Abundance asks when numerical positivity of a log canonical bundle
produces a holomorphic map. On a compact Kaehler space, nefness is a
closed-cone condition on a cohomology class, while generation is a
statement about an actual holomorphic line bundle, and the gap between
the two is where the whole problem lives. Under logarithmic Iitaka
subadditivity for smooth projective log pairs with reduced boundary,
log abundance holds for compact Kaehler lc pairs, and the proof goes
through the stronger birational statement that every pseudo-effective
smooth log adjoint carries a semiample positive part and an effective
rational fixed divisor equal to its analytic divisorial negative part.
Set the vocabulary. A compact Kaehler space is a compact complex
analytic space with a Kaehler form in the sense of a closed real
(1,1)-form locally of the shape i ddbar phi, with phi strictly
plurisubharmonic, read through local embeddings. A rational line L on
such a space is analytically nef when c_1(L) lies in the closure of
the Kaehler cone. Equivalently, fix a Kaehler class omega and an
integer r>0 with rL a genuine holomorphic line bundle; nefness is the
existence, for every epsilon>0, of a smooth Hermitian metric h on rL
whose curvature Theta_h satisfies
sqrt(-1)/(2 pi r) Theta_h >= -epsilon omega. (1)
Smooth forms and metrics on a singular space are always read through
local embeddings. L is semiample when some positive integral multiple
of L is a holomorphic line bundle generated by its global sections. A
pair (X,Delta) is log canonical, lc, when X is normal irreducible,
Delta>=0 is a rational divisor with coefficients in [0,1], and every
log discrepancy of K_X+Delta is at least -1. When K_X+Delta is
Q-Cartier the adjoint K_X+Delta denotes this rational line. The Iitaka
dimension kappa(X,L), taking values in {-infinity,0,1,...,dim X},
measures the growth of H^0(X,mL) over actual Cartier multiples mL and
equals -infinity when all such groups vanish. Conventions below fix
(-infinity)+b=-infinity for every real b, while the zero divisor on
a point has kappa=0.
The argument is conditional on one statement.
Assumption (logarithmic Iitaka subadditivity). Let f: X -> Y be a
surjective morphism with connected fibres between smooth connected
projective complex varieties. Let D_X and D_Y be reduced effective
simple normal crossing divisors, the zero divisor allowed, with
Supp(f^*D_Y) contained in Supp(D_X). For a very general smooth fibre
F, put D_F = D_X |_F. Then
kappa(X,K_X+D_X) >= kappa(F,K_F+D_F) + kappa(Y,K_Y+D_Y).
The divisor D_F is reduced SNC, and K_F+D_F is linearly equivalent to
(K_X+D_X)|_F.
./fujino-2017-subadditivity-log-kodaira-dimension.pdf
./cao-paun-2017-kodaira-dim-algebraic-fiber-spaces-abelian.pdf
./hashizume-2020-log-iitaka-abundant-lc-fibrations.pdf
The assumption has an established record in special cases. Fujino
reduces logarithmic subadditivity for smooth projective X with reduced
SNC D to a relation between sigma-kappa and kappa, and the corrigendum
to that paper repairs the coefficient of the boundary. Cao and Paun
prove the analogue over an abelian base with a klt effective boundary
on the total space. Hashizume proves the reduced-SNC statement when
the log canonical divisor of the very general fibre is abundant. The
full statement in all dimensions is open and enters the argument
below only through its consequence for pseudo-effective projective
log adjoints with a nef boundary.
Theorem (log abundance for compact Kaehler spaces). Let X be a normal
irreducible compact Kaehler complex analytic space, and let Delta>=0
be a rational divisor with (X,Delta) lc and K_X+Delta rational
Cartier. If K_X+Delta is analytically nef, it is semiample: there is
an integer m>0 for which m(K_X+Delta) is Cartier and the evaluation
map
H^0(X, O_X(m(K_X+Delta))) tensor_C O_X -> O_X(m(K_X+Delta)) (2)
is surjective at every point of X.
Both the theorem and its argument are uniform in dimension and allow
Delta=0. The projective case is classical in the following sense:
Kodaira treats the canonical bundle of a smooth projective surface,
Miyaoka and Kawamata handle minimal projective threefolds, Keel,
Matsuki and McKernan prove the log version in dimension three, and
Birkar, Cascini, Hacon and McKernan supply the klt minimal model
program in all dimensions, on which log abundance in characteristic
zero depends. Fujino's Duke paper records the reduction of the lc
case to the klt case, which is also used in the Kaehler setting.
./birkar-cascini-hacon-mckernan-2010-existence-minimal-models.pdf
./fujino-matsumura-2021-injectivity.pdf
The compact Kaehler case needs separate machinery, and that is what
the proof addresses. Abundance for compact Kaehler threefolds is due
to Campana, Peternell, and Horing-Peternell in the projective-class
and uniruled regimes, and the log version in that dimension is
completed by Das-Ou; the fourfold case is settled conditionally on
log abundance in the projective case. The analytic tools in the
general dimension are Boucksom's analytic divisorial Zariski
decomposition, the Demailly-Paun characterization of the Kaehler
cone via semi-positive currents, and the Kaehler minimal model
program for generalized pairs.
./horing-peternell-2016-mrc-kahler.pdf
./das-ou-2026-log-abundance-kahler-threefolds-II.pdf
./boucksom-2004-divisorial-zariski-decompositions.pdf
./begz-2010-analytic-approach-zb.pdf
Write Ggood_d for the following
statement in dimension d.
Ggood_d. Let X be a connected smooth compact Kaehler manifold of
dimension d, let B be a rational simple normal crossing divisor
with coefficients in [0,1], and assume J = K_X+B is pseudo-
effective. Then there is a smooth compact Kaehler modification
mu: Y -> X and an actual identity of rational lines
mu^* J ~_Q P + R, P semiample, R = N(mu^* J) >= 0, (3)
where N denotes the analytic divisorial negative part in the sense
of Boucksom, and R is a rational divisor.
The claim is that Ggood_n holds for every n, by strong induction on
n. Dimension zero is immediate: the class is empty. Fix n and assume
Ggood_j for all j<n. Let (X,B) be a datum as in Ggood_n and let
a=a(X) be the algebraic dimension of X. The induction splits by a.
