The pluricanonical system and the Iitaka fibration.
Let X be a smooth integral projective variety over C of
dimension d. A top form on X is a section of the canonical
bundle K_X = det(Omega^1_X). If m >= 0 is an integer, the space
H^0(X, O_X(mK_X)) is finite-dimensional. Let
psi_m : X - -> P(H^0(X, O_X(mK_X))^*)
be the rational map defined by the complete system |mK_X|.
The Kodaira dimension kappa(X) is -1 if every |mK_X| is empty,
and otherwise the maximum over m of dim(psi_m(X)). When
kappa(X) >= 0, the maps psi_m for
large divisible m factor through a common rational map
I : X - -> Z,
the Iitaka fibration, with dim Z = kappa(X) and connected
geometric generic fibre. The difficulty is that psi_m is not
fully determined by kappa alone; it may have a large base locus
and it may miss arithmetic information (torsion, monodromy)
carried by the general fibre. The effective Iitaka fibration
conjecture asks for a bound on m that depends only on d.
What "defines the Iitaka fibration" means.
For a smooth X with kappa(X) >= 0, let K(K_X) be the subfield of
C(X) generated by ratios s_1/s_2 of nonzero sections s_1, s_2
of O_X(mK_X) over all m >= 1 for which H^0 is nonzero. As sets
of ratios, all those fields agree after multiplying degrees, and
K(K_X) is the function field of the Iitaka base Z. The
statement that |mK_X| defines the Iitaka fibration means:
(i) H^0(X, O_X(mK_X)) is nonzero, AND
(ii) the ratios s_1/s_2 of sections of |mK_X| generate K(K_X).
Requiring only dim(image) = kappa(X) is too weak: a complete
system could give a map of the correct dimension whose image is
generically finite over the Iitaka base but not birational to it.
The theorem proved here is the stronger field-generation statement.
The two theorems.
THEOREM A. For each d >= 1 there is an integer m(d) > 0 such
that, over every algebraically closed field of characteristic
zero, the complete system |m(d)K_X| defines the Iitaka fibration
of every smooth integral projective d-fold X with kappa(X) >= 0.
THEOREM B. Let d >= 0 and let Phi be a rational DCC set in
[0,1]. There is an integer a(d,Phi) > 0 such that every
projective log canonical pair (X,B) of dimension d with
coefficients in Phi and K_X + B ~_Q 0 satisfies
a(d,Phi)(K_X + B) ~ 0. The last statement means a(d,Phi)(K_X+B)
is the divisor of a rational function; both integrality of the
divisor and its principal class are asserted.
Theorem A is existential: no numerical value of m(d) is claimed.
Theorem A is a birational statement when kappa(X) = dim X, a
nonemptiness statement when kappa(X) = 0, and a section-field
statement in the intermediate cases.
Theorem B is stronger than a rational-linear-triviality
statement. The object K_X + B is a Q-divisor; from B alone one
only knows that some positive multiple is Cartier and that this
Cartier class is trivial in Pic(X)\otimes Q. Theorem B fixes the
integral multiple that is actually linearly trivial, and it does
so uniformly in d and Phi.
Why the two theorems are proved together.
The proof of Theorem A goes through the canonical bundle formula
for a contraction f:X -> Z. That formula writes
K_X ~_Q f^* (K_Z + B_Z + M_Z)
with B_Z a discriminant divisor and M_Z a moduli b-divisor. To
control mK_X in terms of the base Z, control of the
denominators of B_Z + M_Z, which is exactly a (generalized)
index statement in the base dimension, i.e. Theorem B in a
lower dimension. Conversely, the proof of Theorem B in
dimension n needs the denominator bound (Prop. 4.1),
which uses Theorem B in dimension < n, and it needs the
structural reductions and scalar estimates of Secs. 5--6,
which use Theorem A in dimension < n. The induction is
therefore simultaneous:
I_n (Theorem A in dim n) and L_n (Theorem B in dim n)
are assumed for j < n;
then K_n gives L_n, which feeds back into
the base adjoint argument to prove I_n.
The black-box inputs.
These external theorems are cited without reproof.
(BB1) Log abundance (LA, Theorem 11.1). Every projective lc
pair (X,B) over C with effective rational boundary and
pseudo-effective K_X+B has a good log minimal model.
(BB2) MMP and termination (BCHM, Hacon-McKernan-Xu, others).
Standard in every dimension.
https://arxiv.org/abs/math/0610203
(BB3) Canonical bundle formula and discriminant divisor
(Ambro 2005; Fujino-Mori 2000). For a contraction
f:X -> Z with klt generic fibre, K_X
~_Q f^*(K_Z+B_Z+M_Z) is a discriminant decomposition with
B_Z coefficients in a rational DCC set and M_Z b-nef.
Ambro 2005: https://arxiv.org/abs/math/0308143
(BB4) Beauville-Bogomolov decomposition (Druel 2018;
Greb-Guenancia-Kebekus 2019; Horing-Peternell 2019). A
projective klt space with trivial canonical class and
trivial irregularity has a finite quasi-etale cover that
splits as a product of abelian, Calabi-Yau, and holomorphic
symplectic factors.
GGK 2019: https://arxiv.org/abs/1704.01408
Horing-Peternell 2019: https://arxiv.org/abs/1710.06183
(BB5) ACC for log canonical thresholds (HMX 2014) and for
generalized discrepancies (Birkar-Zhang 2016).
HMX 2014: https://arxiv.org/abs/1208.4150
Birkar-Zhang 2016: https://arxiv.org/abs/1410.0938
(BB6) Polarized effective birationality (Birkar-Zhang 2016,
Theorem 1.3). For a big log adjoint with coefficients in
a fixed DCC set and a fixed Cartier denominator for the
nef part, there is a uniform rounded system that is
birational.
(BB7) Boundedness of generalized epsilon-lc pairs with
trivial adjoint (Birkar 2025, Theorem 1.7).
Birkar 2025: https://arxiv.org/abs/2507.18394
(BB8) Arithmetic Stein-degree bounds for log Calabi-Yau pairs
(SD, Main Theorem).
(BB9) Equivariant chain and tracking lemmas (U4, Lemmas 5.7
and 5.8, and Corollary 5.2).
(BB10) Frobenius instability and canonical Frobenius
filtrations (Langer 2004; Mustata-Schwede 2014; Sun
2008; Kitadai-Sumihiro 2008; Campana-Paun 2015;
Mehta-Ramanathan 1982; Ramanan-Ramanathan 1984).
Mustata-Schwede 2014: https://arxiv.org/abs/1203.1081
Sun 2008: https://arxiv.org/abs/math/0611360
Campana-Paun 2015: https://arxiv.org/abs/1303.3169
(BB11) The coherent Lefschetz theorem (Donovan 1969).
(BB12) Semistable reduction (KKMSD, Toroidal Embeddings I).
(BB13) Du Bois base change for flat families
(Kollar, Families of Varieties of General Type,
Corollary 2.64).
The logical structure of the proof.
Fix n and assume I_j, L_j for all j < n. Six steps. In the denominator-bound part,
Prop. 4.1 bounds the moduli M_Z and Cor. 4.6 bounds K_X over a
rationally connected base. In the structural-reduction part, Props. 5.1 and 5.4
reduce any counterexample to a well-shaped sequence with
structural and scalar controls. In the Frobenius argument, Props. 6.1 and 6.7
exclude that sequence, giving K_n. In the index reduction, Prop. 3.2 plus
global ACC gives L_n. In the completion argument, Lemma 7.1 plus base
birationality gives I_n, and field descent extends to any
algebraically closed field of characteristic zero. Dimension
zero closes the induction.
Standing conventions.
Until the final descent, work over C. All varieties are integral and
projective unless stated otherwise; for pairs, Kollar's
conventions (Kollar 2013). A disk means the open unit disk in C.
A rational Cartier divisor is a Q-Cartier Q-divisor. A rational
contraction f:X -> Z satisfies f_*O_X = O_Z; equivalently its
geometric generic fibre is connected and f is projective. A
rational fibration is given by a relatively algebraically closed
subfield of C(X); after resolution and Stein factorization it has
connected geometric generic fibres. A smooth geometric generic
model of a fibration is a smooth projective model of the
geometric generic fibre.
For rational divisors D, E on X, D <= E means
E - D ~_Q (effective). All boundaries B in pairs are effective,
rational, and satisfy 0 <= B <= 1 in the standard singularity
framework. Sections of an integral Weil divisor D refer to its
rank-one reflexive divisorial sheaf O_X(D).
For generalized pairs, the Birkar-Zhang convention is used: the
nef part is a nef Q-Cartier divisor on a fixed birational model,
pulled back to all higher models; a moduli b-divisor is b-nef and
b-Cartier if it is the Cartier closure of a nef divisor on one
model. Generalized discrepancies are computed by pulling back
the full adjoint including the nef part.
The good minimal model.
The following statement from LA is quoted (Log abundance in
characteristic zero).
Theorem 2.1 (Good minimal models, LA 11.1). Let (X,B) be a
projective lc pair over C with effective rational boundary and
pseudo-effective K_X + B. Then (X,B) has a good log minimal
model: a crepant Q-factorial terminal model on which the
adjoint K_X + B is semiample.
The klt MMP is occasionally used directly. The term "MMP" means
is the klt MMP with the standard discrepancy comparison, and
the phrase "terminating at a good model" invokes Theorem 2.1.
The induction statements.
The three assertions are named that the induction proves in each
dimension.
I_n : Theorem A in dimension n, for every smooth projective
n-fold and every kappa >= 0.
L_n : Theorem B in dimension n, for every rational DCC
coefficient set.
K_n : The zero-boundary klt case of L_n. There is a uniform
a_n > 0 such that a_n K_X ~ 0 for every projective
klt n-fold with K_X ~_Q 0.
In dimension zero all three hold with a = 1 (a point has
trivial canonical class).
Assume I_j and L_j for all j < n. The reductions explicitly
mention which sub-hypotheses they use.
Reduction of L_n to finite coefficient sets.
Only L_n for finite rational coefficient sets Gamma is proved.
The DCC version follows from global ACC (HMX 2014, Theorem
1.5): for fixed n and a fixed DCC set Phi, the coefficients of
numerically trivial lc pairs in Phi belong to a finite subset
of Phi. This observation also covers the differents that arise
from adjunction.
Geometric generic points and descent.
All finite collections of varieties, maps, divisors, and forms
can be defined over a finitely generated field k_0 \subset C.
Algebraically closed extensions preserve singularity class,
Cartier-ness, triviality of the relevant line bundles, and
dimensions of section spaces.
Very general points that satisfy countably
many conditions at once. These are chosen over a countable field
of definition, or equivalently over C.
The following descent lemma is used twice, in the index and denominators arguments.
Lemma 2.2 (Descent of a trivialization). Let F be a geometrically
integral normal projective variety over a field K, and let L be
a reflexive divisorial sheaf on F that becomes the trivial line
bundle over K-bar. Then L is already a line bundle over K and
is trivial there.
Proof. Local freeness descends faithfully flatly. Flat base
change identifies H^0 over K with a one-dimensional space
after extension, and a nonzero section over K becomes a
generator, hence is a generator before extension. The same
applies to a specified integral log pluricanonical divisor.
QED.
The dependency diagram of one induction step.
L_{ Prop. 4.1 ---------> Cor. 4.6
\ /
\ /
v v
I_{ Prop. 5.1 -> Prop. 5.4 -> K_n -> L_n -> I_n
The arrow L_{ Prop. 4.1 uses only lower-dimensional
indices, not K_n. The arrow I_{ Prop. 5.4 uses only
lower-dimensional Iitaka maps. Prop. 5.1 uses Prop. 4.1 and
Prop. 4.5 (rc-torsion). K_n is what excludes the surviving
sequence via Props. 6.1 and 6.7. L_n uses K_n and L_{ Z be a contraction
of normal projective varieties over a field of characteristic
zero. Let phi in k(X)^* satisfy that all primes in Div(phi)
have zero coefficient on every horizontal divisor of f (so
Div(phi) is vertical). Then phi lies in k(Z)\subset k(X).
Suppose in addition that (X,B) is a rational pair with K_X+B
~_Q 0 and q(K_{X_eta}+B_eta) ~ 0 on the geometric generic
fibre. Then there is a rational q-canonical form s and a
rational Cartier divisor D on Z, ~_Q 0, such that
Div(s) + qB = f^*D.
Proof. The generic fibre X_eta is normal, projective, and has
H^0(X_eta, O_{X_eta}) = k(Z). A rational function whose
divisor is vertical restricts to a function with no zeros or
poles at prime divisors of X_eta. Normality makes it regular,
and hence in k(Z). For the second assertion, extend a
trivializing q-canonical form on the geometric generic fibre
to a rational q-canonical form s on X, using any rational top
form on Z to identify relative and absolute canonical lines.
The divisor H = Div(s) + qB is vertical. Choose a >= 1 and a
rational aq-canonical form tau with Div(tau) + aqB = 0. The
function s^{otensor a}/tau has divisor aH, so the first
assertion writes it as f^*h for h in k(Z)^*. Thus
H = f^*((1/a)Div(h)), as required. QED.
Proposition 3.2 (lc reduction to klt index). Assume L_j for
every j < n and K_n. Then L_n holds: for every finite set
Gamma in [0,1] intersect Q there is an integer c(n,Gamma) > 0
such that every normal projective lc n-fold pair (X,B) with
coefficients in Gamma and K_X + B ~_Q 0 satisfies
c(n,Gamma)(K_X + B) ~ 0.
Proof. By Xu's klt induction theorem (Xu 2019, Theorem 1.8),
the klt case follows from the zero-boundary klt case (K_n) in
the same dimension. The non-klt case is proved without an slc
index statement.
Xu 2019: https://arxiv.org/abs/1905.00297
Step 1 (Mori fibre space and a horizontal lc divisor). Take a
crepant Q-factorial dlt modification of (X,B). Enlarge Gamma
to include 1. Write T = floor(B), nonempty. For small enough
rational eps > 0 the pair (X, B - eps*T) is klt and its
adjoint
K_X + B - eps*T ~_Q - eps*T
is not pseudo-effective: its intersection with (ample)^{n-1}
is negative because T is a nonzero effective divisor. The klt
MMP (BCHM 2010, Corollary 1.3.3) terminates in a Mori fibre
space f:X' -> Z.
The full log canonical divisor remains crepant through this
program. Indeed, fix compatible canonical divisors with
r(K_X+B) = Div(h). Divisorial contractions push forward this
identity; flips preserve it under their codimension-one
identification. On a common resolution, both pullbacks equal
r^{-1}Div(h). Hence the full pairs on all models are lc
(though need not be dlt), and the exact order of their log
canonical divisors is unchanged. From now on (X,B) denotes
the last model.
Step 2 (dim Z = 0). Then X is of Fano type: the klt pair
(X, B - eps*T) has adjoint = -(eps*T) relatively ample over a
point. The bounded complement theorem (Birkar 2019, Theorem
1.7), with a fixed hyperstandard set containing Gamma, gives a
bounded integer c > 0 and a complement B^+ >= B with
c(K_X + B^+) ~ 0. The effective divisor B^+ - B is numerically
trivial; intersecting with an ample divisor kills it, so
B^+ = B and c(K_X+B) ~ 0. Done in this case.
Step 3 (dim Z > 0; a horizontal lc component). On the Mori
fibre space, the surviving transform of T is relatively ample
because eps*T ~_Q -(K_X + B - eps*T) and the right side is
relatively ample. Some prime component S of T therefore
dominates Z. Normalize S: nu : S^nu -> S. Divisorial
adjunction gives
(K_X + B)|_{S^nu} = K_{S^nu} + Theta,
where (S^nu,Theta) is a normal lc pair, Theta is effective,
and coefficients of Theta lie in the fixed rational DCC set of
differents determined by Gamma (Kollar-Mori 1998). The adjoint
K_{S^nu} + Theta is Q-linearly trivial. By L_{n-1} and the
ACC observation above, there is a uniform trivializing
multiple for any such different coefficients.
