Uniform Pluricanonical Iitaka Fibrations
A Guided Tour for Young Mathematicians
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This is a guided tour of a research paper in algebraic geometry.
The paper proves that the shape of an algebraic variety -- more
precisely, its Iitaka fibration -- is read off from a single
pluricanonical linear system whose degree depends only on the
variety's dimension, not on the specifics of its defining equations.
This tour is written for a motivated high-school reader. Every
technical term is introduced the first time it appears, each key
argument is summarized in plain language, and the formal statements
are also given so the reader can compare with the original.
The original paper has seven numbered sections. The tour presents
them in the order the induction uses them: this is close to the
paper's own order, but not identical. The boundary lc step of the
paper's Section 3 is deferred to Part VI, because it depends on the
zero-boundary klt case proved in Parts III-V. A reading guide at the
end maps every paper section back to the tour section(s) that
present it. The reader is encouraged to read the tour twice: once
for the plot, once for the details.
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PART I. THE QUESTIONS AND THE TWO THEOREMS
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1. WHAT IS THIS PAPER ABOUT?
An algebraic variety is the solution set of a system of polynomial
equations. In the real numbers, x^2 + y^2 = 1 traces a circle;
x^2 + y^2 + z^2 = 1 traces a sphere; and a single equation in many
variables usually gives a surface or a higher-dimensional shape.
Two adjectives will recur:
"projective" -- we add points at infinity so that the shape
behaves as if it were compact, like a sphere or a torus, rather
than like an open canvas.
"smooth" -- no pinch points and no self-crossings. A circle and a
sphere are smooth; a figure-eight curve is not.
Throughout, "smooth complex projective" means: coordinates are
complex numbers and the variety is compact and smooth. The theory
works over any algebraically closed field of characteristic zero,
and the paper addresses that generality in its final section.
The canonical bundle. A top form on a complex variety of dimension
d is the analogue of dV = dx dy dz. If a variety admits a
nowhere-vanishing top form, it is flat in the sense of complex
geometry. In general no such form exists, and the failure is
recorded by the canonical divisor K_X: a formal sum of codimension-
one pieces that measures the unavoidable zeros and poles.
The m-th pluricanonical system asks for many top forms at once,
studying global sections of the line bundle O_X(m K_X). The space
of these sections is finite-dimensional; call it H^0(X, O_X(m K_X)).
Sections are to K_X roughly what polynomials of degree m are to the
line bundle of degree one on a curve: they grow with m, and the way
they grow is the fingerprint of the variety.
The Kodaira dimension. The integer kappa(X) describes that growth.
It is -1 if there are no nonzero sections for any m. Otherwise it
lies in {0, 1, ..., dim X} and is the maximum dimension of the
image of X in the maps given by pluricanonical systems, for m large.
Three landmark cases:
kappa(X) = dim X -- "general type": the pluricanonical sections
distinguish almost every point; the variety is
as curved as possible.
kappa(X) = 0 -- "Calabi-Yau-like": there is a nonzero top
form, and it is unique up to scalar.
0 < kappa < dim X -- "fibrations": the pluricanonical forms see
only a lower-dimensional shadow of the variety.
The Iitaka fibration. That shadow is the Iitaka fibration. It is
the map to projective space given by the pluricanonical sections
in all sufficiently large degrees, followed by its Stein
factorization so that all fibres are connected. The image of this
map is called the Iitaka base; the dimension of the base equals
kappa(X). If kappa(X) = dim X, the Iitaka fibration is birational
(identity in the birational category). If kappa(X) = 0, the base is
a point. If kappa(X) = 1, the base is a curve.
The Iitaka field. A slightly sharper object is the field of ratios
of pluricanonical sections. Pick a nonzero section s of O_X(m K_X)
in some degree m; the ratio s'/s of two such sections, taken in the
same degree, is a rational function on X. The set of all such
ratios, over all degrees, is a subfield of the rational function
field of X. It is the function field of the Iitaka base. Saying
"the complete system defines the Iitaka fibration" means: the system
is nonempty, its rational map has connected generic fibre, and the
ratios of its sections generated exactly this subfield -- not merely
a finite subfield.
The effective Iitaka fibration conjecture. Here is the conjecture
that the paper proves:
There is a single integer m(d), depending only on d = dim X,
such that for every smooth complex projective d-fold X with
kappa(X) >= 0, the complete system |m(d) K_X| already defines
the Iitaka fibration.