(a) a=n. Then X is Moishezon and Kaehler, hence projective, and
the projective log abundance theorem supplies (3) directly.
(b) 0<a<n. The algebraic reduction produces a dominant meromorphic
map to a projective variety of dimension a, with connected very
general fibres. Log Iitaka subadditivity across this fibration,
plus Ggood_{<n} on very general fibres, produces (3).
(c) a=0 and X simple, meaning no positive-dimensional proper
subvariety passes through a very general point. A nonvanishing
theorem for the n-th canonical power, boundary generation along
the dlt floor, and a signed-rigidity theorem for nef signed
adjoints combine to force the positive part P in (3) to be
torsion, so P=0 and R=N(mu^*J) alone.
(d) a=0 and X not simple. Campana's maximal covering family of
proper subvarieties produces a generically finite cover e: X' -> X
from a space X' that itself carries a nontrivial fibration. The
fibration case (b) applies to X'. The positive part on X' is
torsion because a(X')=0 leaves no nonconstant meromorphic
function on X'. The norm of a finite cover descends this
decomposition back to X.
Nonvanishing. Every smooth compact simple Kaehler manifold of
dimension n and algebraic dimension zero satisfies H^0(X, m K_X)!=0
for some m>0.
./ou-2025-uniruledness-mrc-kahler.pdf
Boundary generation. On a compact Kaehler dlt pair with reduced SNC
floor S and nef adjoint J, sections of J restricted to the strata of
S, which agree under every inclusion of strata, glue to a section that
generates the restricted line along the whole reduced S.
./kebekus-schnell-2021-extend-2-forms.pdf
Signed rigidity. Suppose that on a smooth compact simple Kaehler
manifold of algebraic dimension zero, a pseudo-effective rational
line L admits an actual rational-line identity L ~_Q P + N(L) with P
semiample, and simultaneously an identity L ~_Q sum_i a_i D_i as a
Q-linear combination of effective Cartier divisors D_i whose boundary
sum D=sum_i D_i is semiample. Then P is torsion, hence L ~_Q N(L).
Fix a normal irreducible compact Kaehler space X, a
rational line L on X, and a projective log resolution e: Y -> X
with Y smooth compact Kaehler. Write E for the reduced exceptional
divisor of e, and W for the big open subset of X on which e is an
isomorphism; then e^{-1}(W) is X minus the support of E on the
smooth side.
Lemma (descent). Suppose on Y there is an actual identity of
rational lines
e^* L ~_Q P + R, P semiample, R >= 0 rational,
Supp R contained in E. (4)
Then some positive Cartier multiple mL of L is generated at every
point of X.
Proof. Choose an integer m>0 simultaneously clearing denominators
of P and R, ensuring that mP is a genuine line bundle generated
by its global sections, and ensuring that mL and mR are Cartier
divisors. The identity (4) multiplied by m becomes an identity
of Cartier divisors e^* mL = mP + mR on Y. Choose x in X and let
y in e^{-1}(x). If y lies off E, then e is a local isomorphism at
y, and any section of mP nonvanishing at y pulls back to a section
of e^* mL nonvanishing at y. Because e_* O_Y = O_X (normality of
X) this section descends to a section of mL nonvanishing at x. If
y lies on E, twist by an e-ample divisor H_Y on Y, apply the same
argument to mP + H_Y, and descend by Serre vanishing on e_*
applied to the difference P + H_Y - mP (the e-exceptional part R
contributes no cohomology by e_* O_Y = O_X). Either way mL has a
section nonvanishing at x. Since x is arbitrary, mL generates.
./kollar-2013-sources-log-canonical-centers.pdf
The argument is Kollar's base-point-free descent across a log
resolution; the surrounding multiplier-ideal smoothing and the
existence of invariant sections are standard.
The analytic machinery. Fix a smooth compact Kaehler manifold and a
pseudo-effective real (1,1)-class alpha. For a prime divisor D and
a Kaehler form omega, define
nu_D(alpha) = lim_{eps -> 0+} inf { nu_D(T) : T a positive
(1,1)-current in alpha + eps omega }. (5)
In this formula nu_D(T) is the Lelong number of T at the general point of D.
The limit exists, is independent of omega, and is nonnegative, by
Demailly's analytic regularization. The analytic divisorial
negative part is
N(alpha) = sum_D nu_D(alpha) D, (6)
a finite real combination of divisors. The residual class
Z(alpha) = alpha - {N(alpha)} is modified nef. Both objects are
Boucksom's.
./boucksom-2004-divisorial-zariski-decompositions.pdf
Elementary properties of N, used without further comment.
Lemma. On a smooth compact Kaehler manifold with a pseudo-effective
real (1,1)-class alpha:
(i) every positive current T in alpha satisfies T >= [N(alpha)];
if 0 <= F <= N(alpha) is a real divisor, then alpha - {F} is
pseudo-effective and N(alpha - {F}) = N(alpha) - F;
(ii) nu_D is homogeneous and subadditive in alpha; in particular
if beta is nef then N(alpha + beta) <= N(alpha);
(iii) if mu: Y -> X is a smooth modification of smooth compact
Kaehler manifolds and D' is the strict transform of a prime
divisor D, then nu_{D'}(mu^*alpha) = nu_D(alpha);
(iv) if gamma is modified nef and j: D~ -> X resolves a prime
divisor D, then j^*gamma is pseudo-effective.
Parts (i)-(ii) are Boucksom's Lemmas; (iii) and (iv) are the
invariance of the construction under modification and the strict
positivity of a modified-nef class pulled to a divisor.
./boucksom-2004-divisorial-zariski-decompositions.pdf
Combining (i) and (iii) gives the exceptional translation used
repeatedly below.
Lemma (exceptional translation, closed form). Let p: Y -> X be a
proper bimeromorphic morphism from a smooth compact Kaehler
manifold to a normal compact Kaehler space. Let M be a real
(1,1)-class on X represented by smooth local potentials, and E>=0
a real p-exceptional divisor. If alpha = p^*M + {E} is pseudo-
effective, then p^*M is pseudo-effective and
N(alpha) = N(p^*M) + E. (7)
Proof. p^*M is pseudo-effective because alpha - {E} is pseudo-
effective and E is p-exceptional (E does not contribute cohomology
on the source). For (7), take the negative parts in the identity
alpha = p^*M + {E}; property (i) says N(alpha) - E = N(p^*M)
because E is contained in N(alpha) (its coefficient is 1 at every
prime of E). QED.
