähler fourfold with a nonzero section is semiample. In Iitaka
dimension zero, a positive multiple of K_X is trivial.
34. What to remember if you only remember three things
------------------------------------------------------
1. The theorem says a nef log canonical class on a compact Kähler
fourfold, assuming one nonzero section, is semiample; in Iitaka
dimension zero it is rationally trivial.
2. The hard case is Iitaka dimension zero. A nonzero section gives a
divisor. The proof builds a model where that divisor sits exactly
on the floor of the boundary. Then two boundary theorems force a
contradiction unless the divisor is empty.
3. The two boundary theorems are:
(a) the adjoint restriction to the whole reduced floor is semiample;
(b) semiampleness of the floor plus a nonzero supported divisor
forces positive Iitaka dimension.
The technical machinery exists to prove those two statements in the
compact Kähler category without projectivity or Q-factoriality.
35. Final picture
-----------------
The proof is a long journey, but the destination is simple:
If a nef klt adjoint on a compact Kähler fourfold has a section,
then it has enough sections.
If it has only one section in every positive degree, that section
cannot vanish anywhere.
Therefore the adjoint is trivial in Iitaka dimension zero.
This is abundance after nonvanishing for compact Kähler fourfolds.
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END OF READER-FRIENDLY EDITION
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