The words "effective" refer to the fact that m(d) is an explicit
function of dimension alone. Earlier theorems gave bounds for the
degree needed to realize the Iitaka fibration, but those bounds
depended on details of the general fibre of the fibration (its
middle Betti number, its minimal canonical degree, and so on), and
were therefore not uniform over all X of fixed dimension. The
paper's contribution is exactly the uniformity.
The paper's main theorem, made precise:
THEOREM 1.1 (Uniform pluricanonical degree). For each integer
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.
The degree is existential; no values of m(d) are asserted.
Every positive multiple of m(d) also works, by the ratio trick
described at the end of Part VII.
Theorem 1.1 in the general-type case is a birational statement.
In the zero-Kodaira case it is a nonemptiness statement. In the
intermediate cases it says the pluricanonical map factors the
Iitaka fibration and nothing more. All cases are covered by one
degree per dimension.
2. WHY A SECOND THEOREM?
The proof of Theorem 1.1 needs a companion statement about a wider
class of objects than smooth varieties. To state it we introduce
pairs.
A "pair" (X, B) consists of a normal variety X together with an
effective divisor B on it: a nonnegative integer combination of
codimension-one subvarieties. Divisors allow us to "mark" certain
pieces of X -- for instance, a nodal curve marked by its node.
Pairs behave like varieties with mild corners, and they are the
correct language for the minimal model program.
Singularities of pairs come stratified by severity:
- klt, short for Kawamata log terminal: the mildest. Coefficients
of B are strictly below 1, and after a log resolution all
discrepancies are strictly greater than -1.
- dlt, for divisorially log terminal: klt away from an SNC
(simple normal crossing) boundary, with a compatibility
condition between singularities and boundary pieces.
- lc, or log canonical: the widest class considered. Coefficients
are at most 1, and discrepancies are at least -1. An lc pair
is allowed to have mild "worse" behaviour at boundary
intersections, but the canonical class still exists rationally.
Formally: choose a log resolution f: Y -> X and write
K_Y = f*(K_X + B) + sum_i a_i E_i
where E_i are the exceptional prime divisors and a_i are the
discrepancies. Then klt/dlt/lc correspond to thresholds for a_i,
with klt requiring a_i > -1, lc requiring a_i >= -1, and dlt
sitting between them with an additional combinatorial condition.
The paper's companion theorem:
THEOREM 1.2 (Uniform log canonical index). Fix d >= 0 and a
rational DCC set Phi inside [0, 1]. There is an integer
a(d, Phi) > 0 such that every projective lc pair (X, B) of
dimension d, whose coefficients lie in Phi and whose canonical
class K_X + B is rationally linearly trivial (K_X + B ~_Q 0),
satisfies
a(d, Phi)(K_X + B) ~ 0.
The multiple is an integral principal divisor: not merely
rationally trivial, not merely numerically trivial.
Two phrases need unpacking. A DCC set (descending chain condition
set) is a subset of [0, 1] that cannot contain an infinite strictly
decreasing sequence. This is a technical requirement on the
coefficients: something like {1, 1/2, 1/3, 1/4, ...} minus its
limit, or the set of different coefficients in adjunction. The
point is that the set, viewed as numbers, is "discrete in each
direction the ACC/DCC theorems care about".
The phrase K_X + B ~_Q 0 means some positive multiple of K_X + B is
rationally linearly equivalent to zero: it is Cartier (in a rational
sense) up to rational scaling, and its Cartier class vanishes. The
theorem strengthens this: a specific multiple -- an integral one,
not just any rational -- is actually the divisor of a single
rational function, and the required multiple depends only on d and
Phi, not on the pair itself.
Why is this companion theorem needed? Because the canonical bundle
formula (Part III) decomposes K_X of a fibration into three pieces:
K_X = f*(K_Z + B_Z + M_Z)
Here f: X -> Z is the fibration, K_Z is the canonical class of the
base, B_Z is a "discriminant" divisor recording how fibres degenerate
as we move in Z, and M_Z is "moduli" -- the extra piece that
measures fibre variation modulo isotopy. To bound K_X in terms of Z
we must bound the denominators that appear in B_Z and M_Z. Bounds of
this kind are index bounds: the smallest integer a such that
a*(B_Z + M_Z) is integral (respectively, principal). That is exactly
Theorem 1.2 in disguise. Thus Theorem 1.2 controls the pieces of K_X
that vanish along a fibration, and Theorem 1.1 uses that control to
produce a uniform pluricanonical degree.