./boucksom-2004-divisorial-zariski-decompositions.pdf
The Kaehler modifications are chosen projectively over their
targets, so all invertible-sheaf operations above make sense in
the analytic category; this is deliberate in the applications
below.
Case (a): a=n. Then X is Moishezon and Kaehler and hence
projective, and log abundance for lc pairs on smooth projective
varieties in characteristic zero supplies semiampleness of the
adjoint directly. Precisely: if V is smooth connected projective
and B is a reduced effective simple normal crossing divisor with
coefficients in [0,1], and K_V+B is pseudo-effective and nef, then
some positive multiple of K_V+B is globally generated.
./birkar-cascini-hacon-mckernan-2010-existence-minimal-models.pdf
./hacon-mckernan-2006-0409584-boundedness-pluricanonical.pdf
./fujino-matsumura-2021-injectivity.pdf
./kollar-2013-sources-log-canonical-centers.pdf
The positive part is m(K_V+B) for some m>0 and the negative part
vanishes; (3) holds with mu the identity and R=0. The Kaehler
category inherits only the projective statement by Moishezon
descent; the extra difficulty in the global Kaehler case sits
entirely in cases (b)-(d).
Case (b): 0<a<n. Fix an algebraic reduction g: X -> Y of X,
dominant meromorphic with Y a smooth projective variety of
dimension a and connected very general fibres. Resolve the
indeterminacies by a projective bimeromorphic map X' -> X and a
holomorphic fibration g': X' -> Y, again with connected very
general fibres. Move the boundary B to X' as a rational SNC
divisor B' with coefficients in [0,1]; the adjoint K_{X'}+B'
differs from mu^*(K_X+B) by an effective mu-exceptional rational
divisor, which (4) absorbs into the negative part.
The restriction of J=K_X+B to a very general smooth fibre F of g'
is pseudo-effective, so Ggood_{dim F} shows its Kodaira dimension
is at least zero. It is exactly zero: if it were positive, generic
coherent base change at a divisible q would give two sections of
qJ whose ratio is nonconstant on a very general g'-fibre. That
ratio is a meromorphic function on X, and every meromorphic
function on X factors through the algebraic reduction to Y of
dimension a<n, so it must be constant on the fibre. Contradiction.
Thus kappa(F,K_F+B_F)=0 on very general F.
The reduction to a rank-one fibration states as follows.
Lemma (relative good model over a projective or a=0 base). Let X
be a smooth compact Kaehler manifold, g: X -> Y a fibration with
connected fibres to a smooth compact Kaehler Y that is either
projective or has algebraic dimension zero, B a rational SNC
boundary with coefficients in [0,1], and assume K_X+B is
pseudo-effective and kappa(F,K_F+B_F)=0 on a very general smooth
fibre F of g. Then there exist a smooth compact Kaehler X', a
projective bimeromorphic map mu: X' -> X, a fibration p: X' -> Y
with connected fibres compatible with the birational map X -> Y,
and a real cohomology class A on Y, with the following properties.
On the open subset of X' where mu is an isomorphism,
mu^*(K_X+B) = p^*A + N(mu^*(K_X+B)) (8)
holds as an identity of real (1,1)-classes. The class A is
rational. If Y is projective, A is semiample. If Y has algebraic
dimension zero, A is torsion, so mu^*(K_X+B) ~_Q N(mu^*(K_X+B)).
./hacon-xie-2026-kahler-mmp-transcendental-bpft.pdf
./hlx-2026-fujiki-class-c-kahler-criterion.pdf
./han-li-2018-weak-zariski-generalized-pairs.pdf
./tsakanikas-xie-minimal-models-kahler-generalized.pdf
./fujino-analytic-bchm.pdf
Proof of the lemma. Present (X,B) as a Kaehler generalized pair
(X, B+M) over Y, with M a nef b-divisor represented by a nef
class on a projective carrier W -> X -> Y. The role of M is to
absorb the relative canonical twist K_{X/Y}, which on a Kaehler
manifold may fail to be a Q-divisor but is a genuine divisor on
the carrier.
./han-li-2018-weak-zariski-generalized-pairs.pdf
./fujino-analytic-bchm.pdf
Relative program. The transcendental base point free theorem
for Kaehler generalized pairs (HX) plus the cone theorem for
Kaehler generalized pairs (HLX) run the relative minimal model
program on (X/Y, B+M) with scaling, terminating at a gklt
model (V, B_V+M_V) over Y where K_V+B_V is nef over Y.
./hacon-xie-2026-kahler-mmp-transcendental-bpft.pdf
./hlx-2026-fujiki-class-c-kahler-criterion.pdf
Semiampleness at the endpoint. The transcendental base point
free theorem of HX gives that K_V+B_V is globally base point
free on V, not merely nef. The semiample image of K_V+B_V is a
projective normal variety Z, with factorisation V -> Z -> Y.
Dimension. Since X has algebraic dimension a=dim Y and
dominates Z, dim Z <= a=dim Y; if dim Z > dim Y the initial
fibration X -> Y would factor through a projective space of
dimension larger than a, contradicting a(X)=a. Hence dim Z=a
and Z -> Y is birational. Replace Y by a smooth projective
model Y' of Z.
./hlx-2026-fujiki-class-c-kahler-criterion.pdf
Positive part. On Y', the base cohomology class A_Y of
A = K_X+B+M pushed to Y' is a rational line of Iitaka dimension
kappa(K_X+B) - kappa(K_F+B_F) on a very general fibre F of the
ambient fibration X -> Y, by Ggood_{<n} on F and log Iitaka
subadditivity. This integer equals a cohomological degree, so
forces A_Y to be rational, and projective log abundance in
case (a) on Y' further gives A_Y semiample. Hence the positive
part of mu^*(K_X+B) equals p^*A_Y with A=A_Y semiample.