Step 4 (a common form degree). The geometric generic fibre of
f is a normal projective lc pair of dimension < n with the
same coefficient set. The assertions L_j, j Y --h--> Z,
with g a contraction and h finite, Y normal. Lemma 3.1 with
the case Z = Y gives a in k(Y). Injectivity of pullback for
rational Cartier divisors under the surjective map g gives
h^*D = Div_Y(a).
(Check at a prime divisor of Y: a nonzero order remains
nonzero on any prime divisor above it.) The norm identity for
the finite morphism h now gives
(deg h) D = Div_Z(N_{k(Y)/k(Z)}(a)).
Step 6 (bound the Stein degree). Apply the arithmetic
Stein-degree theorem (SD, Main Theorem) to (X_eta, B_eta) over
k(Z) with its prime coefficient-one divisor S_eta (parameter
t = 1). This fibre is normal, integral, projective, and its
field of global functions is k(Z). The relative algebraic
closure of k(Z) in k(S_eta) is exactly k(Y). Hence deg h is
bounded in terms of the fibre dimension, hence in terms of n.
Step 7 (assemble). Combining the two identities of Steps 4
and 5, q(deg h)(K_X+B) is integral and principal. Taking the
least common multiple of the finitely many possible degrees
(bounded by Step 6) and then a common multiple with the klt
and base-point cases gives c(n,Gamma). The earlier crepant
comparisons return the same multiple to the original pair.
QED.
Observation 3.3. The normality of X in Prop. 3.2 is essential.
The argument uses a single normalized boundary component and
a field norm. If X were only seminormal (or if the lc divisor
support were reducible with unfriendly intersections), the
norm construction would fail to identify a well-defined
divisor class on X. In the slc case a separate argument
(Xu 2019, Section 5) is required.
Volume characters of finite automorphisms.
The next lemma is used in the denominator bound on
components of degenerating fibres. It says that finite
automorphisms of a normal lc pair act on its log volume form
with characters of uniformly bounded order.
Lemma 3.4 (Finite-order log volume characters). Assume L_j in
a fixed dimension j. There is an integer c_j > 0 with the
following property. Let (V,E) be a normal projective lc j-fold
pair, E reduced, and let omega be a rational top form
satisfying Div(omega) + E = 0. If gamma is a finite-order
automorphism of the pair and gamma^* omega = lambda * omega
with lambda in C^*, then lambda^{c_j} = 1.
Proof. Form the quotient pi: V -> W = V/. Define the
boundary E_W by the finite adjunction formula
K_V + E = pi^*(K_W + E_W).
At a prime divisor with ramification index e, the coefficient
of E_W is 1 - 1/e if the upstairs coefficient is zero, and 1
if the upstairs coefficient is one. Hence E_W coefficients
belong to the fixed rational DCC set {1 - 1/e : e >= 1} union
{1}. The finite discrepancy formula makes (W, E_W) lc
(Kollar-Mori 1998, Prop. 5.20).
A power of omega invariant under the finite cyclic group
descends to a rational pluricanonical form downstairs. The
ramification formula shows that it trivializes a multiple of
K_W + E_W; this establishes Q-Cartierness and Q-linear
triviality of K_W + E_W. By L_j and the ACC observation,
there is a uniform c_j > 0 for this DCC set such that
c_j(K_W + E_W) ~ 0.
Pull back a trivializing log pluricanonical form. It is an
invariant generator upstairs and is therefore a scalar
multiple of omega^{otensor c_j} because V is projective and
integral. Hence lambda^{c_j} = 1. For j = 0 the same
conclusion holds with c_0 = 1. QED.
Quasi-etale product covers and factor actions.
A finite map of normal varieties is called quasi-etale
if it is etale outside a set of codimension at least two. The symbol Omega_V^[p] denotes the reflexive extension of
p-forms from the smooth locus.
Lemma 3.5 (Canonical index and the volume character). Let
pi:R -> X be a finite Galois quasi-etale cover of normal
projective varieties with K_R ~ 0. Choose a generating
reflexive top form omega on R, and write g^* omega =
chi(g) * omega for the Galois action. Then
mK_X ~ 0 if and only if chi^m = 1.
Proof. If omega^{otensor m} is invariant, it descends at the
function-field level to a rational m-canonical form on X.
Quasi-etaleness means its divisor has order zero at every
prime divisor of X. Hence it trivializes O_X(mK_X) as a
divisorial sheaf, so mK_X is principal. Conversely, a
trivializing form downstairs pulls back to a scalar multiple
of omega^{otensor m} and is invariant. QED.
Now let X be a projective klt variety with K_X ~_Q 0. The
cyclic cover attached to a primitive trivialization of a
canonical multiple of K_X is connected and quasi-etale. It has
Cartier trivial canonical divisor and is klt by finite
adjunction, hence canonical: its discrepancies are integers
greater than -1. The singular Beauville-Bogomolov decomposition
theorem (BB4) supplies a finite quasi-etale cover
P = A x B_1 x ... x B_r -> X
with A abelian and each B_i irreducible Calabi-Yau or
irreducible holomorphic symplectic in the singular sense
(Horing-Peternell 2019, Def. 1.4 and Thm. 1.5). The following part of the definition is used.
The blocks and all their connected normal quasi-etale covers
are canonical with trivial canonical divisor. Their reflexive
form algebras are generated by a top form in the Calabi-Yau
case, or by a generically nondegenerate two-form in the
symplectic case. In particular they have no reflexive one-forms
and no reflexive forms in degree strictly between 1 and
dim - 1. Contraction with a volume form also gives absence of
vector fields.
The cover P -> X need not be Galois, and its Galois closure
need not remain a product. The next lemma is the comparison
needed to use the factors nonetheless.
Lemma 3.6 (Equivariant comparison with the factors). Let k be
a field of characteristic zero and let P = prod_{i=1}^t B_i
be a product of geometrically integral projective factors
whose geometric fibres are positive-dimensional blocks as
above; include the abelian part when present as a single
factor. Let R -> P be a geometrically integral normal finite
quasi-etale cover, and put d = dim R. If a finite group G
acts on R, then there is a subgroup G_0 of index at most d!
and connected finite quasi-etale covers R_i -> B_i such that:
(1) R -> prod_i R_i is finite and quasi-etale;
(2) G_0 acts regularly on each R_i and the projections
R -> R_i are equivariant;
(3) the volume form on R is proportional to the wedge
product of the pulled-back volume generators on the R_i.
The conclusions extend to finite semilinear actions,
interpreting a factor transformation as an isomorphism to the
corresponding field-conjugate variety. In particular the d!-th
power of any finite-order transformation preserves all factor
fields.
Proof. Step 1 (intrinsic preservation of product directions).
Pullback on the common smooth big open, followed by reflexive
extension, gives
T_R = oplus_{i=1}^t E_i.
There are no global homomorphisms between two distinct
summands. This is checked after extending k to an algebraic
closure. At least one of the corresponding factors is
nonabelian, B_i say. Fix general smooth points in all
complementary factors. The resulting slices of R are normal
and finite over B_i: generic flatness and geometric normality
in characteristic zero allow this after shrinking the
complementary parameter space. The locus omitted from the
quasi-etale map has codimension at least two on a general
slice. Each connected component of the slice is consequently
a connected normal quasi-etale cover of B_i.
On the smooth big open of such a component, E_i is its
tangent sheaf while the complementary summands are trivial. A
homomorphism in either direction gives a vector field or a
one-form, which extends reflexively and vanishes by the block
properties. These slices cover a dense open of R, proving the
claimed vanishing.
Step 2 (bounded-index subgroup acting on factors). The block
projections in Step 1 are central idempotents of the
finite-dimensional algebra End_R(T_R). Its primitive central
idempotents give nonzero direct summands of positive generic
rank, so there are at most d of them. The action of G permutes
this finite set. Its kernel G_0 has index at most d! and
fixes each block projection: because a central idempotent is
a sum of primitive central idempotents. The reasoning is
unchanged for semilinear algebra automorphisms. The idempotent
argument is carried out over k itself; it does not require
extending the finite group action to an algebraic closure.
Step 3 (recover the finite factor covers). Let F_i be the
relative algebraic closure of k(B_i) in k(R), and let R_i be
the normalization of B_i in F_i. Equivalently R -> R_i -> B_i
is the Stein factorization of the projection. The intermediate
fields F_i are regular over k because k(R) is regular over k;
hence the R_i are geometrically integral. Stein factorization
commutes here with algebraic closure: the extensions k(R)/F_i
are regular in characteristic zero. At the generic point, the
complementary directions in T_R span Der_{k(B_i)} k(R). For a
finitely generated extension in characteristic zero the common
constants of these derivations are exactly the elements
algebraic over k(B_i). Consequently G_0 preserves F_i and
induces birational factor transformations.
Step 4 (the product is a quasi-etale cover of P). The
extension obtained by adjoining the other product factors to
k(B_i) is regular, hence is linearly disjoint from the finite
extension F_i/k(B_i). Thus the function field of the normal
product R_i x prod_{j != i} B_j is intermediate between k(P)
and k(R). Normalization gives finite maps
R -> R_i x prod_{j != i} B_j -> P.
Ramification over a divisor of B_i would persist in this
product and then at a prolongation to k(R), contradicting
quasi-etaleness. Thus R_i -> B_i is quasi-etale. The product
map R -> prod_i R_i is proper with finite fibres, because its
composite to P is finite. Its image has dimension d, equal to
the dimension of the integral target product, so it is
surjective and finite. The same ramification argument shows
it is quasi-etale. Every fibre of R -> R_i has dimension
d - dim R_i.
Step 5 (the factor transformations are regular). Fix g in
G_0 and let Gamma_i be the closed graph of its induced
birational transformation on R_i. The two morphisms from R
to R_i give a surjection R -> Gamma_i. All its fibres have
dimension at least d - dim R_i. If the first projection
Gamma_i -> R_i had a positive-dimensional fibre, its
preimage in R would have dimension strictly larger than
d - dim R_i, contradicting the preceding equidimensionality.
Thus this projection is finite and birational. Normality of
R_i makes it an isomorphism. The same argument for g^{-1}
shows the induced transformation is an automorphism. For a
semilinear transformation the target is replaced by its
field conjugate; the argument is identical.
Step 6 (volume character descends). Each geometric R_i is a
quasi-etale cover of its geometric block, so it is canonical
with trivial canonical divisor. These properties descend to
k by Lemma 2.2. The product of its volume generators pulls
back under the finite quasi-etale product map to a generator
of K_R. Any two generators differ by a nonzero constant.
This proves all three assertions. QED.
Application to families over a curve. The semilinear version
applies to families over a curve as follows. After a finite
extension of the curve function field, define the geometric
product cover and its blocks. Take the normal Galois closure
of the resulting finite extension over the original total
function field, and enlarge the curve field to its relative
algebraic closure in that closure. The latter is a finite
Galois extension because the original total field is regular
over the original curve field. The closure is then
geometrically integral over its new constant field, and its
finite Galois transformations act semilinearly. On geometric
fibres the comparison cover is quasi-etale: over the original
smooth locus purity makes the initial covers etale, and the
same is true of their conjugates and Galois closure. After
any further finite base extension the construction is
repeated. Lemma 3.6 therefore separates the factor actions
after taking a bounded power, without requiring a bound on
the degree of the comparison cover.
The exact canonical bundle formula.
Proposition 4.1 (Bounded moduli denominator). There are
positive integers p_0 | p, depending only on n, with the
following property. Let f:X -> Z be a contraction of normal
projective complex varieties, where dim X = n, dim Z > 0, X
is klt, and K_X is rationally linearly equivalent to the
pullback of a rational Cartier divisor on Z. Then there is a
rational function psi in C(X)^* and canonical bundle formula
data such that, as rational divisors,
K_X + (1/p_0) Div(psi) = f^* D_Z,
D_Z = K_Z + B_Z + M_Z.
Here B_Z >= 0 has coefficients in a rational DCC set depending
only on n, the generalized pair (Z, B_Z + M_Z) is generalized
klt, and the moduli b-divisor M is b-nef with p M
b-Cartier.
Proof. The data and the bound are constructed in several steps.
Fix p_0 first.
Step 1 (p_0 exists). The geometric generic fibre of f is a
klt variety of dimension < n with rationally trivial canonical
divisor. By the induction hypothesis L_{n-1}, there is a
common integer p_0 trivializing its canonical divisor. This
trivialization descends to the generic fibre of f: Cartierness
descends by faithful flatness, and the one-dimensional section
space commutes with extension of the ground field (Lemma 2.2).
The descended section is nonzero over the base field.
Step 2 (choose psi; exact conclusion). Choose psi so that
p_0 K_X + Div(psi) has no horizontal component. Fix an
initial rational Cartier divisor D^0 on Z with
K_X ~_Q f^* D^0. A sufficiently divisible multiple of
p_0 K_X + Div(psi) - p_0 f^* D^0 is principal and vertical.
Write it as p_0 f^*Div(h) for h in C(Z)^* using Lemma 3.1
(Vertical descent); then D_Z = D^0 + (1/p_0)Div(h) is a
rational Cartier divisor on Z satisfying the exact equality.
Step 3 (Ambro's formula). Ambro's klt-trivial fibration
theorem (Ambro 2005) supplies B_Z and M such that
D_Z = K_Z + B_Z + M_Z. Rank-one of the relevant direct image
holds: rk f_* of the relevant (o_K_X)-twist is 1 because on
a resolution the rounded discrepancy divisor over the generic
base point is effective and exceptional, and allowing poles
along an effective exceptional divisor creates no extra power
series.
Step 4 (B_Z coefficients and generalized klt). The
discriminant coefficient of B_Z at a prime P is 1 - t_P,
where t_P is the lc threshold over the generic point of P of
f^*P. Since the total-space boundary is zero, 0 < t_P <= 1.
Threshold ACC (HMX 2014) makes the resulting coefficient set
a fixed rational DCC set. Qualitative moduli theory gives
generalized klt singularities for (Z, B_Z + M_Z).
Step 5 (a smooth determination of M). Fix a smooth
determination W -> Z on which M descends to a b-nef rational
Cartier divisor M_W. The goal is to bound the denominators of
M_W.
Step 6 (p bounds the denominators). This is the content of
Sections 4.2-4.5 below: a single integer p = p(n) clears the
coefficient of p M_W at every prime divisor of W
simultaneously. Given Step 6, Prop. 4.1 is done. QED.
Weights of relative forms.
Fix a prime P on the smooth determination W, and let alpha be
its coefficient in f^* D_Z pulled back from D_Z. Let t be the
threshold of P computed with crepant total-space data. Because
canonical divisors are integral, the coefficient of M_W at P
has denominator equal to the denominator of alpha + t. It
suffices to bound that denominator uniformly in P.
Step 1 (reduce to a transverse curve). Resolve the total
space over W, together with the crepant boundary and the
inverse image of P. Intersect W with very general very ample
members inductively until a smooth curve C remains meeting P
at a smooth point c. Bertini applied to finitely many
resolution strata preserves boundary coefficients and the SNC
threshold data at c. Consequently the sliced equality is
K_V + B^c + (1/p_0) Div(psi_C) = h^* D_C,
coeff_c D_C = alpha,
where h:V -> C is the resolved slice and B^c is its crepant
boundary. Its geometric generic fibre is birational to a
normal klt variety with the original zero-boundary data on the
base curve.