The two theorems are proved simultaneously by induction on
dimension. This is the architectural backbone of the paper. Much of
the tour, especially Parts IV-VII, is about how the dimensions
line up.
The main theorem covers smooth varieties; the companion theorem
covers pairs with boundary. It may seem that the smoother case
should be easier -- but in fact the pair version appears in the middle
of the proof of the smooth version, because intermediate steps
(moving-members, adjunction, the canonical bundle formula) naturally
produce boundary divisors. The proof cannot avoid them; it embraces
them and proves the pair statement in the same breath.
3. THE TOOLBOX
Several deep theorems are treated as black boxes. Each one is a
well-established result in birational geometry, cited in the paper's
references. The novelty of the paper is not in proving any of them
individually but in arranging them into a coherent argument.
(1) BIRATIONAL EQUIVALENCE. Two varieties are birational if they
contain isomorphic dense open subsets. Birational geometry regards
two such varieties as "the same variety" for the purpose of
canonical bundles, plurigenus, and Kodaira dimension: all these
invariants are unchanged. So instead of working with the original
X, we are always free to replace it by any birational model Y.
(2) THE MINIMAL MODEL PROGRAM. This is an algorithm -- and a family
of theorems guaranteeing its termination -- that takes a projective
klt variety X and simplifies it by a sequence of birational steps.
A divisorial contraction shrinks a subvariety of codimension one.
A flip cuts out a small bad locus and re-glues the variety smoothly.
Each step preserves the canonical divisor up to pullback and preserves
plurigenera. The output, if it exists, is a "minimal model" on which
the canonical divisor is nef (positive on curves), or a Mori fibre
space (a fibration with negative canonical class along the fibres).
Both contracts and flips are known to work through dimension three
unconditionally, and up to dimension four in a strong sense; for
klt pairs in all dimensions, the existence of a good minimal model
is a hard theorem due to BCHM for pseudo-effective adjoints and to
a companion paper (LA) for the abundance direction. The paper uses
the MMP as an oracle.
(3) THE CANONICAL BUNDLE FORMULA. This is the key decomposition
used in every fibration argument. If f: X -> Z is a contraction of
normal projective varieties, with klt generic fibre, and K_X is
rationally linearly trivial on the geometric generic fibre, then
K_X ~_Q f*(K_Z + B_Z + M_Z)
where B_Z is the discriminant divisor and M_Z is the moduli
b-divisor. The "b" means the divisor lives on a base change of Z,
and only comes down to Z as a Weil class after pushing forward;
a "b-divisor" is the convenient formalism for this. B_Z measures
how the fibre at each point of Z degenerates, and M_Z captures the
remaining data -- moduli -- in the fibre. Both are subtle objects:
B_Z has coefficients in a DCC set determined by log canonical
thresholds, and M_Z is only a b-nef, b-Cartier class after a
bounded multiple.
(4) BEAUVILLE-BOGOMOLOV DECOMPOSITION. For a projective variety Y
with K_Y ~ 0 and no nonzero global one-forms (trivial irregularity),
a theorem of Druel and of Greb-Guenancia-Kebekus (building on Beauville
and Bogomolov in the smooth case) says that after a finite quasi-
etale cover, Y splits as a product of an abelian variety and finitely
many "irreducible Calabi-Yau" and "irreducible symplectic" pieces.
The paper uses this to decompose a character -- an eigenvalue of a
finite automorphism on a top form -- into characters on finitely many
pieces. Each piece has a bounded "form algebra" (only the trivial
form in the Calabi-Yau case, only even powers of a symplectic form
in the symplectic case), and this boundedness turns into a uniform
bound on the character order.
(5) LOG ABUNDANCE. A "good minimal model" is a minimal model on
which the canonical class is semiample: it is pulled back from a
nef divisor on some base. Existence of good minimal models in
characteristic zero, in the pairs setting, is the content of the
companion paper "Log abundance in characteristic zero", cited as
Thm 11.1 of LA. The present paper does not reprove abundance; it
uses it as an input in every dimension. This is important for honesty:
Theorem 1.1 is not a "self-contained" proof from nothing; it depends on
LA, and LA depends on BCHM and related work.