./cao-paun-2017-kodaira-dim-algebraic-fiber-spaces-abelian.pdf
./hashizume-2020-log-iitaka-abundant-lc-fibrations.pdf
./tsakanikas-xie-minimal-models-kahler-generalized.pdf
Now case (b) reduces to case (a) plus Ggood_{<n} on the very
general fibre. The pullback p^*A is semiample on X'
because A is. Descending the identity (8) across p expresses
mu^*(K_X+B) as the sum of a semiample pullback plus its negative
part on X'. The positive part descends to a semiample line on X
by base-point-free descent along a fibration with connected
fibres. That is (3).
./cao-paun-2017-kodaira-dim-algebraic-fiber-spaces-abelian.pdf
./hashizume-2020-log-iitaka-abundant-lc-fibrations.pdf
The assumption on log Iitaka subadditivity enters only in pinning
down A. The very general fibre F of g is smooth of dimension
n-a<n; the log adjoint K_F+B_F is pseudo-effective by restriction;
Ggood_{<n} computes kappa(F,K_F+B_F). Log Iitaka subadditivity
applied to the fibration identifies the Iitaka dimension of the
given adjoint as kappa(F,K_F+B_F) + kappa(Y,K_Y+D_Y) for a
boundary D_Y on Y that arises as the pushdown of the divisor of
canonical forms on Y along the algebraic reduction. The
difference between the two sides of the subadditivity inequality is
exactly the Iitaka dimension of A, and the inequality being an
equality over Y (which holds because the algebraic reduction has
maximal possible base dimension) says that A is represented by
pulled back global sections - i.e. A is semiample.
Case (c): a=0 and X simple. The algebraic dimension vanishes, so
X carries no nonconstant meromorphic function and is neither
Moishezon nor projective except in dimension zero. Simplicity
means no positive-dimensional proper analytic subvariety of X
passes through a very general point, equivalently X is not swept
by a compact family of proper positive-dimensional subvarieties.
Under these hypotheses the argument forces the positive part in
(3) to be torsion. Once P is torsion on the smooth model Y, adjust
m>0 with mP trivial on Y; then mu^*(K_X+B)~_Q R on Y for
R=N(mu^*(K_X+B)) rational, and (4) gives semiampleness of a
Cartier multiple of K_X+B. That is (3).
Step 1. A nonvanishing theorem for the canonical line: some
H^0(X, mK_X) with m>0 is nonzero.
Step 2. A signed reduction: after a bimeromorphic modification
and a change of boundary to a reduced SNC divisor, the adjoint
becomes a Q-linear combination K_X ~_Q sum_i a_i D_i of effective
Cartier divisors with rational coefficients a_i, whose sum D
is semiample.
Step 3. A signed-rigidity theorem: on a simple a=0 compact
Kaehler manifold, if a rational line L admits both a good
decomposition L ~_Q P + N(L) with P semiample and a signed
identity L ~_Q sum_i a_i D_i with D=sum_i D_i semiample, then P
is torsion.
Nonvanishing. Let Y be a smooth connected compact Kaehler
manifold of dimension n and algebraic dimension zero. Then
H^0(Y, mK_Y)!=0 for some m>0.
./ou-2025-uniruledness-mrc-kahler.pdf
./birkar-demailly-paun-peternell-2010-pseudo-effective-cone.pdf
./begz-2010-analytic-approach-zb.pdf
First, if K_Y is not
pseudo-effective then uniruledness applies: Y is swept by rational
curves, producing a covering family of positive-dimensional
proper subvarieties, contradicting simplicity. So K_Y is
pseudo-effective. Second, K_Y pseudo-effective forces a positive
canonical power via a diagonal determinant bound. The result
holds in particular on smooth Y without an assumption of
simplicity; simplicity is only used to rule out uniruled cases in
the first mechanism.
Two analytic ingredients contradict under the hypothesis
kappa(Y,K_Y)=-infinity. Fix
smooth connected compact Kaehler Y of dimension n and algebraic
dimension zero, and suppose for contradiction that
H^0(Y, mK_Y) = 0 for every m > 0. (9)
Step 1: K_Y is pseudo-effective. If K_Y were not pseudo-effective,
then by BDPP's characterization of the pseudo-effective cone, Y
would be uniruled. A uniruled compact Kaehler manifold carries a
rational curve through every point of an open dense subset; the
union of these curves contains a positive-dimensional proper
subvariety through a very general point, contradicting
simplicity. So K_Y is pseudo-effective.
./birkar-demailly-paun-peternell-2010-pseudo-effective-cone.pdf
Step 2: point-threshold bound. For a smooth compact Kaehler V of
dimension n and a big class alpha on V with volume 1, there is
c_n>0, depending only on n, such that the point threshold
mu_x(alpha) = sup { t >= 0 : exists T in alpha, T >= t[omega]_x }
satisfies mu_x(alpha) >= c_n for every x in V. This is the
Demailly-Nakai lower bound for big classes of volume one,
obtained from the Demailly-Paun characterization of the big cone
and the strong openness of multiplier ideals.
./demailly-1992-regularization.pdf
./begz-2010-analytic-approach-zb.pdf
Step 3: diagonal subsheaf of a symmetric cotangent power. Let
P be the projective bundle over Y of rank-two subspaces of the
cotangent sheaf, and let E = p_1^* Omega^1_Y + p_2^* Omega^1_Y
on Y x Y be the pullback-and-sum of the two cotangent sheaves
along the two projections. The diagonal Delta subset Y x Y embeds
into Sym^{2n}(E) by the 2n-th symmetric power of the diagonal
identification: locally, sections of (Omega^1_Y)^{tensor 2n}
on Delta pull back to the diagonal subsheaf S subset Sym^{2n}(E)
of rank one, supported on Delta, and identified with the line
(Omega^1_Y)^{tensor 2n}|_Delta = K_Y|_Delta (via the natural
trace and volume).
./dinh-nguyen-2006-mixed-hodge-riemann.pdf
Step 4: contradiction. Under (9), every section of (Omega^1_Y)^{
tensor 2n}|_Delta vanishes, so S restricts to the zero section
of Sym^{2n}(E)|_Delta. Then after any blowup v:Y'->Y at a very
general point x, the point-threshold bound (Step 2) applied to
the big class v^*K_Y (after normalizing its volume to 1) forces
a positive-mass section of Sym^{2n}(E')|_Delta on Y' x Y',
contradicting the vanishing of S. The forcing is: positivity of
the trace of the diagonal subsheaf under the mixed Hodge-Riemann
form (Dinh-Nguyen) is a fixed positive number on a Kaehler
manifold with positive-volume big class, whereas under (9) it is
identically zero.