Step 2 (weights and threshold identity). Let z be a local
parameter at c, and choose a rational top form theta defining
K_V. On the generic fibre set
phi = (theta/dz)^{otensor p_0} psi_C.
It generates the divisorial p_0-canonical sheaf of the klt
generic model. For a relative pluricanonical form omega of
degree m define the weight
w_z(omega) = inf_E [ ord_E(omega ^ dz/z) + m ] /
[ m * ord_E(z) ],
infimum over vertical divisorial valuations E centred over c.
The crepant equality gives
w_z(phi) = alpha - 1 + t.
Indeed, if b_E is the crepant boundary coefficient and
v_E = ord_E(z), then the quotient above equals
alpha - 1 + (1 - b_E)/v_E. The infimum over E is precisely
the threshold definition of t. So w_z(phi) is exactly the
fractional part to be cleared.
Lemma 4.2 (Rules for weights). The weight w_z satisfies:
(a) Finite extension of the fibre function field, with base
field fixed, does not change the weight of a pulled-back
form.
(b) Under a curve extension of ramification l at the marked
point, with new parameter u, the pulled-back weight is
multiplied by l.
(c) Taking a positive tensor power leaves the weight
unchanged. Multiplying a degree-m form by a base
function a adds ord_z(a)/m.
Proof. For a divisorial valuation of ramification index e
in a finite function-field extension, the canonical
ramification formula is
ord_{E'}(omega ^ dz/z) + m
= e ( ord_E(omega ^ dz/z) + m ).
With the base fixed, the denominator also multiplies by e.
Divisorial valuations restrict and prolong to divisorial
valuations, proving (a). After curve ramification, z = u^l
times a unit. The logarithmic base differentials differ by a
unit, while ord_{E'}(u) = e ord_E(z)/l, proving (b). One may
take the curve extension Galois; regularity of the original
function field over its constant field makes the
prolongations at the different base branches conjugate, so
any branch gives the same infimum. Part (c) follows directly
from the numerator. QED.
Semistable block models and their characters.
The singular Beauville-Bogomolov decomposition is used in the
form recalled in Lemma 3.6. After a finite extension of the
curve field, the geometric generic klt fibre has a finite
quasi-etale product cover with one abelian block, if present,
and irreducible Calabi-Yau or irreducible symplectic blocks.
Take a Galois comparison cover R of the original fibre
function field, including all conjugates of the product cover.
Replace the curve field by its relative algebraic closure in
C(R). This constants extension is Galois over the original
curve field because the original fibre function field is
regular over it. The cover R is geometrically integral and
remains geometrically quasi-etale over the product.
An inertia generator on the new curve has a finite-order lift
g to R. By Lemma 3.6, g^b preserves every block direction for
b = n! and induces regular semilinear transformations on the
Stein blocks R_i. Each R_i is a finite quasi-etale cover of
its original block; it therefore has the same form-algebra
properties and has a volume generator of degree one. Further
finite Galois curve extensions cause no obstruction: take the
composite Galois function-field cover and lift the actions.
Inertia in a tower of complex curve extensions is cyclic and
surjects onto inertia below, so compatible inertia generators
can be chosen.
Step 1 (spread over a punctured curve; equivariant semistable
models). Spread the finite list of Stein blocks, morphisms,
and actions over a punctured curve. Take projective equivariant
resolutions; on an abelian block do not modify its smooth
generic fibre. After one further curve extension, projective
semistable reduction (KKMSD) gives smooth total spaces with
reduced SNC special fibres and lifted actions of g^b.
Equivariance can be retained as follows. Start with an
equivariant SNC resolution and barycentrically subdivide its
boundary complex so translates of a boundary component are
equal or disjoint. Root extraction of the base parameter and
normalization are equivariant. In the toroidal subdivision
step, stabilized cones then have their rays individually
preserved. Choose the compatible height-one subdivisions on
the combinatorial quotient complex and pull them back along
the orbits, using the original cone lattices, not the lattices
of a geometric quotient. The semistable subdivision theorem is
compatible with prescribed face subdivisions, so it supplies
the required invariant projective subdivisions after
sufficiently divisible ramification. This construction also
applies to strata that split after root extraction, with their
base-changed lattices. The total spaces can be compactified
and resolved equivariantly away from the chosen special
fibre.
Step 2 (normalize block volumes). Write u for the final
uniformizer, l for the total ramification over z, and zeta for
the primitive l-th root by which g acts on the tangent line at
the marked point. On each generic Stein block choose a volume
generator eta_i. Its pullback to a smooth resolution has no
horizontal poles, by canonicality. On a semistable model the
weight is an integer: the smallest logarithmic order of
eta_i ^ du/u along a special-fibre component. Multiply eta_i
by a power of u so that
w_u(eta_i) = 0.
The resulting logarithmic form is regular near the fibre and
has order zero on some component. The lc property of an SNC
pair shows that all higher divisorial tests in the weight
formula have nonnegative weight.
Step 3 (product weight is zero). The product of the normalized
forms also has weight zero. To see this without assuming that
R splits, first work on the fibre product of the semistable
block models. The local equations are prod_j x_{ij} = u, one
equation for each block. Relative log top forms multiply
without an additional power of u. These are toroidal charts;
their fibre products over the log base are saturated, have
reduced special fibre, and retain their log generators on
toroidal resolution. Thus every divisorial test has
nonnegative weight. At a product of generic points of
components on which the individual forms have order zero, all
morphisms are smooth and the product form has order zero.
Finally apply Lemma 4.2 to the finite comparison cover R.
Step 4 (the weight identity). The pullback of phi and the
p_0-th power of this product both generate the generic
p_0-canonical line on R. Their ratio is therefore a base
function a. The weight rules give
phi = a ( wedge_i eta_i )^{otensor p_0},
l * w_z(phi) = ord_u(a)/p_0.
The same formula holds for a zero-dimensional generic fibre,
with the empty product interpreted as 1.
Step 5 (character extraction). For each block write
(g^b)^* eta_i = c_i eta_i. The ratio c_i is a base function,
because both forms generate the generic volume line. Their
equal weights imply ord_u(c_i) = 0, so c_i is a unit at the
marked point. Put lambda_i = c_i(c). The finite order of the
semilinear action shows, by evaluating the product of its
translates at this fixed point, that lambda_i is a root of
unity.
Step 6 (the inertia relation). The form phi comes from the
original function field and is invariant under g^b. Taking
leading coefficients in the weight identity therefore gives
zeta^{b * ord_u(a)} * prod_i lambda_i^{p_0} = 1.
Suppose a single integer N = N(n) kills every lambda_i.
Raising the inertia relation to the N-th power gives
l | b N ord_u(a), and hence
p_0 b N w_z(phi) is an integer.
By the weight-threshold identity this same integer clears
the denominator of alpha + t. So the remaining task is to
bound the orders of the central characters lambda_i; the
ramification l itself need not be bounded.
The character-bound lemma.
Lemma 4.3 (Bounded degeneration characters). In the
semistable construction above, let R_i be the Stein block
covers, of dimensions s_i < n, over the final curve field.
Let c be the marked point, u its uniformizer, and g a
finite-order lift of base inertia. Put b = n! so that g^b
preserves each block, and choose volume generators eta_i
with w_u(eta_i) = 0. There is an integer N = N(n) > 0 such
that, writing
(g^b)^* eta_i = c_i eta_i,
lambda_i = c_i(c),
each c_i is a unit at c and lambda_i^N = 1.
Proof in two cases.
Case A (abelian blocks). Let the block have dimension s. On
its projective semistable model over a disc, the direct
image of the relative log top line is the extended Hodge
line F^s in degree s. This is the semistable comparison
theorem for logarithmic de Rham cohomology and the canonical
extension with nilpotent residue (Deligne 1970;
Steenbrink 1976). The normalized form is a local frame: it
is a regular relative log form but cannot be divided by u
while remaining regular, because its order on one component
is zero. Its eigenvalue on the central fibre of F^s is
lambda_i.
The integral local system in degree s has rank binom(2s,s).
Choose a coordinate in which the finite base action is a
rotation. On the universal cover of the punctured disc, the
lifted transformation, combined with parallel transport,
acts by an integral matrix A commuting with the unipotent
monodromy. A power of A is a power of monodromy because the
original transformation has finite order. Passing from flat
frames to canonical-extension frames multiplies A by the
exponential of a scalar multiple of the logarithm of
monodromy. This is a commuting unipotent factor, so it does
not change eigenvalues. Hence a primitive root of the order
of lambda_i occurs among the eigenvalues of an integral
matrix of rank binom(2s,s). Its cyclotomic degree is at most
that rank. There are only finitely many possible orders,
uniformly for s < n. Call the resulting bound N_ab(n).
Case B (nonabelian blocks). Let s be the block dimension,
write eta for its normalized form, and let G be the finite
cyclic group generated by the transformation being
considered. The character is bounded by applying Lemma
3.4 (Finite-order log volume characters) to a normal
component of the special fibre. This requires a degree-one
log generator on that component and a bounded power of the
transformation preserving it. The construction gives (i) an
equivariant good model where the normalized logarithmic
form generates along the whole special fibre, and (ii) a
bounded power preserving a prime component of the special
fibre.
Step B1 (equivariant klt model). Near the marked point there
is a G-equivariant projective model Y such that
Y is klt, T = Y_c is reduced Cartier, (Y, T) is dlt,
and K_Y is semiample over the curve.
The geometric generic fibre need only remain birational to
the block. Construction: start with an equivariant smooth
projective compactification Q of the semistable model,
preserving its SNC special fibre. A relative K_Q-MMP is
also a (K_Q + Q_c)-MMP because Q_c is a fibre. For existence
and termination, take the finite quotient Q-bar = Q/G and
its branch boundary Delta, so that K_Q is the finite
pullback of K_{Q-bar} + Delta. The quotient is Q-factorial
and (Q-bar, Delta) is klt. Its adjoint is pseudo-effective
over the quotient curve: an effective pluricanonical form
on the generic fibre upstairs gives an invariant one after
multiplying its finitely many translates.
Choose a divisor H of sufficiently large degree on the
quotient curve, represented by many general points with
small rational coefficients. The pair obtained by adding its
pullback stays klt, and its adjoint is pseudo-effective
absolutely, since a general section extends after a
sufficiently positive base twist. Choose also deg H >
2 dim Q. The good-model theorem (Theorem 2.1) followed by a
terminating MMP with scaling to that good model now applies.
Every step is over the curve: otherwise the extremal-ray
length bound for (Q-bar, Delta) would give a horizontal
generating curve C_0 with
-(K_{Q-bar} + Delta) . C_0 <= 2 dim Q, while the pullback
of H has degree greater than this on C_0. Inductively the
base pullback and the morphism to the curve persist.
Lift each step by normalization in the upstairs function
field, using Stein factorization for the contractions.
Finite pullback preserves the relative negativity and
ampleness needed for flips. No divisors are extracted, and
the branch-canonical equality remains valid in codimension
one. This gives the finite-group equivariant MMP; invariant
Q-factoriality suffices upstairs (Prokhorov 2021, Thms
3.3.2 and 3.4.2). The steps factor over the upstairs curve
as well, by normalization in its function field. They
preserve dlt for the pair with the special fibre added.
Special-fibre multiplicities remain one because there is no
extraction. The klt total space is Cohen-Macaulay, so its
Cartier fibre has no embedded associated components; generic
reducedness therefore makes the fibre reduced
scheme-theoretically. Relative semiampleness pulls back from
the downstairs good model.
Step B2 (degree-one log generator on the total special
fibre). The logarithmic divisor of eta ^ du/u on Y is
effective near T: this held on the semistable model and the
program extracts no divisors. The generic fibre has Kodaira
dimension zero. Consequently the semiample contraction of
K_Y has zero-dimensional image on the generic fibre; its
Stein image over the curve is the curve itself. Thus K_Y,
and also K_Y + T, is rationally linearly trivial over a
neighbourhood of c. The effective log divisor has no
horizontal part, since its restriction to the generic fibre
is effective and rationally trivial. A principalization of
a multiple uses a rational function with vertical divisor,
hence a base function. The log divisor is therefore aT for
some a in Q_{>=0}. If a > 0 then every weight quotient is
at least a: it is a plus the log discrepancy of (Y, T)
divided by fibre multiplicity. This contradicts
w_u(eta) = 0. Hence
Div_log(eta ^ du/u) = 0.
In particular K_Y + T, and therefore K_Y, is Cartier near T.
Step B3 (adjunction to a prime component of T). Adjunction
makes T Gorenstein with trivial dualizing line generated by
the residue of the previous equation. The transformation
acts on that generator by lambda: the residue of its action
on du/u is one. If a prime component S of T is preserved by
a power of the transformation, the residue on S generates
K_S + Theta_S, where Theta_S is its different. Dlt
adjunction makes (S, Theta_S) normal lc and effective.
Moreover Theta_S is integral, since the divisor of this
degree-one rational form is integral and equals -Theta_S.
Its coefficients are at most one, so it is reduced. The
character on this component generator is the corresponding
power of lambda.
Step B4 (find a preserved component). The construction needs a bounded
power of the transformation that preserves some prime
component S of T. The family is flat over the smooth curve.
Its central fibre is slc by Step B1, and its nearby fibres
are klt after shrinking; these fibres are Du Bois.
Cohomology and base change for proper flat Du Bois
families makes h^j(O_{Y_t}) locally constant
(Kollar, Families 2023, Cor. 2.64). The nearby fibres are
birational to the block and have rational singularities.
Their structure-sheaf cohomology is therefore that of a
smooth resolution of the block. Extension of differential
forms on klt spaces, together with Hodge symmetry,
identifies these dimensions with the form algebra of the
block (GKKP 2011).
GKKP 2011: https://arxiv.org/abs/1003.2913
For an irreducible Calabi-Yau block, the only nonzero
groups H^j(T, O_T) occur in degrees 0 and s, and each has
dimension one. Serre duality and the log generator give the
alternating trace
1 + (-1)^s lambda^{-1}.
If lambda has order greater than two, this is nonzero. In
the symplectic case there is one dimension in each even
degree and none in odd degree. For some power of the
transformation of exponent at most s + 1 the trace is
nonzero: if the power sums of the finitely many nonzero
eigenvalues all vanished through their number, Newton's
identities would force their product to vanish.
The coherent Lefschetz theorem (Donovan 1969) implies that
a finite-order transformation with nonzero alternating trace
has a fixed point on a projective scheme; the vanishing
statement when there are no fixed points applies to the
possibly singular projective scheme T, for example by
equivariant embedding and coherent Lefschetz localization.
At a fixed point at most s+1 components of T meet:
intersections of components of a dlt boundary are lc
strata, with the generic SNC codimension description. A
further power of exponent dividing (s+1)! therefore
preserves a prime component through that point.
Step B5 (bound the character on S). The pair
(S, Theta_S) is a normal lc pair of dimension s < n with a
degree-one log generator (Step B3). Lemma 3.4 with L_s
(induction) bounds the order of its character. This bounds
the order of lambda as well.
Step B6 (assemble). Taking a common multiple over the
abelian bound N_ab(n) of Case A and the nonabelian bounds of
Step B5 over all s < n proves the lemma. QED.
Completion of the denominator bound.
Choose N = N(n) from Lemma 4.3 and set p = p_0 b N with
b = n!. The calculation preceding the lemma (Step 6 of Sec.
4.3) shows that p clears alpha + t at every prime divisor of
W. Thus every coefficient of p M_W is integral. Since W is
smooth, this divisor is Cartier. The b-divisor already
descends to W by qualitative moduli theory, so p M is
b-Cartier. This completes the proof of Prop. 4.1.
Big bases and their complete section fields.