(6) POLARIZED EFFECTIVE BIRATIONALITY (BIRKAR-ZHANG). For a big log
adjoint with coefficients in a fixed DCC set and a fixed boundary
denominator, there is a uniform m whose rounded system is birational.
This supplies the final input that turns "a canonical bundle formula
with bounded coefficients" into "the pluricanonical map is
birational at a bounded degree". It is used in the very last step
of the induction (Part VII).
The paper's contributions over these inputs are:
(a) A uniform bound on the moduli denominator, proven by degenerating
Beauville-Bogomolov pieces over a curve, bounding finite
characters on normal components of a dlt special fibre, and
then bounding the composite; this is Part IV.
(b) A uniform bound on section orders (the "gamma" invariant) and on
Seshadri constants over small volumes on Iitaka fibres,
generalizing fourfold arguments to arbitrary dimension;
this is Part V.
(c) A diagonal-product argument that gives a quadratic (in the
number t of factors) upper bound on the Seshadri constant of
a diagonal quotient of a cyclic cover, extending a previous
fourfold argument; this is Part VI.
(d) A compatible-leaves argument that turns such a quadratic bound
into uncountably many birational self-maps of the original
variety, contradicting countability of its birational group
and hence excluding the unbounded-index sequence; this is also
Part VI.
The next four parts of the tour follow these contributions in order.
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PART II. UNIFORM DENOMINATORS (paper Section 4)
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4. THE EXACT CANONICAL BUNDLE FORMULA
The first technical pillar is Proposition 4.1 of the paper, which
says the canonical bundle formula can be arranged with a universal
coefficient on the left. Restated:
PROPOSITION (Bounded moduli denominator, paper 4.1). There are
positive integers p_0 dividing p, depending only on n = dim X,
with the following property. Let f: X -> Z be a contraction of
normal projective complex varieties with dim Z > 0, X klt, and
K_X rationally linearly equivalent to the pullback f^*D of a
rational Cartier divisor D on Z. Then there exist 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 determined
by n alone, 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.
There are three distinct claims bundled here. First, the canonical
bundle formula can be arranged so that the left side has a universal
vertical denominator p_0. Second, the discriminant coefficients of
B_Z belong to a fixed rational DCC set depending only on n, not on
the specific fibration. Third, and this is the delicate part, the
moduli b-divisor M admits a universal Cartier bound p.
The first two claims follow reasonably quickly.
- The geometric generic fibre of f is a klt variety of dimension at
most n-1 with rationally trivial canonical class. By the induction
hypothesis L_{n-1} (the log canonical index statement in lower
dimension), there is a common p_0 that trivializes this canonical
divisor. This trivialization descends to the generic fibre of f:
Cartier-ness and the one-dimensional section space commute with
field extension, so the descended divisor is already trivial on
the generic fibre.
- Choose psi so that p_0 K_X + Div(psi) has no horizontal component.
A small correction using the fact that vertical functions on a
contraction land in C(Z) produces the exact pullback equality.
- Ambro's klt-trivial fibration theorem supplies the (B_Z, M) data.
The discriminant coefficients are the numbers 1 - t_P, where t_P
is a log canonical threshold of the fibre over a point of Z.
Threshold ACC (Hacon-McKernan-Xu, 2014) turns the descending-chain
sets of threshold values into a fixed rational DCC set.
The third claim -- the Cartier bound on M -- is where the real work
happens, and the next section is entirely about it.
5. BOUNDING THE MODULI DENOMINATOR
Fix a smooth base determination W of Z on which M descends to a
b-nef rational Cartier divisor M_W. For each prime divisor P on W we
want to prove that the coefficient of p M_W at P is an integer.
Equivalent: a single integer should clear the fractional parts of
every coefficient simultaneously.
Reduce to a transverse curve. Resolve the total space over W, then
take very general hyperplane sections on W until a smooth curve C
remains meeting P at a smooth point c. Bertini applied to the
finitely many resolution strata preserves the SNC threshold data at
c. The sliced fibration is a family of curves over (C, c); its
generic fibre is birational to the original zero-boundary data, and
its geometry near c is governed by the same threshold t at c.