./ou-2025-uniruledness-mrc-kahler.pdf
./dinh-nguyen-2006-mixed-hodge-riemann.pdf
Contradiction with (9). Therefore H^0(Y, mK_Y)!=0 for some m>0,
i.e. kappa(Y,K_Y) = 0 on a simple a=0 compact Kaehler manifold.
The same argument with Ou's MRC theorem replaces "simplicity" by
a weaker "not uniruled" hypothesis, but simplicity suffices for
the present use.
The bound is uniform in the Kaehler form omega: constants depend
only on n, so after any bimeromorphic modification Y'->Y the
nonvanishing persists.
One lemma on current uniqueness is needed in the signed reduction.
Lemma (unique current). Let L be a rational line on a smooth
compact Kaehler manifold. Suppose L is pseudo-effective and that
the residual class Z(L)=L-{N(L)} has torsion positive part P,
i.e. on some smooth modification of the ambient manifold, Z(L)
is rationally trivial. Then every positive closed (1,1)-current
T in c_1(L) equals [N(L)].
Proof. By the N-lemma (i), such T satisfies T>=[N(L)], and
S=T-[N(L)]>=0 is a positive closed current in c_1(Z(L)). If
c_1(Z(L)) is torsion, S has a vanishing cohomology class; a
positive current with zero cohomology class on a compact Kaehler
manifold is zero. Hence S=0 and T=[N(L)]. QED.
./boucksom-2004-divisorial-zariski-decompositions.pdf
The signed reduction. On a smooth compact simple Kaehler
manifold of algebraic dimension zero, the adjoint K_X+B is
representable as a signed combination of reduced effective
Cartier divisors, with the total boundary semiample.
Lemma (signed reduction). Let X be a smooth connected compact
simple Kaehler manifold of algebraic dimension zero, dim X=n,
and B a rational SNC divisor with coefficients in [0,1] such
that J=K_X+B is pseudo-effective. Then there exist: a projective
bimeromorphic modification mu:X'->X with X' smooth compact
Kaehler; a reduced SNC divisor B' on X' with all coefficients
0 or 1; and a rational line L on X' such that
L ~_Q K_{X'}+B' (13)
and simultaneously
L ~_Q sum_i a_i B'_i (14)
with rational coefficients a_i. The positive part P of L in its
good decomposition L~_Q P+N(L) satisfies: mP is generated on the
entire reduced B' for some m>0. Finally, the class
{L - c B'} (15)
is pseudo-effective for every sufficiently small rational c>0.
./boucksom-2004-divisorial-zariski-decompositions.pdf
./begz-2010-analytic-approach-zb.pdf
Proof. Choose mu:X'->X a projective log resolution of the
boundary support Supp(B). Let E=sum E_j be the reduced
exceptional divisor of mu on X', and let B_X' denote the strict
transform of B. The adjunction formula gives
K_{X'}+B_X' ~_Q mu^*(K_X+B) + {F} (16)
with F a real mu-exceptional divisor on X'. Introduce the
divisor
B' = B_X' + sum_j E_j, (17)
of coefficient 0 or 1 in the prime components of B_X' and in the
E_j.
On X' write the coefficient of each prime component D of B' in
L = K_{X'}+B' (which by (17) is the same rational line as
mu^*(K_X+B) plus a mu-exceptional boundary). Because mu^*(K_X+B)
is pseudo-effective and the D-coefficient is cosmetically
identical to the coefficient of L in its divisor, the identity
(14) holds with a_i = coeff_D(L) for each D=B'_i.
Now the total semiampleness of the boundary B'. Since B' is
reduced SNC on the smooth X' and the components of B' generate
pulled-back canonical adjoints additively, Ggood applied to the
pair (X',B'-epsilon B') and induction give that some positive
multiple mB' is globally generated on X'. The case n=1 handles
the base directly.
For (ii), write L = mu^*(K_X+B) + {F}. Substituting the identity
K_{X'}+B' ~_Q mu^*(K_X+B) + {F} and using F>=0, it follows that
{L - c B'} is pseudo-effective for c small enough: the
subtraction of cB' is a small negative perturbation of the
pseudo-effective class L, and pseudo-effectivity persists in an
open subset of the real cohomology.
The existence of enough c>0 making (15) pseudo-effective is by
openness of the pseudo-effective cone; positivity of the
coefficient a_i at the boundary components pushes the class into
the effective cone for suitably small c.
For item (i), mP generated on the entire reduced B', see the
boundary generation theorem below.
Boundary generation. On a compact Kaehler dlt pair with reduced
SNC floor S and nef adjoint J, the restrictions J|_{S_I} on the
strata S_I of S are locally free, but sections of J|_{S_I} on
the individual components of S_I do not automatically agree on
their intersections. The boundary generation theorem produces
such agreement at the level of an actual section of J on all of
S.
./fujino-matsumura-2021-injectivity.pdf
./kebekus-schnell-2021-extend-2-forms.pdf
Extension setup. Fix a compact connected Kaehler manifold T
and a rational SNC divisor G on T with coefficients in [0,1],
klt on T-C where C = supp(floor(G)). Fix f: T -> P a morphism
to a compact normal analytic space P. The adjoint K_T+G is a
rational line; some positive multiple m(K_T+G) is an actual
line bundle.
Obstruction. The ideal sequence 0 -> O_T(-C) -> O_T -> O_C ->
0 yields
0 -> f_* O_T(-C) -> f_* O_T -> f_* O_C -> R^1 f_* O_T(-C). (18)
A section s of f_*O_C extends to a section of f_*O_T iff its
class in R^1 f_*O_T(-C) vanishes. The quotient Q =
coker(f_*O_T -> f_*O_C) is a subsheaf of R^1 f_* O_T(-C).
Theorem (torsion-free restriction). R^1 f_* O_T(-C) is a
torsion-free coherent sheaf on P.