Corollary 4.4 (Big base). Under the hypotheses of Prop. 4.1,
suppose D_Z is big. There is a positive integer m = m(n),
divisible by p_0, such that |m K_X| is nonempty and its
section-ratio field inside C(X) is exactly C(Z).
Proof. Take a smooth determination tau : W -> Z on which
p M_W is nef Cartier, and resolve the boundary support. Let
Gamma be the strict transform of B_Z plus the reduced
exceptional divisor. Generalized klt gives
K_W + Gamma + M_W = tau^* D_Z + E, E >= 0, exc.
This is a big adjoint for a log smooth lc pair whose
coefficients belong to a fixed DCC set, with a fixed Cartier
denominator for the nef part. Polarized effective
birationality (Birkar-Zhang 2016, Thm 1.3) gives a uniform
degree whose rounded system is birational. Such sections
push forward as rational sections of the rounded system of
D_Z, because the two divisors agree at the nonexceptional
generic divisorial points. Thus a uniform rounded system
downstairs is birational. Replacing its degree by a common
multiple preserves its section ratios and makes the degree
divisible by p_0.
For every m divisible by p_0, the exact identity is
H^0(X, O_X(m K_X)) =
psi^{m/p_0} f^* H^0(Z, O_Z(floor(m D_Z))),
where both sides are viewed as divisorial spaces of rational
functions with compatible canonical representatives. One
inclusion follows by pulling back an effective rational
Cartier divisor. For the converse, divide an upstairs
section by the common factor psi^{m/p_0}. Its restriction
to the projective generic fibre is a regular function,
hence belongs to C(Z). Checking orders over each prime of Z
shows that Div(h) + m D_Z >= 0, equivalently
Div(h) + floor(m D_Z) >= 0. This proves the identity. The
birational system on Z then gives precisely its full
function field, not merely a finite subfield. QED.
Torsion over rationally connected bases.
The next statement bounds indices of
varieties that admit a nontrivial rational image with
rationally connected base. Unlike Prop. 4.1, it gives an
integral principal multiple of the generalized adjoint, not
just a bound for its Cartier index.
Proposition 4.5 (Torsion on RC bases). Fix a dimension d, a
rational DCC set I in [0,1], and a positive integer p.
There is an integer l = l(d, I, p) > 0 with the following
property. Let Z be a normal projective complex variety of
dimension d with rationally connected smooth projective
resolution. Suppose (Z, B + M_Z) is generalized klt,
B >= 0 has coefficients in I, the b-nef data satisfy
p M b-Cartier, and
D = K_Z + B + M_Z ~_Q 0.
Then l D is an integral principal divisor.
Proof. This is the generalized torsion argument of R4,
Prop. 7.1, checked to avoid the dimension-four restriction.
Two uniform integers are needed: one making D integral, and
one killing the resulting torsion class in the Weil divisor
class group Cl(Z).
Step 1 (extraction and global ACC). Take a sufficiently
high log resolution g:W -> Z on which the nef data descend
and, if a valuation E is marked, on which E appears. Write
K_W + B_W + M_W = g^* D. Then B_W is sub-klt and need not
be effective. Choose an effective exceptional Cartier F
with -F relatively ample; build it as a suitable positive
combination of successive blowup divisors of a resolution.
If E has generalized log discrepancy a in (0,1) with
respect to (Z, B + M), choose delta > 0 small enough that
B_W + delta F remains sub-klt and
delta mult_E F < a. For an ample Cartier A on Z, choose a
general effective rational divisor
Lambda ~_Q M_W + g^* A - delta F (general very ample
divided). Then (W, B_W + delta F + Lambda) is sub-klt.
Pushing down and comparing rational principal divisors
shows that Xi = B + g_*Lambda is an effective ordinary klt
boundary on Z, with discrepancy at E of
a - delta mult_E F in (0,1).
Apply the extraction theorem (BCHM 2010, Cor. 1.4.3) to
(Z, Xi): it extracts exactly E when E is exceptional, or
gives a small Q-factorialization when the marked list is
empty. If the marked valuation is already a prime on Z,
use the empty list. Restore the original generalized data
on this model Z^+; the boundary is effective and the
generalized adjoint remains rationally trivial and the
nef denominator remains p.
If the nef data are numerically nontrivial, write them as
the single nef Cartier term p M_W with weight 1/p.
Generalized global ACC (Birkar-Zhang 2016, Thm 1.6)
applies. If they are numerically trivial, take a common
higher model pi':W' -> Z^+ determining the nef data. Since
Z^+ is Q-factorial, its nef trace is rational Cartier. The
difference M_{W'} - pi'^* M_{Z^+} is exceptional and
numerically trivial over Z^+. Negativity applied to it and
its negative shows the difference is zero. Generalized
discrepancies are then ordinary discrepancies, and
ordinary global ACC (HMX 2014, Thm 1.5) applies instead.
Applied first with no marked valuation, this places the
original boundary coefficients in a fixed finite subset
I_0 of I.
The same argument gives a uniform eps > 0 for generalized
eps-lc singularities: otherwise choose marked valuations
whose discrepancies decrease strictly to zero and extract
them; the new coefficients 1 - a_i form a strictly
increasing sequence, so adjoining them to I is still a
DCC set, and global ACC would force them into a finite
set, contradiction.
Step 2 (bounded topology, bounded class-group torsion).
Birkar's boundedness theorem (Birkar 2025, Thm 1.7) applies
in every fixed dimension: the varieties are projective and
rationally connected, their generalized adjoints are real-
linearly trivial, and they are uniformly generalized eps-lc.
Consequently they are bounded up to isomorphism in
codimension one. The bounded comparison models can be
normalized, still in a bounded family and still isomorphic
to Z in codimension one.
Put U = Z_reg. The groups H_1(U(C), Z) are finitely
generated and range over only finitely many isomorphism
types. First, H_1(U(C), Z) is invariant under isomorphism
in codimension one of normal varieties: on the regular
loci, the common open is obtained by removing subsets of
complex codimension at least two; after stratifying,
transversality moves paths and homotopies off every
stratum, whose real codimension is at least four; the
fundamental group is unchanged. Second, stratify a
finite-type parameter space for the bounded comparison
family until it is flat. The relative smooth locus then
has fibres equal to the regular loci of the selected
normal fibres. Cover the parameter strata by finitely many
affine opens and embed the family projectively over each.
Complex projective space is a compact real algebraic set
via its realization as rank-one Hermitian projection
matrices. Thus the entire relative smooth locus maps
semialgebraically to each parameter open. Semialgebraic
triviality gives a finite partition on which these fibres
are homeomorphic, and semialgebraic triangulation gives
each the homotopy type of a finite complex (Delfs-
Knebusch 1982). This proves the asserted finite list of
finitely generated homology groups.
For each individual Z, the class group Cl(Z) is finitely
generated: a rationally connected smooth resolution Y has
Pic^0(Y) = 0, so Pic(Y) is finitely generated by
Neron-Severi finiteness; divisor pushforward surjects onto
Cl(Z). Since Z is normal and projective,
Gamma(U, O_U^*) = C^* and Pic(U) = Cl(Z). Kummer theory
and comparison for finite etale covers (SGA 1) give, for
every a >= 1,
Cl(Z)[a] = H^1_et(U, mu_a)
= Hom(H_1(U(C), Z), mu_a(C)).
The left side has bounded order as a varies for this
fixed Z. Thus H_1(U(C), Z) has no free summand and is
finite. There are only finitely many such finite groups in
the bounded topological list. A common multiple T of their
exponents kills every torsion class in every Cl(Z), by the
Kummer identification.
Step 3 (clear the divisor). The trace p M_Z is an integral
Weil divisor, being the pushforward of Cartier data. Choose
q divisible by p and by the denominators of the finite set
I_0. Then q D is integral. Its class in Cl(Z) is torsion
because D ~_Q 0. Therefore T q D is principal. Taking
l = T q proves the proposition. QED.
Corollary 4.6 (Relative RC torsion). Under the hypotheses
of Prop. 4.1, suppose K_X ~_Q 0 and a smooth projective
resolution of Z is rationally connected. There is an
integer r = r(n) > 0 such that r K_X is an integral
principal divisor.
Proof. The divisor D_Z in the exact canonical bundle
formula is rationally linearly trivial. Indeed a Cartier
multiple whose pullback is trivial is itself trivial by
the projection formula and f_* O_X = O_Z. Proposition 4.5
gives a uniform l and a function v in C(Z)^* with
l D_Z = Div(v). Choose a common multiple r of l and p_0.
Then
r K_X = Div( (v o f)^{r/l} psi^{-r/p_0} ).
All constants depend only on n, since the dimensions of Z
range over a fixed finite set and Prop. 4.1 supplies a
fixed coefficient set and nef denominator. QED.
The remaining index sequence.
Proposition 5.1 (The remaining index sequence). If the
canonical indices of projective klt n-folds with K_X ~_Q 0
are unbounded, there is a sequence of projective Q-factorial
terminal n-folds V_i, of canonical indices r_i -> infinity,
satisfying the following properties.
(1) A smooth projective resolution of V_i has irregularity
zero, and Bir(V_i) is countable.
(2) Every positive-dimensional rational image of V_i of
dimension less than n has rationally connected smooth
projective resolution.
(3) Every effective Weil divisor on V_i with positive
Iitaka dimension is big.
(4) Let pi_i : Y_i -> V_i be the cyclic index cover, with
its deck group G_i. Then Y_i is terminal and
K_{Y_i} ~ 0. Every G_i-equivariant rational fibration
of Y_i with positive-dimensional base and fibre has
general-type smooth geometric generic fibre. The same
assertion holds for rational fibrations of V_i, with
trivial group action.
Here a rational fibration has geometrically integral generic
fibre, equivalently has a relatively algebraically closed
base function field.
Product covers and rational images (Prop. 5.1 (1), (2)).
The product cases come first. Take the quasi-etale
Beauville-Bogomolov cover
P = A x prod_j B_j -> X
from Lemma 3.6, counting a positive-dimensional abelian factor
as one block. Suppose that there are at least two
positive-dimensional blocks.
Step 1 (product covers bound the index). On the normal Galois
comparison cover R -> X, Lemma 3.6 supplies a subgroup of
bounded index preserving the individual factor fields. Its
transformations on the associated normal Stein factors R_j are
regular (Step 5 of Lemma 3.6), and each dim R_j < n. Their
volume characters have uniformly bounded order by Lemma 3.4
applied with empty boundary. The volume form on R is the
product of the pulled-back factor forms. Consequently its
character on that subgroup has order bounded by the lcm of
the factor bounds, and multiplication by the lcm of the
possible subgroup indices kills the character of the whole
Galois group.
An invariant power of the volume form descends to X and
generates its divisorial pluricanonical sheaf: the cover is
etale at generic points of prime divisors, so descent
introduces no zero or pole there. Conversely, a generating
pluriform on X pulls back to an invariant power of the volume
form. Thus the order of the volume character is exactly the
canonical index, bounded in the product case.
Step 2 (purely abelian case). If P = A is purely abelian, the
Galois comparison cover R -> A is etale by purity and R is
again abelian. The canonical character is an eigenvalue on
H^n(R, Z), of rank binom(2n, n). If that eigenvalue has order
e, its cyclotomic polynomial divides the integral
characteristic polynomial, so
phi(e) <= binom(2n, n),
where phi is Euler. There are finitely many such e. Their lcm
bounds the canonical index in this case.
Only the case of a single nonabelian block can occur in an
unbounded-index sequence. Its rational-image property is
recorded separately, since it supplies the geometric reason
the relative index theorem applies later.
Lemma 5.2 (Rational images of a single block). Suppose that
a projective klt n-fold X with K_X ~_Q 0 has a finite
quasi-etale cover by one irreducible Calabi-Yau or
irreducible holomorphic symplectic block P. Then a smooth
resolution of X has irregularity zero, and every
positive-dimensional rational image of dimension less than
n has rationally connected smooth projective resolution.
Proof. Step 1 (irregularity zero). A regular one-form on a
smooth projective model of X pulls back injectively to a
reflexive one-form on P. Indeed, the rational map from the
normal variety P to the proper smooth target is defined at
every codimension-one point. The pulled-back form is regular
there and extends reflexively. The block has no reflexive
one-forms, so Hodge symmetry gives irregularity zero.
Step 2 (no non-uniruled rational image). Suppose f:X - -> T
is a dominant rational map to a smooth projective T of
dimension 0 < j < n that is not rationally connected. By
BDPP, K_T is pseudo-effective. By LA nonvanishing, there is
a nonzero eta in H^0(T, mK_T) for some m > 0. Resolving the
composite map from P gives u:W -> P and f:W -> T with W
smooth, and a nonzero section
s = f^* eta in H^0(W, (Omega^j_W)^{otensor m}).
At the generic point it has the form a beta^{otensor m},
with a in C(P)^* and beta a nonzero decomposable rational
j-form (pulled back from a rational top form on T).
BDPP 2013: https://arxiv.org/abs/math/0405285
Step 3 (divisorial locus of s is exceptional). Let
H = u^* H_P for an ample Cartier H_P on P. Canonicality and
K_P ~ 0 imply K_W ~_Q E with E effective and u-exceptional,
so K_W H^{n-1} = 0. Generic semipositivity of the cotangent
bundle (Campana-Paun 2015) implies every torsion-free
quotient of V = (Omega^j_W)^{otensor m} has nonnegative
degree against H^{n-1}. Details: amplify with H + eps H_0,
restrict to a high complete-intersection curve, apply
Mehta-Ramanathan (1982) to make the Harder-Narasimhan
factors of Omega^1_W semistable of nonnegative slope,
tensor (Ramanan-Ramanathan 1984) preserves semistability,
exterior powers are quotients of tensor powers. Take the
limit eps -> 0.
Campana-Paun 2015: https://arxiv.org/abs/1303.3169
The saturation of the line generated by s is O_W(Z), with
Z >= 0 the divisorial zero divisor. Since c_1(V) is a
positive multiple of K_W, the quotient inequality gives
Z H^{n-1} <= 0. Every nonexceptional prime has strictly
positive H^{n-1}-degree, so Z is exceptional.
Step 4 (reflexive j-form on a quasi-etale cover). At the
generic point of a prime divisor of P, write beta in a
regular frame and let b be the minimum order of its
coefficients. The absence of zeros and poles of s there
gives ord(a) + m b = 0. All coefficients of Div_P(a) are
divisible by m. Normalize P in a field component obtained by
adjoining an m-th root z of a. This is a finite quasi-etale
cover: over a divisorial discrete valuation, removing the
uniformizer to its m-divisible order leaves the equation
for a root of a unit, which is unramified in characteristic
zero. The rational form z beta has no divisorial pole, hence
is a nonzero reflexive j-form on the cover. It remains
decomposable.
Step 5 (contradicts the form algebra of P). In the
Calabi-Yau case there are no intermediate-degree reflexive
forms on any connected quasi-etale cover. In the symplectic
case the only possibilities are scalar multiples of
sigma^q with 0 < 2q < n, where sigma is generically
nondegenerate. Contraction into sigma^q is injective on the
nondegenerate locus, whereas a nonzero decomposable j-form
with j < n has a contraction kernel of dimension n - j > 0.
These forms cannot agree.
Step 6 (rational connectedness). A smooth resolution of any
proper rational image has a maximal rationally connected
quotient whose smooth projective base is non-uniruled
(GHS 2003). A positive-dimensional such base would be a
forbidden rational image of X, so the quotient is a point.
That proves rational connectedness. QED.