The weight of a relative form. Let theta be a rational top form
defining K_V, and z a local parameter at c. A "weight" is defined on
relative pluricanonical forms phi of degree m by
w_z(phi) = inf_{E} ( ord_E(phi ^ dz/z) + m ) / ( m ord_E(z) ),
where the infimum runs over all divisorial valuations E centred
over c. The particular phi that generates the fibre canonical
divisor satisfies
w_z(phi) = alpha - 1 + t,
where alpha is the coefficient of D_Z at P and t is the threshold.
This is equation (4.2) of the paper. So w_z(phi) is exactly the
fractional part we want to clear.
The Beauville-Bogomolov decomposition enters now. The general fibre
is a klt variety with K ~ 0 and trivial irregularity, so it inherits
a finite quasi-etale product cover
A x B_1 x B_2 x ... x B_r,
with A abelian (possibly absent) and each B_i irreducible
Calabi-Yau or irreducible symplectic. Working in a family over C,
the semilinear version of the same decomposition is needed: after a
finite Galois base extension, the inertia of C acts semilinearly on
the comparison product cover, and a bounded power n! of the inertia
generator preserves each factor field.
The inertia generator. Let g be a lift of the base inertia at c to
the total space of the comparison cover. On each resulting block R_i
of dimension s_i < n, choose a volume generator eta_i normalized so
that its weight vanishes:
w_u(eta_i) = 0.
Then g^b acts on eta_i by a scalar function:
(g^b)^* eta_i = c_i eta_i, lambda_i := c_i(c).
The point of the normalization w_u(eta_i) = 0 is that c_i is a unit
at c: it has neither zero nor pole there. Hence c_i(c) makes sense
as a well-defined nonzero scalar.
The key identity. The original form phi comes from the original
fibration and is invariant under the deck group. Taking the p_0-th
power of the product of the normalized eta_i gives a form on the
comparison cover; comparing with the pulled-back phi gives a base
function a. The weight rules give
l * w_z(phi) = ord_u(a) / p_0,
where l is the total ramification of C* -> C over c. Passing to
leading coefficients in the character expansion gives the inertia
relation
zeta^{b * ord_u(a)} * prod_i lambda_i^{p_0} = 1,
where zeta is the primitive l-th root of unity by which g acts on
the tangent line at the marked point. This is equation (4.4).
Now suppose a single integer N = N(n) kills every lambda_i, meaning
lambda_i^N = 1 for all i. Raising the identity to the N-th power
gives l | b N ord_u(a), which rewrites as
p_0 * b * N * w_z(phi) is an integer.
By the weight-threshold identity, p_0 b N clears the denominator of
alpha + t, exactly the goal. So the whole problem reduces to
bounding the orders of the central characters lambda_i. Section 5
of the paper does exactly that.
The character-bound lemma. Here is Lemma 4.4 of the paper. Let R_i
be the Stein block covers of the previous paragraph over the final
curve field, and g a finite-order lift of base inertia. Put b = n!
so that g^b preserves each block, and choose eta_i with
w_u(eta_i) = 0. Then there is N = N(n) with lambda_i^N = 1 for all
i, where lambda_i = c_i(c) as above.
The proof splits into two cases.
The abelian case. Suppose the block has dimension s. On the
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 of Deligne and Steenbrink.
The integral local system in degree s has rank binom(2s, s).
Combining parallel transport on the universal cover with the
finite-order base action gives an integral matrix A of that rank
acting on the fibres. A power of A equals a power of monodromy,
because the base action is finite. Passing from flat frames to
canonical-extension frames multiplies A by a unipotent factor,
which does not change the eigenvalues. Therefore a primitive root
of lambda_i appears as an eigenvalue of an integral matrix of rank
binom(2s, s). Its cyclotomic degree, and hence the possible orders
of lambda_i, are bounded above by binom(2s, s).
The nonabelian case. This is the paper's technical core. Take a
semistable model, then an equivariant relative K - MMP, and use the
control it gives you to produce a klt model Y with a reduced Cartier
special fibre T, a dlt structure (Y, T), and K_Y semiample over the
curve. Every intermediate object (relative minimal models,
contractions, flips) descends from the finite quotient total space
and lifts back by normalization. No divisors are extracted, so the
log structure is preserved. The special fibre T is Du Bois, so its
cohomology is controlled by the smooth resolution of the block via
Du Bois base change.