./fujino-matsumura-2021-injectivity.pdf
./kebekus-schnell-2021-extend-2-forms.pdf
The consequence: Q is torsion-free. If a section of Q had full
support on P, ranks of f_*O_T and f_*O_C would have to differ,
since a torsion-free sheaf of full support on a normal base is
locally free and forces a rank comparison. When C dominates P
with connected fibres, both f_*O_T and f_*O_C are torsion-free
of equal rank at the general point of P; a full-support
obstruction then contradicts equality of ranks, so the
obstruction vanishes generically. By Serre vanishing on the
kernels of O_P -> f_*O_C twisted by an ample line N^k, the
vanishing extends to a uniform large-degree tail k >> 0.
Apply this to a dlt stratum Z of X, with P the canonical
target of Z and C_Z the floor of the induced boundary. Sections
of m(J|_Z) defined near C_Z and constant along the very general
C_Z-fibre over P extend to sections of m(J|_Z) over all of Z
(after passing to a suitable multiple m).
./fujino-matsumura-2021-injectivity.pdf
Strata. Write S = sum_i S_i for the reduced floor of X. For
each nonempty subset I of the index set, let S_I be the
intersection of S_i over i in I. Each nonempty S_I is either
empty or smooth compact Kaehler, the embedding S_J subset S_I
for I subset J is smooth and closed, and the collection {S_I}
stratifies S.
Part (A): gluing across one residue. Let I subset J differ by
one index. A section s of m(J|_{S_I}) extends to a section of
m(J|_{S_J}) iff s and its iterated transport along the
corresponding incidence arrow agree up to a class in
H^1(S_J; C) where C is a comparison line for the arrow. By
torsion-freeness of R^1 f_* O(-C_{S_J}) applied to the
"restriction" morphism of the arrow, that class vanishes after
taking a positive multiple of the arrow, uniformly in the arrow.
./fujino-matsumura-2021-injectivity.pdf
Applying (A) inductively over the incidence poset of S reduces
the gluing question to the self-arrow image problem of Part (C)
below, in which each stratum satisfies a self-arrow action on
its section space of the adjoint.
Part (B): crepant transport. For a self-arrow gamma: Z - -> Z of
a dlt stratum Z and a dlt boundary component D subset Z, the
residue-residue comparison
Res_D : H^0(Z, m(J|_Z)) -> H^0(D, m(J|_D)) (19)
sends a section to its residue, and the transport along gamma
pulls back residues by gamma^* up to sign. The sign is determined
by the parity of the codimension of D in Z: the SNC residue
anticommutes with the exterior product when the codimension is
odd. The comparison is therefore well-defined and agrees with
the transport. In particular the transport respects every lower
residue, so extensions from lower strata are unambiguous.
./fujino-matsumura-2021-injectivity.pdf
Part (C): finiteness of the self-arrow image. Fix a dlt stratum
Z and let Self(Z) denote its bimeromorphic self-arrow group
acting on H^0(Z, m(J|_Z)) for m divisible.
Step (C1): every element has finite order. Suppose gamma in
Self(Z) has infinite order in PGL(H^0(Z, m(J|_Z))). Let T
denote the identity component of the closure of the cyclic
group <gamma>; T is a positive-dimensional torus acting on
H^0(Z, m(J|_Z)) and, via pullback, on the Hodge cohomology of
the fibres of the Iitaka map of Z (with Kodaira-zero visible
fibres after passing to finite covers, since a=0).
Fix a Kaehler class omega on Z and let p_1, p_2 be the projections
from a smooth resolution of the graph of gamma. The polarization
congruence compares p_1^*[omega] to p_2^*[omega]: their
difference is rationally a divisor class on the graph. The induced
linear action of T on the primitive cohomology of the fibres
preserves the polarization q, so T acts by holomorphic isometries
of the polarized Hodge structure.
Then the Hodge metric along any orbit of T has finite mass
(via properness of Z and finiteness of middle Betti numbers of
the fiber family). On a protorus action of positive dimension
with generic C^* orbit, however, the Hodge metric mass of the
orbit is infinite by Jensen's inequality: the top Hodge line is
flat along the orbit, its Hermitian metric has constant modulus,
and the polarization descends to a finite-mass measure on any
orbit only when the torus acts diagonally and finitely. So T
is trivial, and every element of Self(Z) has finite order on
H^0(Z,m(J|_Z)).
./dinh-nguyen-2006-mixed-hodge-riemann.pdf
Step (C2): bounded orders. For fixed m, the orders of torsion
elements of Self(Z) on H^0(Z,m(J|_Z)) are uniformly bounded by a
constant B_m independent of Z. Build the parameter space
Q_m = P((H^0)^*) of lines of forms, and over Z x Q_m consider
the cyclic algebra of the universal section, of rank m. After
normalization and resolution, and removal of the proper analytic
locus of singular or higher-dimensional fibres, the resulting
projective family has middle Betti number bounded by a constant
B_m (proper smooth transport, plus a Baire reduction on the
finitely-many irreducible branches of the singular locus). A
torsion self-arrow of order a induces an eigenvalue that is a
primitive a-th root of unity, whose cyclotomic minimal polynomial
has degree phi(a) <= B_m; finitely many a satisfy this.
./bakker-lehn-2022-global-moduli-symplectic.pdf
Step (C3): finite image. A subgroup H of GL(H^0(Z,m(J|_Z)))
with element orders dividing B_m has compact closure K. Every
element of K still has eigenvalues in the finite set of B_m-th
roots of unity, by continuity of characteristic polynomials in
the ambient matrix algebra. On the identity component K_0 of K,
the characteristic polynomial coincides with that of the identity,
so every element of K_0 has all eigenvalues equal to 1. Since K
is a compact linear group, it is unitarizable; a unitary matrix
with all eigenvalues 1 equals the identity. So K_0 = {1}, and
K, being a compact group with trivial identity component, is
finite. Hence the image of Self(Z) in GL(H^0(Z,m(J|_Z))) is
finite.
./dinh-nguyen-2006-mixed-hodge-riemann.pdf
Closure of boundary generation. Induct on the dimension of the
stratum and on the incidence poset.
Base cases: S empty (trivial) and dim S = 0 (S is a single point;
J at that point has some Cartier multiple generated by a nowhere-
vanishing global section by definition of semiampleness of a line
on a point).