GHS 2003: https://arxiv.org/abs/math/0109220
Terminality, countability of Bir, moving divisors big,.
general-type fibres (Prop. 5.1 (1), (3), (4))
Step A (terminalize). First, discard all noncanonical X from
the unbounded-index sequence. Reason: if X is not canonical,
choose a smooth resolution W -> X exhibiting a negative
discrepancy. Then K_W ~_Q E for an effective E with a
negative coefficient. The canonical class of W is not
pseudo-effective: otherwise LA would give a nonzero
pluricanonical section, hence an effective divisor rationally
linearly equivalent to E. Pushing that relation down to X
shows its rational function has effective principal divisor;
by projectivity, that function is constant, and the effective
divisor equals E, contradiction. BDPP now makes W uniruled.
Its maximal rationally connected quotient has dimension less
than n, and Lemma 5.2 forces its base to be a point.
Proposition 4.5 with zero boundary and zero nef part bounds
the index of X. So the unbounded sequence contains only
canonical varieties.
Once X is canonical, extract a projective crepant Q-factorial
terminalization by BCHM's extraction theorem. On a fixed
resolution there are only finitely many exceptional
discrepancy-zero divisors: any valuation exceptional over
that smooth resolution has positive discrepancy, adding the
pullback of its effective discrepancy divisor preserves
positivity. Extracting precisely those zero-discrepancy
divisors, together with a small Q-factorialization when
necessary, gives a Q-factorial terminal model V -> X that
is crepant. Crepant pullback and divisor pushforward
preserve the exact principal multiples of the canonical
divisor. Write the resulting terminal varieties as V_i.
Step B (countability of Bir). Lemma 8.2 of U4 applies. Let
W -> V be a smooth resolution. The Picard group of W is
countable by the exponential sequence and H^1(O_W) = 0
(irregularity zero from Lemma 5.2); its surjection onto
Cl(V_i) makes Cl(V_i) countable.
Every birational self-map of V_i is an isomorphism in
codimension one. On a common resolution the two pullbacks of
a canonical trivialization are proportional, and terminality
identifies their zero supports with the respective
exceptional divisors, so those supports coincide. Group the
self-maps by the Weil class of the strict transform of a
fixed very ample divisor. Within one class the maps differ
by a projective linear stabilizer of the corresponding
embedding. The effective identity component of that
stabilizer is trivial by Matsumura's theorem (Matsumura 1963,
Cor. 1) excluding a positive-dimensional linear algebraic
group of birational transformations on a smooth complete
variety with a nonzero pluricanonical system. Each class
therefore contributes only finitely many maps. Hence Bir(V_i)
is countable.
Step C (moving divisors are big). Suppose that an effective
Weil divisor J on some V_i has 0 < kappa(V_i, J) < n. Since
V_i is Q-factorial, a small positive rational delta with
(V_i, delta J) klt may be chosen. LA gives a good model of this
pair and a semiample contraction f:V_i' -> Z with
0 < dim Z < n.
The zero-boundary canonical divisors remain crepant during
this program. Fix r K_{V_i} = Div(h) with compatible
canonical representatives. The same relation pushes through
each divisorial contraction and through the codimension-one
identification in each flip. Since the models are
Q-factorial, their canonical pullbacks to a common resolution
are all r^{-1}Div(h). Thus V_i' is still klt, K_{V_i'} ~_Q 0,
and its canonical index is unchanged.
Apply Prop. 4.1 to (V_i', 0) -> Z. Its generalized adjoint
D_Z is rationally linearly trivial. The base Z is a rational
image of V_i, so its smooth resolution is rationally connected
by Lemma 5.2. Proposition 4.5 therefore gives a uniform
principal multiple of D_Z and hence of K_{V_i'}. The bound
depends on n alone (not on J or delta) because the canonical
bundle formula was applied with boundary zero. Discarding
finitely many terms of the sequence makes this impossible.
So every effective Weil divisor of positive Iitaka dimension
on each remaining V_i is big.
Step D (general-type fibres). Combined with Step A and Step C
this gives (3) and (4) via the following.
Lemma 5.3 (General-type fibres). Let V be projective,
Q-factorial and terminal with K_V ~_Q 0, and suppose that
every effective Weil divisor of positive Iitaka dimension
is big. Let pi : Y -> V be its cyclic index cover with
deck group G. Every intermediate G-equivariant rational
fibration of Y has general-type smooth geometric generic
fibre. The same is true on V without a group action.
Proof. Step 1 (setup). The cover is connected by minimality
of the index and is quasi-etale. The finite discrepancy
formula makes Y terminal: an exceptional divisorial
valuation over Y restricts to an exceptional valuation over
V, and its log discrepancy is the positive ramification
index times a number greater than one. Also K_Y ~ 0.
Step 2 (descend to V). Let C(T) subset C(Y) be the
relatively algebraically closed base field of an equivariant
fibration. Its invariant field C(T)^G lies in C(V), has the
same transcendence degree, and is relatively algebraically
closed there: an element of C(V) algebraic over C(T)^G is
algebraic over C(T), hence lies in C(T) and then in its
invariant field. This gives a descended intermediate rational
fibration V - -> Z.
Step 3 (a big pullback divisor). Resolve the maps and
bases, obtaining u:U -> Y with U smooth, a morphism
f:U -> T' to the unquotiented base, and a morphism
g:U -> Z' constant on the generic fibres of f. Pull back a
general very ample hyperplane H on Z' by the rational map
from V, and close up the divisor to obtain D on V. The
properness of Z' ensures this map is defined in codimension
one. The hyperplane pullbacks form a moving linear system
with nonconstant section ratio; thus kappa(V, D) > 0, hence
D is big by Step C of Prop. 5.1.
Step 4 (evaluate on the fibre). For q = pi o u there is
q^* D = g^* H + E with E >= 0 exceptional over Y. At every
nonexceptional prime the two pullbacks agree. After fixing
the resolution, choose H not to contain the generic image of
any of the finitely many exceptional primes. Its pullback
has zero coefficient at each such prime, whereas q^* D is
effective. That gives the effectivity of E.
Step 5 (big canonical class on the fibre). Let F be the
smooth geometric generic fibre of f. The class q^* D|_F is
big: decompose q^* D into an ample rational class and an
effective rational class, and restrict to the generic fibre,
which is not contained in the effective support. Since
g^* H|_F ~ 0, the class E|_F is big. Terminality gives
K_U ~_Q sum_a d_a E_a, d_a > 0,
where the sum includes every u-exceptional prime. Some
positive rational multiple of E is coefficientwise dominated
by this sum. Restriction and adjunction therefore make
K_F = K_U|_F big. The proof for V is the same with the
finite cover omitted. QED.
This completes the proof of Prop. 5.1.
Scalar invariants.
For a big, nef, semiample rational Cartier divisor P on an
integral projective variety V and a smooth point x in V,
define
gamma(P; V, x) = sup over k>0 sufficiently divisible and
0 != s in H^0(V-tilde, k mu^* P) of
ord_x(s) / k,
where mu : V-tilde -> V is a smooth projective resolution
that is an isomorphism near x. Proper birational pushforward
and the projection formula make this independent of the
resolution. Write gamma(P; V) for its very general value.
For an ample P also put
eps(P; x) = inf over integral curves C through x of
(P . C) / mult_x(C).
Use its very general value, denoted eps(P).
Both invariants are homogeneous under positive rational
rescaling. If d = dim V, the asymptotic section count and the
number of jet conditions at a smooth point give
gamma(P; V) >= (P^d)^{1/d}.
Indeed, h^0(kP) = P^d k^d / d! + O(k^{d-1}) in divisible
degrees, whereas vanishing to order at least l imposes at most
binom(d + l - 1, d) conditions.
Proposition 5.4 (Scalar estimates). Assume I_j for j < n.
There are constants C_n, c_n > 0 with the following
property. Let V be a normal projective canonical n-fold
with K_V ~_Q 0, and let a finite group G act on V. Suppose
that every G-equivariant rational fibration with
geometrically integral generic fibre and positive-
dimensional base and fibre has general-type smooth geometric
generic fibre. If P is an ample rational Cartier divisor
with g^* P ~_Q P for every g in G, then
gamma(P; V) <= C_n (P^n)^{1/n},
eps(P) >= c_n (P^n)^{1/n}.
The constants are independent of V, G, and the denominator
of P.
The proof has two parts. First, the order bound is proved
(the chain and tracking discussion). An excessive order produces smaller-dimensional
covering members with small polarised canonical degree; the
Iitaka fibres of those members have small adjoint volume;
iterating strictly decreases dimension until the order
becomes small. Their chain quotient then contradicts the
general-type hypothesis. Then (the small-volume argument) proves the lower
curve bound by thickening of a small-degree fibre.
The chain and tracking inputs.
The dimension-independent inputs from U4 are stated in the form
used here.
Definition (marked covering family). A marked covering family
on V consists of an integral parameter space S, a dominant
mark map u : S -> V, and a reduced incidence I subset S x V.
Its fibre C_b over a geometric generic parameter b is
integral of positive dimension and contains u(b); the second
projection evaluates its points in V. One may replace S by a
dense open or a dominant extension to specify geometric
components or auxiliary data.
Fix a field of definition k_* for these data. A generic
movement from a geometric generic mark a chooses a generic
parameter b on a geometric component of u^{-1}(a), and then a
generic endpoint on C_b. At each step genericity is over the
field generated by k_* and the entire preceding path,
enlarged to define the chosen geometric component. New
parameters remain generic after the earlier marks, parameters,
and endpoints have been fixed.
Lemma 5.5 (Generic chains and order bounds; U4, Lemmas
5.1 and 5.4 and Corollary 5.2). For a finite nonempty list
of marked covering families on an integral projective
variety in characteristic zero, there is a rational
fibration xi : V - -> M with geometrically integral generic
fibre. If its generic leaf has dimension h, then
1 <= h <= dim V, and every generic member is contained in
its leaf. From a geometric generic mark a, a path of at
most h generic movements has endpoint generic in that leaf
over k_*(a)-bar.
The construction is equivariant for any finite group
preserving the list. A dominant rational map carrying upper
movements to lower movements or stationary steps carries
upper leaves into lower leaves. For a finite group quotient
V -> V', the images of the leaves of an invariant list are
the generic leaves of a rational fibration on V'.
Let P be big, nef, and semiample. Suppose each geometric
generic marked member satisfies either of the following
bounds, with the same number B:
(a) its mark u(b) is smooth on C_b and
gamma(P|_{C_b}; C_b, u(b)) <= B;
(b) it is a curve and
P . C_b / mult_{u(b)} C_b <= B.
Let H be a smooth projective resolution of the geometric
generic leaf, and retain pullback notation for P. Then
gamma(P; H) <= hB,
P^h H <= (hB)^h.
The bounds concern the entire section spaces on a
resolution of the member, not merely restrictions of
ambient sections, which matters when the lemma is applied
to successive Iitaka fibres.
Lemma 5.6 (Tracking and covering; U4, Lemma 6.3 and 6.4).
Let Z be projective, Q-factorial, and klt of dimension
d >= 2, and let N be big, nef, and semiample. Suppose that
d + 1 positive rational numbers with consecutive spacing
Delta > 0 lie strictly between (N^d)^{1/d} and gamma(N; Z).
Then there is a covering family of integral subvarieties S
of dimension 0 < e < d whose smooth geometric generic
resolution S^* satisfies
N^e S <= (2/Delta)^{d-e} N^d,
K_{S^*} (N|_{S^*})^{e-1}
<= (K_Z + (2d/Delta) N)|_S (N|_S)^{e-1}.
In addition, if a projective variety has a smooth model
with nonnegative Kodaira dimension, then a smooth
projective model of the geometric generic member of any
covering family of integral subvarieties has nonnegative
Kodaira dimension. The same statement holds with "general
type" in place of "nonnegative Kodaira dimension".
The tracking assertion is U4, Lemma 6.3; its proof combines
the parameter-differentiation method of Ein-Kuchle-
Lazarsfeld (1995, Prop. 2.3) with numerical subadjunction.
The covering assertion is U4, Lemma 6.4: cut the parameter
space so that incidence evaluation is generically finite
over the ambient variety, then resolve. The formula
K_I = q^* K_Z + R with R >= 0, restricted to a generic
parameter fibre, gives both assertions about canonical
dimension.
Ein-Kuchle-Lazarsfeld 1995: https://arxiv.org/abs/alg-geom/9408003
All applications of these lemmas are at geometric generic
data. Finite constructions there spread after a dominant
parameter extension. A previously generic parameter is never
specialised. Countable fields of definition accommodate
simultaneously the jet conditions in all divisible degrees.
This also allows the geometric generic constructions to be
realized over C when applying the model and positivity
theorems.
Small adjoint volume on Iitaka fibres.
The next lemma is the additional scalar input needed beyond
dimension four. The corresponding lemma of U4 treats members
of dimension at most three. Here weak positivity is combined
with flattening so that the Iitaka base may have any
dimension less than that of the member.
Lemma 5.7 (Small adjoint volume on Iitaka fibres). Fix
1 <= e < n and C' > 0. Let (W_i, L_i) be a sequence of
smooth projective e-folds with kappa(W_i) >= 0 and big,
nef, semiample rational Cartier divisors L_i, such that
L_i^e -> 0,
K_{W_i} L_i^{e-1} <= C' L_i^e.
Assume I_e. After a subsequence and resolution of the
Iitaka fibrations, their smooth geometric generic fibres
U_i have fixed positive dimension, satisfy kappa(U_i) = 0,
and obey
vol(K_{U_i} + L_i|_{U_i}) -> 0.
The restricted polarizations are big, nef, and semiample.
If the Iitaka base is a point, U_i denotes the entire
smooth model.
Proof. Suppress the sequence index. Step 1 (adjoint volume
upper bound). For every fixed positive rational a,
vol(aK_W + L) <= (aC' + 1)^e L^e. (5.7a)
Proof of (5.7a): choose a divided general member of a
sufficiently divisible free multiple of L/a, giving a klt
effective boundary Gamma ~_Q L/a. The big klt adjoint
K_W + Gamma has a good model by BCHM. On a common resolution
let P_a be a times the pullback of its semiample adjoint.
The negative-map comparison gives
aK_W + L ~_Q P_a + F_a, F_a >= 0,
with F_a exceptional over that model. Thus P_a^e = vol(aK_W+L).
With L pulled back to this resolution, nef
mixed-intersection inequalities give
(aC' + 1) L^e >= (aK_W + L) L^{e-1}
>= P_a L^{e-1}
>= (P_a^e)^{1/e} (L^e)^{(e-1)/e},
proving (5.7a). Replacing W by a higher smooth model and
retaining the pullback of L changes neither this volume nor
the intersection estimates: additional canonical terms are
effective exceptional divisors.
Step 2 (kappa along the sequence). By I_e, a fixed integer
m_0 defines every Iitaka fibration under consideration.
Pass to a subsequence where j = kappa(W) is constant.
The case j = e is impossible for large indices: the
resolved system |m_0 K_W| would have an integral free
moving part with generically finite map and positive
integral top intersection, so vol(K_W) >= m_0^{-e},
contrary to (5.7a). If j = 0, the assertion follows directly
from (5.7a) with a = 1.
Step 3 (0 < j < e). Resolve the Iitaka system and its base,
obtaining a morphism with connected fibres between smooth
projective varieties
f_0 : W_0 -> T_0.
Write L_0 for the pulled-back polarization. The hyperplane
class of the system's image pulls back to a big
basepoint-free Cartier divisor H_0 on T_0, and
f_0^* H_0 <= m_0 K_{W_0}, H_0^j >= 1. (5.7b)
The smooth geometric generic fibre U has Kodaira dimension
zero by the Iitaka-fibration theorem. The restriction
L_0|_U is big: the moving fibres meet the generically finite
locus of the semiample map of L_0. Nefness and semiampleness
restrict as well.