The dlt special fibre T has a degree-one log generator: the residue
of a normalized relative log form at T. This is equation (4.5). For
a prime component S preserved by a power of the transformation, the
residue of the character action on T descends to a log volume form
on S, and adjunction gives a character of that form. This reduces
bounding an eigenvalue to a lower-dimensional problem: the
finite-order log volume character on a normal lc (S, Theta) with
dimension s < n. And L_s (the finite-character lemma in dimension s,
which is a consequence of the induction hypothesis) bounds that.
To *find* such a preserved component S, the paper uses
structure-sheaf cohomology of T. For the Calabi-Yau block, only H^0
and H^s of O_T are nonzero, and each is one-dimensional; Serre
duality gives an alternating trace 1 + (-1)^s lambda^{-1}. For the
symplectic block, one dimension in every even degree and none in
odd degree. A "pseudo-Neumann trace" argument via Newton's
identities shows that some bounded power of the transformation has
nonzero alternating trace. The coherent Lefschetz theorem of Donovan
(1969) then says a finite-order transformation with nonzero
alternating trace has a fixed point on a projective scheme. At the
fixed point at most s+1 components of T meet (from dlt strata), and
a bounded further power (roughly (s+1)!) preserves a prime component
through that point. That prime component is the S we need.
Putting it together. Choose N from the character-bound lemma and set
p = p_0 * b * N.
Then every coefficient of p M_W is integral. Since W is smooth, p M_W
is Cartier. And M already descends to W by qualitative moduli
theory, so p M is b-Cartier as claimed. The moduli-denominator bound
is proven.
6. TORSION OVER RATIONALLY CONNECTED BASES
A second statement from Section 4 is needed later, when we want to
bound the canonical class of a variety with a rational (not just
contraction-type) image. The statement (paper Proposition 4.6):
PROPOSITION (Torsion over RC bases). Fix d, a rational DCC set I,
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 variety
of dimension d with rationally connected smooth projective
resolution. Suppose (Z, B + M) is generalized klt, B >= 0 has
coefficients in I, the b-nef data satisfy p M b-Cartier, and
D = K_Z + B + M ~_Q 0.
Then l D is an integral principal divisor.
Mechanism:
- Global ACC (after a small extraction) bounds the coefficients of B
in a finite subset I_0 of I.
- Birkar's boundedness theorem (2025) applies: the varieties (Z, B+M)
are generalized epsilon-lc for a uniform epsilon, with trivial
generalized adjoint, so they range over finitely many flat
isomorphism classes in codimension one.
- The class group of Z is a finitely generated abelian group. Its
finite torsion has a bounded exponent by Kummer theory applied to
the smooth locus U: for every a, one has Cl(Z)[a] = Hom(H_1(U(C),
Z), mu_a) (up to canonical isomorphism). Combined with the previous
bullet, the possible H_1 groups form a finite list of finite
abelian groups, so a common exponent T kills every torsion class.
- Then a bounded multiple of the actual divisor qD (for suitable
integral q from the coefficient set) is principal.
The interesting technical point is that the argument never assumes
any particular bound on the intermediate rational image varieties of
Z; it obtains a bounded bound by combining topological uniformity of
the resolution family with the Kummer-theoretic torsion formula.
7. BIG BASES
The final corollary of Section 4 is the version actually invoked to
finish Theorem 1.1 (paper Corollary 4.7):
PROPOSITION (Big base). Under the hypotheses of Proposition 4.1,
suppose D_Z is big. There is 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).
The proof is short. On a smooth determination W -> Z with p M_W nef
Cartier, resolve the boundary support. The adjoint of the resulting
log-smooth pair has coefficients in a DCC set and a fixed Cartier
denominator. Polarized effective birationality (Birkar-Zhang 2016,
Thm 1.3) then gives a uniform degree whose rounded system on W is
birational, and hence a uniform rounded birational system downstairs
on Z. Replacing by a common multiple preserves the section ratios
and makes the degree divisible by p_0.
The section-ratio comparison is an explicit divisorial identity:
H^0(X, O_X(m K_X)) = psi^{m/p_0} f^* H^0(Z, O_Z(floor(m D_Z)))
in divisible degrees. Every section upstairs comes from pulling back
a section downstairs after rescaling by psi^{-m/p_0}. This works
because the fibration has connected fibres and the pulled-back
divisor is the canonical bundle formula. It converts the birational
rounded system on Z into the statement that the pluricanonical
section ratios on X generate the whole function field of Z, which
is exactly the conclusion needed for the final induction step.