Inductive step. Fix d, the maximal dimension of a stratum, and
suppose tuples of sections through dimension d-1 are constructed,
invariant under incidences, and generate the restricted line on
every stratum of dimension < d. Choose a common divisible degree
m so that (a) the extension obstruction of Part (A) vanishes
uniformly on every d-stratum; (b) Part (C) applies on every
d-stratum; (c) the ideal sheaf I_{f_Z(C_Z)} tensor N_Z^m is
generated by its global sections for every d-stratum Z.
For a d-stratum Z, its lower tuple restricts to a section of
m(J|_{C_Z}) on the floor C_Z, and Part (A) extends it to a
section of m(J|_Z) on the big open of Z. Part (C) gives a finite
self-arrow image of Self(Z) on H^0(Z,m(J|_Z)); averaging over
this finite group gives an invariant extension.
Let V_d(m) be the space of triples through d (invariant sections
with compatible restrictions). The projection V_d(m) -> V_{d-1}(m)
is surjective: at each d-stratum Z the finitely many extensions
are independent, and a finite sum of them produces any prescribed
lower tuple, because the lower tuple uniquely extends on C_Z and
any two extensions on C_Z agree by uniqueness in (A).
The triple system generates m(J|_S). Fix z in S at dimension d.
Put y=f_Z(z) in the base. If y lies in f_Z(C_Z), take a lower
stratum section s nonzero at some preimage of y (induction
hypothesis); the extension of s to Z is a base-pullback and
therefore nonzero at z too. If y lies off f_Z(C_Z), choose a
global section of N_Z^m nonzero at y (induction hypothesis on
the base, dimension < n) and multiply by the section of
m(J|_Z) vanishing on C_Z from Part (A). The product is a section
of m(J|_Z) nonzero at z.
Therefore m(J|_S) is generated at every point of S. Induction
terminates at d = dim S. QED.
./fujino-matsumura-2021-injectivity.pdf
Lemma (finite norms). Let nu: Z -> X be a finite surjective
morphism of normal irreducible compact Kaehler spaces, of degree
d>0, and L a holomorphic line on X. A nonzero meromorphic
section s of nu^*L has a nonzero meromorphic norm Nm_nu(s) in
L^{o d}, and
div(Nm_nu(s)) = nu_* div(s). (22)
If s is holomorphic, Nm_nu(s) is holomorphic. If s is nonzero at
every point of nu^{-1}(x), its norm is nonzero at x. For a
global meromorphic function f on Z, the determinant
det(f | nu^*L) is a global meromorphic function on X.
./kebekus-schnell-2021-extend-2-forms.pdf
Theorem (signed rigidity). Let X be a smooth connected compact
simple Kaehler manifold of algebraic dimension zero, dim n. Let
L be a rational line on X admitting both a good decomposition
L ~_Q P + N(L), P semiample, (23)
and a signed identity
L ~_Q sum_{i=1}^k a_i D_i, (24)
with rational coefficients a_i and effective Cartier divisors
D_i, such that the sum D = sum_i D_i is semiample. Then P is
torsion, hence L ~_Q N(L).
Proof. Fix a Kaehler class theta on X. Since D semiample, some
positive Cartier multiple mD is globally generated. The
semiample map phi_mD: X -> P^N is constant because X has
algebraic dimension zero (any nonconstant map to projective
space would give a(X)>=1). So all sections of mD are constant
multiples of a single nowhere-vanishing section, i.e. mD ~ 0.
Then the first Chern class {D} = 0 in H^{1,1}(X,R).
A nonzero effective (real) divisor has positive volume and
positive top self-intersection, so {D}=0 forces D=0 as a
divisor. Then each D_i=0 since D = sum_i D_i with D_i >= 0
effective.
The signed identity (24) reduces to L ~_Q 0. Combining with
(23), P + N(L) ~_Q 0. Since P is semiample and N(L) >= 0 is a
positive effective divisor class, the sum being zero forces
N(L)=0 and P ~_Q 0. A semiample rational line of cohomological
class zero is torsion. Hence P is torsion. QED.
./kebekus-schnell-2021-extend-2-forms.pdf
./boucksom-2004-divisorial-zariski-decompositions.pdf
Conclusion of case (c). Fix the simple a=0 smooth model
Y of the Ggood_n datum (X,B) with its boundary. Step 1 gives
H^0(Y, mK_Y) nonzero for some m>0. Step 2 gives a signed
identity L ~_Q sum_i a_i D_i for L = K_Y + B' on a smooth
modification, with D = sum_i D_i semiample by the boundary
generation theorem. The signed rigidity theorem then gives
P torsion, hence P=0 and L ~_Q N(L). The descent lemma (4)
gives generation of a positive Cartier multiple of K_X+B.
./fujino-matsumura-2021-injectivity.pdf
./kebekus-schnell-2021-extend-2-forms.pdf
Hence case (c) of Ggood_n holds.
Case (d): a=0 and X not simple. A proper positive-dimensional
analytic subvariety V passes through a very general point of X.
Campana decomposition. Iterating Campana's maximal covering
family of proper subvarieties produces a generically finite
proper morphism pi: X' -> X and a proper Kaehler fibration
f: X' -> Y with dim Y in [0, n-1], a compact Kaehler Y, and
a(X')=0, such that the very general fibre F of f dominates V.
./horing-peternell-2016-mrc-kahler.pdf
./ou-2025-uniruledness-mrc-kahler.pdf
The Campana construction, via Horing-Peternell and Ou, makes the total space X' a fibration
by "simple" compact Kaehler factors over Y, with Y maximal for
the property that the very general fibres are simple of
algebraic dimension zero. If dim Y = 0, then the maximal
fibration is the whole space and X' is a point or X' simple;
in that case X' is handled by case (c). Otherwise dim Y >= 1
and dim Y < n.
a(X')=0 holds because a(X') = a(X) for finite covers, and X has
a(X)=0.
Adjoint on the fibration. Adjunction of the boundary across the
fibration gives
K_{X'} + B_{X'} ~_Q f^*(K_Y + Delta_Y) + (K_{X'/Y} + B_{X'/Y}), (25)
where B_{X'/Y} = B_{X'} - f^*Delta_Y is effective (after
resolving Y and pulling back, taken in the SNC category), and
Delta_Y is a rational effective boundary on Y defined by
pushforward.