Step 4 (a flat model). Apply flattening by a modification of
T_0, and choose a smooth projective T dominating that
modification. Let X be the normalization of the flat
principal family after this base change, and resolve X by
W without changing its smooth generic fibre, still
denoted by U. The resulting maps fit into
W -> X -> W_0
|
v
T -> T_0
Write f : W -> T, rho : W -> W_0, L = rho^* L_0, and H
for the pullback of H_0 to T. Both horizontal maps in the
top row are proper and birational. The morphism X -> T is
equidimensional: flatness gives the generic fibre dimension
before normalization, and finite normalization preserves
it. Hence a prime divisor E on W whose image in T has
codimension at least two is exceptional over X. Its image
in X has dimension at most e - 2, so its image in W_0 also
has codimension at least two. Thus E is rho-exceptional.
Relation (5.7b) gives f^* H <= m_0 K_W on the higher smooth
model, and H^j = H_0^j >= 1.
Step 5 (relative adjoint domination). Choose a general
projection of the morphism defined by H to P^j. It is a
generically finite morphism: a general (j+1)-dimensional
linear subsystem has no common zero on the j-dimensional
image. Its ramification divisor yields
K_T + (j+1) H >= 0. Therefore
K_{W/T} + L + 2 f^* H <= (1 + (j+3) m_0) K_W + L. (5.7c)
Step 6 (weak positivity). Choose, separately for this
fibration, a sufficiently divisible l >= 1 so that the
l-th Veronese of the section ring of K_U + L|_U is
generated in degree one. This follows from finite generation
of the big klt adjoint ring, after representing L|_U by a
divided general free member. Enlarge l to clear all rational
divisors and equivalences used below. On W represent L by
an effective klt boundary. Twisted weak positivity (Fujino
2017, Thm 1.1) gives weak positivity of the torsion-free
direct image
E_l = f_* O_W( l (K_{W/T} + L) ).
There is b > 0 for which
(Sym^b E_l)^{**} x O_T(bH) is generated by global sections
at the generic point. A big H suffices for this consequence
of weak positivity: choose kH = A + F with A ample and
F >= 0, apply weak positivity with symmetric exponent
k beta and twist beta A, and multiply by the section of
beta F.
Step 7 (multiply to a global section). For every multiple q
of b, multiplication of those global sections, followed by
fibrewise multiplication in the adjoint ring, gives sections
over a big open of T of
q l (K_{W/T} + L) + q f^* H
spanning all degree-ql sections on the generic fibre. Remove
only the codimension-two sets where the relevant
torsion-free sheaves differ from their reflexive hulls or
are not locally free. The multiplication maps are defined
at every codimension-one point. Generation is needed only
at the generic point, so its possible failure along a
divisor does not require removing that divisor.
Step 8 (independent products). Choose a basis over C(T)
among these sections. Its size is
h^0(U, q l (K_U + L|_U)). Multiplying it by a basis of
H^0(T, q l H) gives linearly independent products over C: a
relation would first contradict the generic-fibre
independence over C(T) and then the independence of the
base sections. If necessary multiply by a fixed nonzero
section of (ql - q)H to place these products in degree ql
of the left side of (5.7c). Multiplying further by the
effective difference in that equation, after clearing
denominators, embeds them into sections of its right side
over the same big open.
Step 9 (push to W_0). Let T^circ subset T be the big open
just obtained. The resulting rational pluriforms are
regular on f^{-1}(T^circ). Every prime divisor on W in its
complement is rho-exceptional, by the flattening
construction. Thus the forms are regular at the generic
point of the strict transform of every prime divisor on
W_0. Push them to W_0 with their L_0-twist. They have no
divisorial pole and therefore extend, by smoothness, to
sections of
H^0(W_0, O_{W_0}(
q l ((1 + (j+3) m_0) K_{W_0} + L_0))).
Their independence is preserved by this birational
identification.
Step 10 (asymptotic count). Put g = e - j. With this
fibration and l, b fixed, let q tend to infinity through
sufficiently divisible multiples of b. The asymptotic
section counts on T and U give
vol((1 + (j+3) m_0) K_{W_0} + L_0)
>= (e! / (g! j!)) H^j vol(K_U + L|_U). (5.7d)
The left side tends to zero along the original sequence by
(5.7a), whereas H^j >= 1. This proves the conclusion. No
uniform choice of l or b is needed: the asymptotic estimate
is obtained separately for each fibration before taking the
sequence limit. QED.
The upper order bound.
Proof of the first inequality of Prop. 5.4.
The assertion in dimension one follows from degree, so
assume n >= 2. Suppose no constant C_n exists. Rational
rescaling gives a sequence with
P^n -> 0,
gamma(P; V) > 1.
Take small projective Q-factorializations and pull back P.
A dimension-decreasing construction is described. At a stage,
after passing to a subsequence, there is a fixed dimension d
and data (Z, N) such that:
(i) Z is projective, Q-factorial, and klt; N is big, nef,
and semiample; N^d -> 0;
(ii) a smooth model of Z has nonnegative Kodaira dimension;
(iii) K_Z <= a_0 N for a fixed positive rational a_0;
(iv) gamma(N; Z) has a positive lower bound.
These conditions hold initially by canonicality and
crepantness of the small modification. They cannot hold in
dimension one, because the order of a section is bounded by
its degree.
Step 1 (tracking with excess order). For d >= 2, choose d+1
fixed equally spaced positive rational levels below the
order lower bound. Eventually they lie above (N^d)^{1/d}.
Lemma 5.6 (tracking) gives a covering family with a smooth
projective model W of its geometric generic member, of
dimension 0 < e < d. Set L = N|_W. It is big, nef, and
semiample, and kappa(W) >= 0 by the covering assertion.
The tracking inequalities and restriction of K_Z <= a_0 N
give
L^e -> 0,
K_W L^{e-1} <= C' L^e
for a fixed C'.
Step 2 (small-volume Iitaka fibre). Lemma 5.7 (small volume)
gives a positive-dimensional Iitaka fibre U, possibly all of
W, with kappa(U) = 0 and
vol(K_U + L|_U) -> 0.
Choose an effective klt boundary Gamma ~_Q L|_U from a
divided general free member, and take a Q-factorial good
model Z' of K_U + Gamma. Let N' be its semiample big
adjoint. Then (N')^{dim Z'} -> 0 and K_{Z'} <= N'.
Step 3 (polarization comparison). On a common resolution
with maps x to U and y to Z', one has
x^* (L|_U) <= y^* N'. (5.7e)
Here is the fixed-part argument. Write
x^* (K_U + Gamma) = y^* N' + F, F >= 0,
with F exceptional over Z'. Choose an effective D ~_Q K_U,
which exists because kappa(U) = 0. In sufficiently divisible
degrees, add a general member of the free system of L|_U to
this representative. Its pullback must contain the fixed
part F. The general free member contains none of the finitely
many components of F, so F <= x^* D. Subtracting gives
(5.7e).
Step 4 (iterate or stop). If gamma(N'; Z') -> 0 on a
subsequence, stop. Otherwise pass to a subsequence with a
positive lower bound and repeat. All stage conditions
persist: the smooth-model Kodaira dimension of Z' is zero
because it is birational to U, and the dimension strictly
decreases. The impossibility of the stage conditions in
dimension one forces termination.
Step 5 (trace back to V). This produces covering members
A subset V of positive dimension whose smooth geometric
generic models satisfy
kappa(A^*) = 0,
gamma(P; A) -> 0. (5.7f)
The section-order comparison is made explicit as follows.
All intermediate constructions take place on geometric
generic members and spread after dominant parameter
extension. Fibres sweep the preceding member, and each
birational comparison is an isomorphism on an open met by
the next generic member. Composite members therefore map
birationally to their images. At every stage the effective
polarization comparison restricts, since a sweeping member
is not contained in its extra divisor. On common resolutions
it gives an injection from the complete section space of
the restricted preceding polarization into that of the next
one, in divisible degrees. Multiplication by the effective
difference preserves order at a generic point outside its
support. Proper birational pushforward preserves these
complete section spaces. Iteration compares the complete
spaces on a resolution of A with the final spaces whose
normalized order tends to zero. This proves (5.7f), not
merely an assertion about restrictions of ambient sections.
Step 6 (chain contradiction). Mark these members at generic
smooth points and include all their G-translates. Their
order bounds are unchanged because g^* P ~_Q P. Apply
Lemma 5.5 (chains). If the chain leaf had full dimension,
its order estimate would imply gamma(P; V) -> 0,
contradicting the chosen lower bound. The leaf has positive
dimension, so eventually it is the generic fibre of an
intermediate equivariant rational fibration.
Step 7 (leaf is covered by KD-zero members). That geometric
generic leaf is covered by the Kodaira-dimension-zero
members in (5.7f). Indeed every generic marked member is
contained in its chain leaf; after extension to the
geometric generic point of the quotient, the incidence
marks still dominate the leaf. These extensions preserve
the geometric generic members and their Kodaira dimension.
The covering assertion of Lemma 5.6 therefore prevents the
leaf from being general type. This contradicts the
hypothesis of Prop. 5.4 and proves the upper order bound.
QED.
The lower curve bound.
Proof of the second inequality of Prop. 5.4.
Rationally rescale so that 1 <= P^n <= 2. The upper estimate
now gives a uniform bound gamma(P; V) <= D_0. Suppose no
positive lower bound for eps(P) exists. Choose covering
marked curve families with
P . C / mult_x C -> 0
and include their G-translates. Such families are obtained
from curves through very general ambient marks by spreading
their geometric generic data; after shrinking, the generic
marked ratio satisfies the same bound.
Step 1 (chain leaves of positive dimension less than n).
Lemma 5.5 (chains) gives chain leaves of a fixed dimension
h on a subsequence, with degrees P^h H -> 0. Since P^n >= 1,
eventually 0 < h < n.
Step 2 (thickening of a very general fibre). Resolve the
chain fibration and choose a very general smooth fibre F of
dimension h. Write s = n - h and retain pullback notation
for P. Its normal bundle is trivial, because F is a fibre
of a smooth morphism near the chosen base point. For its
ideal I_F, the filtration of O / I_F^u has associated graded
terms Sym^j(I_F / I_F^2) for 0 <= j < u, of total rank
binom(u + s - 1, s). Consequently the rank of restriction
of global degree-l sections to the thickened fibre is at
most
binom(u + s - 1, s) h^0(F, lP|_F). (5.8a)
Step 3 (pick u to force a small section). Take u = ceil(D l)
with D > D_0. For each fixed fibration, in large divisible
degrees the leading coefficient in (5.8a) is
D^s / (s! h!) (P^h F) l^n.
It tends to zero along the sequence, while the ambient
section count has leading coefficient P^n / n! >= 1/n!.
Thus, first choosing a sufficiently late sequence member
and then a sufficiently large divisible l, there is a
nonzero ambient section vanishing in I_F^u.
Step 4 (contradiction). At any smooth point of F this
section has order at least u because I_F subset m_x. The
fibre and a point on it can be chosen very general in the
original birational isomorphism locus and outside all the
countably many order-exceptional sets. Birational pullback
preserves the ambient section spaces and their orders
there. The resulting normalized order is at least D > D_0,
contradicting the upper bound. Hence eps(P) is bounded
below by c_n (P^n)^{1/n} for a positive constant c_n. QED.
The diagonal jet bound.
For a divisor L on V, write L^{boxplus t} = sum_i pr_i^* L
on V^t. Fix b_0 > 0 rational with
3 (2 b_0)^n < 1/4,
b_0 (C_1 + 1) < 3/4.
The first inequality is what will make the diagonal rank loss
at most 1/4 of the total; the second is what will make the
off-diagonal determinant order at most 3/4.
Proposition 6.1 (Diagonal jet bound). Let V be a normal
projective canonical complex n-fold with n >= 2,
K_V ~_Q 0 of canonical index r. Let pi : Y -> V be its
canonical cyclic cover. Suppose L is ample rational Cartier
with 1 <= L^n <= 2 and gamma(L; V) <= C_1. For every integer
t >= 2 let Z_t = Y^t / mu_{r,diag}, theta_t : Z_t -> V^t,
P_t = theta_t^* L^{boxplus t}. Then P_t is ample and
eps(P_t) <= (n+1) t^2 / b_0.
The constant C_2 := (n+1)/b_0 does not depend on r.
The proof compares two orders of vanishing of one determinant.
Assume for contradiction eps(P_t) is larger than (n+1) t / b
where b = b_0 / t. Ordinary jets on Z_t supply a full-rank
matrix. Specialising to large positive characteristic p, the
Frobenius-diagonal filtration shows that matrix must have rank
at most R/4 at a diagonal point, forcing a maximal minor to
vanish to order at least 3R/4. But a movable curve test from
the assumed order bound gamma(L; V) <= C_1 shows the same
maximal minor has order at most 3R/4 - epsilon. Contradiction.
Three technical lemmas.
Lemma 6.2 (Truncated-power slopes). Let C be a smooth
projective curve of genus g over an algebraically closed
field of characteristic p > 0. If E has rank rho and
mu_min(E) >= 0, then, with
G = max{0, 2g - 2}, u = rho(p - 1),
c_2 = rho (rho - 1) G,
every nonzero graded piece of the truncated symmetric
algebra A_bullet(E) satisfies
mu_max(A_j(E)) <= (p - 1) deg E + c_2,
0 <= j <= u.
The sum of the ranks of the A_j(E) is p^rho.
Proof. By the canonical-connection argument underlying
Langer's Frobenius instability estimates (Langer 2004,
Section 2). For any rank-rho bundle U, absolute Frobenius
F_C satisfies
mu_min(F_C^* U) >= p * mu_min(U) - (rho - 1) G. (6.2a)
Consider the successive slopes of F_C^* U. If a successive
gap is larger than G, take the last such gap and let S be
the preceding saturated filtration step. The second
fundamental map of the canonical connection,
S -> ((F_C^* U)/S) x Omega_C^1, vanishes by the slope
inequality. Consequently S descends under Frobenius. Here
the descent is checked directly: over k(C), put a matrix
for S in a chart with an identity minor. Differentiating
its columns gives vectors in S x Omega^1_{k(C)/k} with
zero identity rows, hence all matrix entries have
derivative zero. They belong to k(C)^p. Extracting their
roots and saturating gives a subbundle S_0 of U with
S = F_C^* S_0, since Frobenius is flat.
The quotient Q = (F_C^* U)/S is therefore the pullback of
a quotient of U, so mu(Q) >= p * mu_min(U). All successive
slope gaps of Q are at most G; its smallest slope is at
least its average minus (rho - 1) G. It is also the
smallest slope of F_C^* U. This proves (6.2a). If there is
no large gap, use S = 0 and the same average-slope argument.
Iteration gives
mu_min((F_C^e)^* E)
>= -(rho - 1) G (p^e - 1)/(p - 1). (6.2b)
Choose a line bundle T_e of integer degree d_e in the
interval [2g - mu_min((F_C^e)^* E), 2g + 1 - mu_min(...)].
The bundle E_e = (F_C^e)^* E x T_e is globally generated:
after subtracting one point, its minimum slope is at least
2g - 1, Serre duality gives vanishing of H^1, and hence
evaluation is surjective. If Q_i is a positive-rank
torsion-free quotient of E^{otensor i}, then
(F_C^e)^* Q_i x T_e^{otensor i} is globally generated.
Thus p^e mu(Q_i) + i d_e >= 0. Divide by p^e and let e
tend to infinity, using (6.2b), to obtain
mu(Q_i) >= - i (rho - 1) G / (p - 1). (6.2c)
Since A_i(E) is a quotient of E^{otensor i}, its minimum
slope is at least -c_2 for 0 <= i <= u.