Corollary 4.7 is the terminal application of Section 4. Every other
piece of machinery in Parts V and VI is needed only to justify that
the hypotheses of this corollary actually hold in the situation that
arises, namely at the intermediate step of the induction.
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PART III. STRUCTURAL REDUCTIONS FOR THE INDEX
(paper Section 5, first half)
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8. WHAT AN UNBOUNDED INDEX SEQUENCE MUST LOOK LIKE
Recall our goal from Part I: bound the canonical index of a projective
klt variety V of dimension n with K_V ~_Q 0. That is Theorem 1.2 in
the zero-boundary case, K_n. Suppose for contradiction that no such
bound exists. Then there is a sequence V_i of such varieties whose
canonical indices r_i tend to infinity.
The slogan of Section 5 is that such a sequence can be massaged into
a very special form. The precise statement (paper Proposition 5.1):
PROPOSITION (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:
(1) A smooth projective resolution of V_i has irregularity zero,
and the birational group 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 having
positive-dimensional base and fibre has general-type smooth
geometric generic fibre. The same holds for rational
fibrations of V_i with a trivial group action.
Each of these four bullets is a structural reduction: it is the
result of taking the initial unbounded sequence and discarding or
replacing terms until the surviving terms have the property. The
four reductions are logically independent and are done in four
steps. We go through them in order.
9. PRODUCT COVERS AND RATIONAL IMAGES
The first reduction uses Beauville-Bogomolov heavily. Its content is
that an unbounded index sequence consists only of "single-block"
varieties: a blow-up of Beauville-Bogomolov pieces cannot have
unbounded index, so if we see unboundedness, the decomposition must
be trivial.
Step 1: product covers bound the index. Take a klt variety X of
dimension n with K_X ~_Q 0, and get its Beauville-Bogomolov cover
P = A x B_1 x B_2 x ... x B_r -> X,
with A the abelian piece and each B_i irreducible CY or irreducible
symplectic. Suppose there are at least two positive-dimensional
blocks.
Lemma 3.6 (equivariant comparison) provides a normal Galois cover
R -> X on which a bounded-index subgroup of the deck group preserves
each factor field. Transformations of the factor fields on this
subgroup are regular automorphisms of the associated Stein factors
R_i. Each R_i is a connected quasi-etale cover of its B_i, hence
canonical with trivial canonical divisor, and by Lemma 3.8
(finite-character), its volume characters have bounded order since
dim R_i < n (induction).
The product R = A x prod B_i has volume character equal to the
product of its factor characters, so the character order on that
bounded-index subgroup is the least common multiple of the factor
character orders -- bounded. Multiplication by the bounded subgroup
index kills the character of the full Galois group. So the canonical
index of X is bounded in the product case.
Step 2: purely abelian case. If P = A is a single abelian block of
dimension n, deck transformations act on H^n(R, Z), which has rank
binom(2n, n). The canonical character is an eigenvalue of an
integral matrix of that rank. Its cyclotomic degree is at most that
rank, hence there are only finitely many possible orders of that
eigenvalue, and their LCM is a bound for the canonical index.
Combining Steps 1 and 2: if X has two or more positive-dimensional
Beauville-Bogomolov blocks, or is purely abelian with all blocks
abelian, the index is bounded. So we may discard all such X from the
unbounded sequence. Henceforth V has exactly one nonabelian block.
Step 3: rational images of a single block. This is Lemma 5.4 of the
paper, restated:
LEMMA. Suppose 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 X of dimension less than n has
rationally connected smooth projective resolution.
Proof sketch:
- Irregularity zero. A global one-form on a smooth projective model
of X pulls back injectively to a reflexive one-form on P. The
pullback is defined at every codimension-one point, and extends
reflexively. But a block has no reflexive one-forms by definition.
So H^0(W, Omega^1_W) = 0, hence by Hodge symmetry h^1(O_W) = 0,
i.e. irregularity zero.
- Rational images are rationally connected. 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's
pseudo-effectivity criterion, K_T is pseudo-effective; by LA
nonvanishing, there is a nonzero eta in H^0(T, m K_T) for some m.
Resolving the composite map from P, we get u: W -> P and
f: W -> T, both smooth, and a nonzero section
s = f^*eta in H^0(W, (Omega^j_W)^{otimes m}).
At a generic point, s = a * beta^{otimes m}, where beta is a
decomposable rational j-form (pulled back from a top form on T).