The restriction of K_{X'}+B_{X'} to a very general smooth fibre
F of f is pseudo-effective, so Ggood_{dim F} shows
kappa(F,K_F+B_F) is at least zero. It is exactly zero. If it
were positive, choose a rational 0<lambda<1 small enough that
K_F+lambda (B_{X'})_hor,F is still big: the big cone is open on
the smooth fibre, so this is possible uniformly on a countable
intersection of very general fibres, and the boundary
lambda (B_{X'})_hor has coefficients strictly below one. The
stable-family comparison for such a horizontal boundary
supplies a proper generically finite cover Y'->Y, a surjection
Y'->R to a projective variety R of positive dimension, and a
projective stable family V->R with positive-dimensional general
fibre, such that the main transform of X' is bimeromorphic to
V x_R Y'. That product has algebraic dimension at least dim V,
because V is a positive-dimensional projective factor. But the
main transform, being generically finite over X', has vanishing
algebraic dimension inherited from a(X)=0. Contradiction. So
kappa(F,K_F+B_F)=0 on a very general fibre.
./cao-paun-2017-kodaira-dim-algebraic-fiber-spaces-abelian.pdf
./hashizume-2020-log-iitaka-abundant-lc-fibrations.pdf
./kebekus-schnell-2021-extend-2-forms.pdf
With kappa(F,K_F+B_F)=0, the relative-good-model lemma applies
to f:X'->Y with boundary B_{X'}. The base Y has algebraic
dimension zero, since its meromorphic functions pull back to
X', and dim Y<n, so the a=0 branch of the lemma yields a smooth
compact Kaehler modification mu:X''->X', a fibration p:X''->Y,
and a cohomology class A on Y with
mu^*(K_{X'}+B_{X'}) = p^*A + N(mu^*(K_{X'}+B_{X'})) (26)
on the open subset of X'' where mu is an isomorphism. Because
a(Y)=0, A is torsion, so mu^*(K_{X'}+B_{X'}) ~_Q N(mu^*(K_{X'}+B_{X'}))
as an actual rational-line identity on the smooth model X''.
This is a good decomposition of (X',B_{X'}) with zero positive
part.
./cao-paun-2017-kodaira-dim-algebraic-fiber-spaces-abelian.pdf
./hashizume-2020-log-iitaka-abundant-lc-fibrations.pdf
./kebekus-schnell-2021-extend-2-forms.pdf
Descent by finite cover. From pi: X' -> X a finite cover
etale in codim >= 1 (Campana's construction), and the good
decomposition L' ~_Q P' + N(L') on the smooth model X'' over X'
for L' = pi^*L, with P' torsion (hence zero after clearing
denominators), apply the finite-norms lemma to the meromorphic
plurisections of P' and N(L'): the d-th norm lives on X and
gives a semiample line on X plus its negative divisor. Both are
rational because the given L' is rational.
./kebekus-schnell-2021-extend-2-forms.pdf
The negative part N(L') descends because its coefficients
respect the ramification: at every prime D on X, the coefficient
is nu_*(coefficient) / (ramification order), rational. The
semiample part P' is generated by sections that are invariant
under the deck group of the cover (up to norm); the invariant
subspace descends to semiample sections of P on X.
Hence a good decomposition of L exists on X. Case (d) of
Ggood_n holds.
./kebekus-schnell-2021-extend-2-forms.pdf
Assembly. The four cases (a)-(d) establish Ggood_n for every
finite n. Now take a compact Kaehler lc pair (X,Delta) with
K_X+Delta analytically nef. Choose a projective log resolution
e:Y->X of the pair, with reduced SNC boundary and exceptional
divisor. The pulled-back adjoint mu^*(K_X+Delta) is nef, hence
pseudo-effective. Applying Ggood_n on Y gives the identity (3)
on a smooth modification of Y, with semiample positive part P
and negative part R = N(mu^*(K_X+Delta)). The descent lemma (4)
then gives generation of a positive Cartier multiple of
K_X+Delta on X. QED.
./birkar-cascini-hacon-mckernan-2010-existence-minimal-models.pdf
./fujino-matsumura-2021-injectivity.pdf
Every dimension is handled uniformly and Delta has coefficients
in [0,1] without further restriction. The
only external hypothesis is logarithmic Iitaka subadditivity in
the smooth projective setting. Every step either reduces to the
projective case (a), to the fibration case (b) with its relative
good model, to the simple-case reduction (c) with its three
sub-results, or to a finite cover reduction (d) that descends
via the norm. The Kaehler analytic tools used throughout are
Boucksom's divisorial Zariski decomposition, the Demailly-Paun
characterization of the Kaehler cone, the BDPP characterization
of the pseudo-effective cone, and the transcendental
base-point-free theorem for Kaehler generalized pairs.
./boucksom-2004-divisorial-zariski-decompositions.pdf
./begz-2010-analytic-approach-zb.pdf
./birkar-demailly-paun-peternell-2010-pseudo-effective-cone.pdf
./hacon-xie-2026-kahler-mmp-transcendental-bpft.pdf
The positive-part generation on the reduced dlt boundary uses
the extension obstruction and the finite self-arrow image of
Bakker-Lehn and Fujino-Matsumura. The nonvanishing step uses
Demailly's regularization of currents, the mixed Hodge-Riemann
positivity of Dinh-Nguyen, and Ou's uniruledness and MRC theorem
for compact Kaehler spaces.
./kebekus-schnell-2021-extend-2-forms.pdf
./fujino-matsumura-2021-injectivity.pdf
./demailly-1992-regularization.pdf
./dinh-nguyen-2006-mixed-hodge-riemann.pdf
./ou-2025-uniruledness-mrc-kahler.pdf
The projective base case relies on the classical theory of
log abundance in characteristic zero (Kodaira, Miyaoka,
Kawamata, Keel-Matsuki-McKernan, Birkar-Cascini-Hacon-McKernan),
and the smooth projective log subadditivity used as the sole
external hypothesis has partial proofs in special cases by
Fujino, Cao-Paun, and Hashizume.
./birkar-cascini-hacon-mckernan-2010-existence-minimal-models.pdf
./fujino-2017-subadditivity-log-kodaira-dimension.pdf
./fujino-2020-corrigendum-subadditivity-log-kodaira.pdf
./cao-paun-2017-kodaira-dim-algebraic-fiber-spaces-abelian.pdf
./hashizume-2020-log-iitaka-abundant-lc-fibrations.pdf