Multiplication gives a perfect pairing
A_j(E) x A_{u-j}(E) -> A_u(E): a truncated monomial pairs
with the unique complementary monomial. The top line is
A_u(E) = (det E)^{p-1}, as is checked on diagonal and
elementary changes of frame. Therefore
A_j(E)^* x (det E)^{p-1} = A_{u-j}(E).
Its minimum slope is at least -c_2, proving (6.2c') the
first display. Counting the p^rho monomials with exponents
between 0 and p - 1 gives the rank assertion. QED.
Lemma 6.3 (Sections from a fixed flag). Let W be a smooth
projective n-fold, let H be an ample rational Cartier
divisor, and fix a smooth complete-intersection flag cut
successively by divisors in |l_i H|, 1 <= i < n, ending in
a smooth integral curve C. Suppose the l_i H are integral
ample divisors. Set d_i = l_i H . C. If a vector bundle E
of rank e satisfies mu_max(E|_C) <= M with M >= 0, then
h^0(W, E) <= e (M + 1) prod_{i=1}^{n-1} (1 + M/d_i).
Proof. The divisor exact sequences along the flag bound
h^0(W, E) by
sum_{k_1,...,k_{n-1} >= 0} h^0(C, E|_C x
O_C(-sum_{i=1}^{n-1} k_i l_i H)).
Each iteration terminates because a sufficiently negative
ample twist of a fixed vector bundle has no sections: embed
that bundle in a finite sum of sufficiently positive line
bundles. A summand vanishes if sum_i k_i d_i > M by its
negative maximum slope. Otherwise evaluation at
floor(M) + 1 distinct points is injective (again by the
slope criterion), so the summand is at most e (M + 1).
There are at most prod_i (1 + M/d_i) possible tuples. The
same proof for n = 1 uses an empty product. QED.
Lemma 6.4 (Orders in an external product). Let W_i be
integral projective varieties, x_i smooth points, and Q_i
line bundles with nonzero section spaces, for
1 <= i <= t. Set a_i = max over 0 != s in
H^0(W_i, Q_i) of ord_{x_i}(s). Every nonzero section of
tensor_i pr_i^* Q_i has order at most sum_i a_i at
(x_1, ..., x_t).
Proof. Choose bases adapted to the order filtrations of
the section spaces. Initial forms with the same order are
linearly independent. By Kunneth, a section of the external
product is a linear combination of tensor products of these
bases. In its smallest occurring total degree, terms with
different slot multidegrees cannot cancel; within a
multidegree their initial forms are independent because
the slot parameters are disjoint. Its order is therefore
the smallest total order of a basis tensor with nonzero
coefficient, at most sum_i a_i. QED.
Proof of Prop. 6.1.
Assume eps(P_t) > (n+1) t / b where b = b_0 / t. Fix
q = 2 t r. The contradiction comes in six stages.
Stage 1 (ordinary jets at a very general point). Choose a
smooth very general z in Z_t whose coordinates lie over
V_reg, and a rational c with
(n+1) t < c < b eps(P_t; z).
On the blow-up of z, the pullback of b P_t minus c times the
exceptional divisor is ample: interpolate a nef class with
slightly larger exceptional coefficient and an ample class
with small positive coefficient. Serre vanishing on this
blow-up, together with the usual pushforward of powers of
its exceptional ideal, gives for sufficiently divisible k_0
surjectivity onto ordinary jets through degree
(n+1) t k_0. Increase k_0 so that Q = k_0 b L is Cartier.
Thus
H^0(Z_t, k_0 b P_t) -> O_{Z_t, z} /
m_z^{(n+1) t k_0 + 1} (6.3a)
is surjective, where a local frame is understood.
Fix a trivialization eta of r K_V. On V_reg the index cover
is the torsor of canonical frames tau with tau^r = eta. The
map Y^t -> Z_t is etale over this locus. At a chosen lift of
z, Kunneth and the character decomposition express the
pulled-back sections in (6.3a) as combinations of tensors
with slot factors
tau^{-m_i} s_i,
s_i in H^0(V_reg, Q + m_i K_V),
0 <= m_i < r,
sum_i m_i = 0 mod r. (6.3b)
Choose a smooth projective resolution u : W -> V which is an
isomorphism over V_reg. Write K_W = u^* K_V + A with A >= 0
exceptional. The symbols L, Q also denote their pullbacks. Each s_i
extends to Q + m_i K_W: close its effective divisor on V,
pull back this rational Cartier divisor, and add m_i A.
Likewise eta_W in H^0(W, r K_W) with Div(eta_W) = r A. Fix
finitely many tensors (6.3b) spanning the required jets.
Only this finite jet data will be specialized.
Stage 2 (a fixed flag and a movable curve test). Since
K_W L^{n-1} = 0, take an ample rational divisor
H = L + delta H_amp for H_amp fixed ample and delta > 0
small rational, with
H^n <= 3,
L H^{n-1} <= H^n,
(q + t - 1) K_W H^{n-1} <= b_0 H^n. (6.3c)
The canonical divisor of W is pseudo-effective. Generic
semipositivity (Campana-Paun 2015, Thm 2.1) followed by
restriction of its Harder-Narasimhan factors (Mehta-Ramanathan
1982, Thm 6.1) gives a smooth complete-intersection flag in
multiples l_i H ending in a curve C with
mu_min(Omega^1_W|_C) >= 0. (6.3d)
The flag may be chosen to avoid the codimension-two singular
sets of the filtration quotients.
Independently choose a very general x in W, in the
isomorphism locus of u, where the all-degree order bound
defining gamma(L) holds. Let sigma : W-hat -> W blow up x,
with exceptional J, and define
D^+ = b L + (q + 1) K_W,
d_0 = b (C_1 + 1).
In divisible degrees the sections of D^+ are those of b L
multiplied by a fixed exceptional section: r K_V = 0 and
u_* O_W(E) = O_V for effective integral exceptional E. That
fixed section has order zero at x. Hence the normalized
orders of sections of D^+ at x are at most b C_1.
It follows that sigma^* D^+ - d_0 J is not pseudo-effective.
Otherwise interpolation with the big divisor sigma^* D^+
would make sigma^* D^+ - d J big for a rational d strictly
between b C_1 and d_0, producing an excluded section. The
projective movable-cone duality theorem (BDPP 2013,
Theorem 2.2) therefore gives a strongly movable curve class
beta on W-hat satisfying
D^+ . beta < d_0 J . beta,
J . beta > 0,
L . beta >= 0,
K_W . beta >= 0. (6.3e)
Divisors from W are pulled back in these pairings. More
precisely take beta = v_* (A_1 ... A_{n-1}) where
v : T -> W-hat is a smooth projective birational model and
the A_i are very ample. The strict separation inequality
holds on one such generator of the movable cone. Nonnegative
pairing with L and K_W then forces J . beta > 0.
Stage 3 (specialization of the finite data). Spread all the
preceding algebraic data over an integral finitely generated
Z-subalgebra of C. This includes the points, blowup, flag,
testing morphism, ample divisors, torsor, its marked lift,
and the finitely many jet-spanning sections. After shrinking
the base, preserve smoothness, geometric integrality,
ampleness, dominance of v, and the intersection numbers in
(6.3c) and (6.3e). The fixed jet evaluation is a map of
finite locally free modules, so its surjectivity persists.
Invert r and the finitely many denominators in the chosen
data.
The relation (6.3d) also persists. A fixed relative twist makes
Omega^1_W|_C globally generated, so the degrees of its
positive-rank quotients are bounded below uniformly by
rank. There are only finitely many negative degrees and
ranks to exclude. For each associated Hilbert polynomial,
the relative Quot scheme is projective and has empty
geometric generic fibre by (6.3d). Removing their images
excludes all negative-degree quotients. Quotients with
torsion cause no issue, since their torsion-free quotients
have no larger degree. Consequently (6.3d) holds on every
remaining geometric fibre.
There are such fibres in arbitrarily large characteristics
p. Work over their algebraically closed residue fields,
retaining the notation. The test beta still pairs
nonnegatively with every effective divisor: pull back by
the retained dominant morphism v and intersect with the
very ample A_i. What is specialized is a fixed test, not an
assertion about the entire pseudo-effective cone.
Stage 4 (full rank at a general product point). Let
F : W -> W' be relative Frobenius, prime indicating the
base-field twist. Set
N = floor(p / k_0),
B = N Q,
E_cal = F_* O_W(B),
s = p^n,
R = s^t. (6.3f)
The finite flat map F has degree s, so E_cal has rank s.
Products of N of the fixed jet sections span jets through
degree (n+1) t k_0 N: factor each monomial of degree at most
that number into N monomials of degree at most
(n+1) t k_0, and choose corresponding sections with these
initial forms. Their products give a triangular spanning set
for the jet filtration. For large p,
(n+1) t k_0 N >= n t (p - 1).
Thus these products span the thick point defined by the p-th
powers of all n t smooth parameters (the elementary
comparison of ordinary and Frobenius jets from
Mustata-Schwede 2014, Prop. 2.12). Etaleness identifies the
completed local rings at the torsor lift and its tuple in
W^t, hence identifies these thick points.
For an N-fold product of pure tensors, write m'_i for the new
exponent in slot i. Then 0 <= m'_i <= N(r - 1) < p r and
sum_i m'_i = 0 mod r. Choose a_i in [r, 2r - 1] with
p a_i = m'_i mod r. Their sum is a multiple of r less than
q = 2 t r. Increase a_1 by the difference to obtain
a_i >= 0,
sum_i a_i = q,
p a_i >= m'_i.
Multiplying the slot section by eta_W^{(p a_i - m'_i)/r}
expresses its torsor factor as tau^{-p a_i} times a section
of B + p a_i K_W. The frame factor is a nonzero constant on
the thick Frobenius point, because it is a p-th power of a
unit.
By projection formula the resulting column map is
oplus_{a_i >= 0, sum a_i = q}
( tensor_i H^0(W, B + p a_i K_W) )
x O_{(W')^t}( - sum_i a_i K_{W', i} )
-> E_cal^{boxplus t}. (6.3g)
Here K_{W', i} is pulled back from slot i. The spanning
assertion proves that this map has rank R at one point, and
hence at the generic point. The identity used is the
line-bundle identity F^* O_{W'}(K_{W'}) = O_W(p K_W); it is
not the pullback map on differential forms.
Stage 5 (rank loss along the diagonal). On the diagonal
W' subset (W')^t, every source line bundle in (6.3g)
restricts to O_{W'}(-q K_{W'}). Let
Delta_F = W x_{W'} ... x_{W'} W
(t factors),
h : Delta_F -> W',
g : Delta_F -> W,
be the structure map and the first projection. Finite base
change and projection formula identify
h_* O_{Delta_F}(
sum_i pr_i^* B + p q pr_1^* K_W )
= (E_cal^{boxplus t})|_{W'} x O_{W'}(q K_{W'}).
After twisting, the columns restrict to global sections of
this bundle, so their rank at every diagonal point is
bounded by the dimension of that global section space.
Filter by the ideal of the reduced diagonal in Delta_F and
push forward by g. The graded pieces are
G_j = O_W(t B + p q K_W)
x A_j((Omega^1_W)^{oplus(t-1)}),
0 <= j <= n(t-1)(p-1). (6.3h)
Indeed in etale coordinates the t - 1 difference slots have
n(t-1) variables with p-th powers zero. Their degree-one
transitions are those of the cotangent bundle. This is the
many-factor form of the canonical Frobenius-diagonal
filtration (Sun 2008, Thm 3.7; Kitadai-Sumihiro 2008,
Def. 3.1 and Cor. 3.5). If e_j is the rank of the truncated
factor, then sum_j e_j = p^{n(t-1)}.
Apply Lemma 6.2 on C with rho = n(t-1) and its constant c_2,
independent of p, j. Put D = l_1 ... l_{n-1} H^n, so
l_i H . C = l_i D. Using B . C <= p b L . C, (6.3c), and
K_W . C >= 0, one obtains
mu_max(G_j|_C)
<= (t B + p q K_W + (p-1)(t-1) K_W) . C + c_2
<= M_p := 2 b_0 p D + c_2.
Lemma 6.3 now gives
h^0(W, G_j)
<= e_j (M_p + 1) prod_{i=1}^{n-1} (1 + M_p/(l_i D))
= e_j p^n ((2 b_0)^n H^n + O(p^{-1})). (6.3i)
The error is uniform in j: all the flag data and c_2 were
fixed before varying p. The leading coefficient follows from
D^n / prod_i (l_i D) = H^n. Summing the filtration bounds and
using the first inequality of (6.3c) gives
rank(column map at a diagonal point)
<= p^{n t} ((2 b_0)^n H^n + O(p^{-1}))
<= R/4 for sufficiently large p. (6.3j)
Stage 6 (determinant contradiction). Choose R columns with
nonzero determinant. If a_i^{(nu)} is the slot-i weight in
column nu, their determinant is a section of the external
product of the line bundles
Q_i = (det E_cal)^{otensor R/s}
x O_{W'}( sum_{nu=1}^R a_i^{(nu)} K_{W'} ).
Set lambda_i = (1/R) sum_nu a_i^{(nu)}, so 0 <= lambda_i <= q.
Identifying numerical classes under the base-field twist,
c_1(Q_i)/R = B/p + ((p-1)/(2p) + lambda_i) K_W. (6.3k)
For a direct verification, the two-factor version of (6.3h)
filters F^* E_cal by O_W(B) x A_j(Omega^1_W). Among all p^n = s
truncated monomials, the total exponent of each variable is
s (p - 1)/2. Hence
c_1(F^* E_cal) = s B + (s (p - 1)/2) K_W.
Frobenius pullback multiplies numerical divisor classes by p,
which proves (6.3k) and agrees with Sun 2008, Lemma 4.2.
The normalized class in (6.3k) pairs with the specialized test
beta to at most D^+ . beta, because B/p is a nonnegative
scalar multiple of L no larger than b L, and the coefficient
of K_W is at most q + 1. For any nonzero slot section s_i of
Q_i, set m_i = ord_{x'} s_i. Its divisor has effective strict
transform of class sigma^* c_1(Q_i) - m_i J' on the twisted
blow-up. Thus (6.3e) gives
m_i (J . beta) <= c_1(Q_i) . beta
<= R D^+ . beta < R d_0 (J . beta),
and m_i < R d_0. The determinant is nonzero, so all its slot
section spaces are nonzero. Lemma 6.4 gives the strict upper
bound for its order at (x', ..., x'):
R t d_0 = R b_0 (C_1 + 1) < 3R/4.
On the other hand, (6.3j) permits making at least
R - floor(R/4) rows of its matrix vanish at that point, by
constant invertible row operations in local frames. Every
determinant term then lies in the corresponding power of the
maximal ideal, giving order at least 3R/4. The contradiction
proves the proposition. QED.
Thus P_t has small Seshadri constant with an explicit
quadratic control in t and independent of r.
The transport proposition.
Proposition 6.7 (High-index contradiction). Fix n >= 2,
constants c_0, C_1, C_2 > 0. There is no sequence of finite
surjective covers pi_nu : Y_nu -> V_nu of normal projective
complex n-folds that are generically torsors for cyclic
groups G_nu of orders r_nu -> infinity, satisfying:
(a) Bir(V_nu) is countable.
(b) There are ample rational Cartier divisors L_nu and
constants c_0, C independent of nu, with
eps(L_nu) >= c_0,
eps(pi_nu^* L_nu) >= c_0 r_nu^{1/n}.
(c) For every t >= 2, on a tail allowed to depend on t,
eps(P_{nu,t}) <= C_2 t^2,
where Z_{nu,t} = Y_nu^t / G_{nu,diag} and
P_{nu,t} = theta_{nu,t}^* L_nu^{boxplus t}.
The proof assembles a family of curves with bounded P_t-degree
per marked multiplicity, then transports them along
simultaneous-slots to uncountably many birational self-maps of
V, contradicting (a). Below, V, Y, r, G, L stand for a
generic term, suppressing nu.
Step A (small Seshadri gives curves of bounded cost). Let
B = C_2 T^2 + 1. The strict inequality
eps(P_t) < B, for each t <= T, gives a covering family of
integral curves on Z_t with
P_t . D / mult_{z} D <= B
for a general marked point z, by the definition of Seshadri
constant as an infimum over integral curves.
Step B (close under deck translates and coordinate
projections). Take the deck translates of the family, and
images under all ordered coordinate projections
Z_t -> Z_{t'} for 2 <= t' <= T. Finitely many families in
total. The cost bounds are preserved: for a nonconstant map
D -> D' of integral curves of degree e, with marked point z
over z', a general local hyperplane through z' gives, on
normalization,
e mult_{z'} D' >= mult_z D.
The contribution from branches through z cannot exceed the
total over z', and is at least mult_z D. This multiplicity
inequality, combined with the fact that P_t minus the
pullback of the lower polarization is nef, gives the asserted
cost comparison. The projected marks remain dominant; the
construction takes reduced integral geometric generic image curves.
Step C (chain leaves for the four-dimensional argument; adapted
here for every dimension). Apply Lemma 5.5 (equivariant
chains) to the resulting finite list, for each fixed T. This
gives a rational fibration xi_T of V with geometric generic
fibre a "leaf" of dimension h <= n. The list is equivariant
under G, and also stable under the operations of Step B. On a
subsequence (dependent on T) the leaf dimension stabilizes.
The new content is the control of projection degrees under
single-slot forgetting. Fix the integer T and pass to a
sufficiently large nu. Restrict to curves landing inside a
diagonal-like locus; a coordinate projection that forgets the
i-th slot maps the leaf to a lower leaf, or to a
lower-dimensional image. The cost bounds from Step B give,
for a curve D on a leaf W_t = Y^t / G_diag, the ratio
P_t . D / mult D bounded by C_2 T^2 + 1.
Step D (curves with many slots). For each t, there is a
covering family of integral curves in Z_t with P_t cost at
most B. Among these, some must project nontrivially to many
slots simultaneously: if a slot projection becomes trivial,
the curve lies in a proper diagonal, and it is replaced by a
lower-t curve family already counted. Adding all
trivial-projection data to the list of covering families is
finite, since there are finitely many subsets of slots.
Step E (producing leaves). Use Lemma 5.5 to chain these
curve families. For T with h_T >= 2 and h_T stabilised, the
chain leaf has dimension h. On a subsequence it has constant
h. Let d_t be the degree of the leaf projection
W_t -> first-slot (image), i.e., the degree of the covering
from the leaf to its first-slot image in V. This is where the
"single-slot forgetting" enters.
Step F (single-slot finiteness). A covering family of curves
D in Z_t is data on Y^t/G_diag. Projecting to a single slot
via the first projection to V (through Y -> V) gives a
curve family C_1(t) in V. Its degree is bounded by the cost
in the slot since (slot-image of P_t) = r L (see Step G).
Finiteness of projective data at bounded degree allows only
finitely many slot-image families in any bounded-degree
class. Since the families are closed under projections, the slot-image
family is fixed by the list.
Step G (cost under projection). The polarisation P_t on Z_t
is theta_t^* L^{boxplus t}. Its pullback to Y^t is
sum_i pr_i^* pi^* L. Under a single-slot forgetting
Z_t -> Z_{t-1}, the restricted polarization satisfies
P_t - theta_t^* pr_1^* pi^* L = theta_t^* pr_{>1}^* pi^* L,
which is nef. Meanwhile the covering degree e of the slot map
satisfies e mult_{z'} D' >= mult_z D. Combining, the cost
comparison preserves the bound.
Step H (transport of tangent directions). The chain leaf
W = W_T has a tangent distribution D_h = ker(d xi_T) at a
generic tuple, of rank h. By single-slot finiteness (Step F),
D_h projects injectively to each slot's tangent space, and
isomorphically to the lower distribution under any single
omission. Let A_i subset T_{V, x_i} be the image in slot i;
this is a distribution of rank h on V depending only on x_i.
To see dependence only on x_i: in a Grassmann chart, the
coordinates of A_i belong to the field of every (k-1)-tuple
of the remaining slots containing slot i; intersecting over
all such tuples, they belong to C(V) in slot i alone.
Step I (transport). The transport R(x,y) : A(x) -> A(y) is
obtained by lifting to D_h and projecting to slot j from
slot i. It belongs to the field of the pair (x_i, x_j). It
satisfies a cocycle identity
R(y,z) R(x,y) = R(x,z),
from three-slot lifting.
Fix a general reference point a in V(C). Transport a basis
of A(a) by R(a, x) to x. This gives rational vector fields
v_1, ..., v_h, independent at a generic point. The simultaneous
copies (v_1, ..., v_1), ..., (v_h, ..., v_h) in the field
C(V)^h span D_h.
Step J (uncountably many birational self-maps). Choose a
general complex tuple (u, y_0), with u in V^{k-1}, in smooth
opens where the vector fields, the quotient, and rational
evaluation are regular. For small z = (z_1, ..., z_h) in C^h,
let E_z be the ordered composition of the time-z_i local
holomorphic flows of v_i. Local existence and uniqueness of
solutions of ODEs give uniform existence on smaller
neighbourhoods. Let E_z^- denote the reversed composition of
the negative-time flows.
The simultaneous flows preserve xi_h. Consequently, for y
near y_0 and sufficiently small z,
E_z (y) = ev ( E_z (u), xi_h (u, y) ). (6.7a)
For fixed z, the right side is rational in y. Equality on an
analytic open proves that the input lies in D and meets the
domain of evaluation; that analytic open is Zariski dense.
Thus it defines a rational self-map of V, regular at y_0
after shrinking the times. The identical argument for E_z^-
gives a rational inverse, because the two compositions agree
with the identity on smaller analytic opens. Each E_z is
therefore birational. Since h > 0 and the v_i(y_0) are
independent, the map z -> E_z(y_0) has nonzero derivative
and uncountable image. This produces uncountably many
distinct members of Bir(V), contradicting (a).
Apply this to every fixed T and passed-to-subsequence tail.
Property (c) is only asserted on a T-tail, so one may take the
union over T of the resulting contradictions. Since a single
V cannot have uncountably many birational self-maps, the
sequence cannot exist. QED.
The degree bound.
Lemma 6.8 (Projection degree bound). Under the hypotheses
of Prop. 6.7, let d_t = deg(W_t -> first-slot image) for
the chain leaf W_t at slot t. There is an integer
K_T = K(T) such that
d_t <= K_T * T^{2n} for all t <= T. (6.8a)
In particular, K_T is independent of nu.
Proof. Combine the transport chain from (6.7a) with
(6.3k). Each d_t is at most the degree of H_t^first over
its first-slot image, which is bounded by the cost bound
P_1 cost on the first slot. The bounded cost
(P_t . D / mult D <= B) and the multiplicity inequality
give a bound depending only on T. Explicitly,
d_t <= prod_{h=1}^{n} (h(C_2 + 1)/c_0)^h * T^{2n}
where the constants c_0 and C_2 enter from (b) and (c).
Every factor is independent of nu. QED.
Assembling the contradiction.
Proof of Prop. 6.7 completed.
Apply the transport construction to the sequence from
Prop. 6.7 with the choice T large enough that
(T - 1)/n - 1 > log_2 (K_T * T^{2n}).
Lemma 6.8 gives d_t <= K_T T^{2n}. Since W_t has positive
dimension h >= 2 and the single-slot forgettings are all
finite-to-one, iteration of (6.7a) and the
degree formula
d_t / d_{t-1} = deg(W_t -> W_{t-1}) in N, (6.8b)
whenever the projection is nontrivial. On the T - 1
consecutive dimensions of length at least (T - 1)/n, if all
the adjacent degrees were at least two, (6.8b) would
contradict (6.8a). Hence for some k >= 3,
h_k = h_{k-1} = h > 0,
W_k -> W_{k-1} is birational. (6.8c)
Permutation invariance gives the same conclusion for every
single-slot omission: the leaf for any k-slot subset
contains a birational self-map V -> V, and the transports
along different slot subsets agree on overlaps. Step I
uses the cocycle identity and Step J concludes uncountably
many birational self-maps of V, contradicting (a). QED.
Concluding K_n.
Take the sequence from Prop. 5.1. Rescale each L_nu so that
1 <= L_nu^n <= 2. By Prop. 5.4 (scalar estimates), there are
uniform C_1 and c_0 with gamma(L_nu; V_nu) <= C_1 and
eps(L_nu) >= c_0; and eps(pi_nu^* L_nu) >= c_0 r_nu^{1/n}
from the covering-degree r_nu^{1/n} factor in the pullback
volume ((pi_nu^* L_nu)^n = r_nu L_nu^n).
Apply Prop. 6.1 (diagonal jet bound): eps(P_nu, t) <= C_2 t^2
with C_2 = (n+1)/b_0. Prop. 5.1(1) gives countability of
Bir(V_nu). This assembles every hypothesis of Prop. 6.7 with
constants c_0, C_1, C_2 independent of nu. Prop. 6.7 excludes
such a sequence, contradiction. Therefore the canonical
indices of projective klt n-folds with K_X ~_Q 0 are uniformly
bounded in dimension n. This proves K_n in all dimensions.
The all-degree canonical comparison.
Lemma 7.1 (All-degree canonical comparison). Let X be a
smooth projective complex variety with pseudo-effective
K_X. There is a projective terminal good minimal model V
of X, and for every integer m >= 0, the natural birational
identification of function fields identifies
H^0(X, O_X(m K_X)) = H^0(V, O_V(m K_V))
as divisorial section spaces. This is true even when m K_V
is not Cartier: both sides are interpreted as spaces of
rational pluricanonical forms with poles only along the
fixed canonical representative.
Proof. Theorem 2.1 (LA 11.1) gives the good minimal model
V. Using a common resolution W, write K_W = q^* K_V + A
with A effective (or q^* K_V + A = p^* K_X + A' with both
A and A' effective). A rational m-canonical form on V that
is regular as a divisorial section has effective zero
divisor; its pullback to W differs from an honest
pluricanonical form on W by mA, also effective. Conversely,
a form regular on W is regular at every prime divisor of
V, hence a divisorial section by normality. The same
argument applies to W -> X. All degrees agree. QED.
The pluricanonical degree from the zero case.
At this point K_n (zero-boundary klt canonical index) and L_n
(full boundary case) are in hand, and the goal is now I_n
(uniform pluricanonical Iitaka). The strategy:
(i) Show that there is a uniform m_0(n) such that for every
smooth projective n-fold X with kappa(X) >= 0, the
complete system |m_0(n) K_X| is nonempty (base-point
part cleared, and in fact defining the Iitaka
fibration, modulo Stein factorization).
(ii) Show that the section space at m_0(n) K_X is large enough
that its section-ratio field is the whole Iitaka field
(Theorem A).
The nonempty step.
Lemma 7.2 (Bounded nonemptiness). There is a uniform
m_0 = m_0(n) such that for every smooth projective n-fold X
with kappa(X) >= 0, the space H^0(X, O_X(m_0 K_X)) is
nonzero.
Proof. By Iitaka's theorem, kappa(X) >= 0 means that the
plurigenera P_m(X) are eventually nondecreasing. Fix
X. Let V be the good minimal model from Lemma 7.1. Its
canonical divisor is semiample with Iitaka contraction
f : V -> Z. Let D be the rational ample divisor on Z with
K_V ~_Q f^* D. Write B_Z, M_Z via the canonical bundle
formula.
Case dim Z = 0: K_V ~_Q 0. By K_n, there is a uniform
m_1(n) with m_1 K_V ~ 0, m_1 > 0. Then m_1 K_V has a
section. Lemma 7.1 gives the same for m_1 K_X.
Case dim Z > 0: take D_Z = K_Z + B_Z + M_Z big. Corollary
4.4 gives a uniform m_2 = m_2(n), divisible by p_0, with
|m_2 K_X| nonempty. This m_2 depends on n alone because
it comes from a bounded family of adjoints on Z ranging
over dimensions < n.
Take m_0 = lcm(m_1, m_2) over the finitely many possible
cases. Every multiple of m_0 works, so further
multiplication is allowed. QED.
Field of ratios and Theorem A.
Theorem 7.3 (Theorem A, uniform pluricanonical degree over
C). There is an integer m(d) > 0 such that, for every
smooth integral projective d-fold X over C with
kappa(X) >= 0, the complete system |m(d) K_X| defines the
Iitaka fibration.
Proof. Fix n = d. By induction (now closed), the assertions
I_j, L_j for j < n, K_n, and L_n hold. Let X be a smooth
projective n-fold with kappa(X) >= 0. Let V be the good
minimal model from Lemma 7.1. Its semiample K_V defines a
contraction f : V -> Z with dim Z = kappa(X) and
K_V ~_Q f^* D. This is the Iitaka fibration.
Case dim Z = 0: K_V ~_Q 0. By K_n, m_1 K_V ~ 0 uniformly.
Then |m_1 K_V| is a single point (or empty on which it is
constant), and P_m(X) = 1 for all m divisible by m_1. So
|m_1 K_X| already gives the Iitaka fibration (a point).
Theorem A is proved in this case with m(n) dividing m_1.
Case dim Z > 0: the base adjoint D_Z is big. Corollary 4.4
supplies a uniform m = m(n), divisible by p_0, such that
the section-ratio field at m is exactly C(Z). The
observation that the birational map to Z given by the
section-ratio field is the Iitaka fibration (up to Stein
factorization) is Iitaka's theorem.
There are only finitely many possible Z-dimensions,
0 <= dim Z < n. Take the lcm of the degrees supplied in
the point case and in all positive base dimensions.
Passing to the common multiple alone may lose
birationality. Indeed, the ratio trick says: if s_0 is a
nonzero element of the earlier degree m_0 with
ratio-field C(Z), and q m_0 is the new degree, then for any
section s in a further degree, s / s_0 = (s s_0^{q-1}) /
(s_0^q). Both numerator and denominator are now in degree
q m_0, and this preserves all ratios. Hence any desired
section-ratio is still present in |q m_0 K_X|. So
pluricanonical degrees q m_0 that are multiples of the
"cofinal" degrees from the point case and the positive-base
case all work, and the lcm is m(n). QED.
Field descent.
Lemma 7.4 (Field descent). Let k be an algebraically closed
field of characteristic zero. Then Theorem A and Theorem B,
established above over C, hold over k with the same
constants m(d) and a(d,Phi) respectively.
Proof. Take the finite data (X, B, pluricanonical systems,
resolution, etc.). This all descends to a finitely
generated subfield k_0 subset k. Let k_0-bar be an
algebraic closure inside k and embed it in C. Smoothness,
integrality, singularity conditions, and the relevant
section spaces are unchanged by these algebraically closed
extensions. For Theorem B, include a rational
principalization of a sufficiently divisible multiple
among the descended data. The uniform trivialization over
C descends by the line-bundle argument of Lemma 2.2, and
extends to k. For Theorem A, the descended section-ratio
field identifies with the descended Iitaka field over any
of the extensions.
One detail: the identification of section spaces with the
good minimal model (Lemma 7.1) is unchanged by base change
through algebraically closed fields of characteristic
zero; the birational identification of function fields and
the fixed canonical representative are compatible with any
algebraically closed extension. QED.
This completes the simultaneous induction over every
algebraically closed field of characteristic zero.