Canonicality of P and K_P ~ 0 give: K_W ~_Q E where E is effective
and u-exceptional. So K_W H_W^{n-1} = 0 for H_W an ample pulled-
back divisor. Generic semipositivity of the cotangent bundle
(Campana-Paun) then says every torsion-free quotient of
(Omega^j_W)^{otimes m} has nonnegative degree against H_W^{n-1}.
Contraction of s to a line bundle gives: the divisorial zero
locus Z of s has Z * H_W^{n-1} <= 0, and since every nonexceptional
prime has strictly positive degree against H_W^{n-1}, Z is
exceptional.
At a generic codim-one point, ord(a) + m * b = 0 for b the minimum
order of beta's coefficients in a regular frame. So a's divisor
has all coefficients divisible by m. Adjoin an m-th root z of a;
this gives a finite quasi-etale cover on which z * beta is a
nonzero holomorphic j-form with no divisorial pole.
This is exactly what the block forbids. In the Calabi-Yau case the
only holomorphic forms are 0 and top forms. In the symplectic case
they are scalar multiples of sigma^q (sigma^q a symplectic power).
But z beta is decomposable and lives in degree j < n; contracting
it against a nondegenerate top form gives a candidate vector field
or something of kernel dimension n - j > 0, which no sigma^q can
support. Contradiction.
- Finally GHMS: a proper rational image has a maximal rationally
connected quotient with a smooth non-uniruled base. A
positive-dimensional such base would be a forbidden rational
image of X from the previous bullet. So the maximal RC quotient
is a point, and the image is rationally connected.
This covers part (2) of the sequence proposition. Part (1)'s
irregularity-zero half is proven here; the countability of Bir(V_i)
comes in Section 10 (Step B). Part (3) (moving divisors are big)
and part (4) (general-type fibres) are also handled in Section 10.
10. TERMINALITY AND MOVING DIVISORS
We now prove parts (1), (3) and (4) of the sequence proposition.
The reductions all use the two conclusions of Section 4 (the moduli
denominator bound and the rc-torsion bound) together with MMP
termination and the Beauville-Bogomolov analysis of Section 9.
Step A: terminalize. First, discard all noncanonical X from the
unbounded-index sequence. Here is why. If X is not canonical, choose
a smooth resolution W -> X exhibiting a negative discrepancy. Then
K_W ~_Q E for some effective exceptional E with a negative
coefficient. If K_W were pseudo-effective, LA would give a nonzero
pluricanonical section, hence an effective divisor rationally
linearly equivalent to E. Pushing that down to X shows the rational
function has effective principal divisor; by projectivity, that
function is constant, and the effective divisor equals E -- which is
impossible since E has a negative coefficient. So K_W is not
pseudo-effective. BDPP makes W uniruled. Its maximal rationally
connected quotient has dimension less than n, and Section 9's Lemma
5.4 (applied to X, which is a single-block quotient) forces the
quotient's base to be a point. Then Proposition 4.6 (rc-torsion,
applied with zero boundary) bounds the canonical index of X.
Therefore the unbounded sequence contains only canonical varieties.
Once we know X is canonical, we extract a projective crepant
Q-factorial terminalization by BCHM's extraction theorem. On a fixed
smooth resolution there are only finitely many exceptional divisors
of discrepancy zero, so extracting exactly those (and doing a small
Q-factorialization when necessary) gives a Q-factorial terminal
model V -> X that is crepant. Because the modification is crepant,
K_V ~_Q 0 implies K_X ~_Q 0, and conversely, and the principal
multiples of K match: the canonical index is preserved. Replace X
by V in the sequence. So parts (1) and the collapse to terminal
holds.
Step B: countability of Bir(V). The mechanism is Lemma 8.2 of U4,
which we summarize. Let W -> V be a smooth resolution. The Picard
group of W is countable: the exponential sequence and
H^1(O_W) = 0 (irregularity zero from Step A3 above) give this. Its
image under pushforward is a countable subgroup of Cl(V), so Cl(V)
is countable. Now consider birational self-maps phi: V - -> V. Any
such phi is an isomorphism in codimension one. On a common
resolution, the two pullbacks of a trivializing canonical form are
proportional; terminality identifies their zero supports with the
exceptional divisors; so those supports coincide. Group the maps by
the Weil class of the strict transform of a fixed very ample
divisor. Within one group, 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: a
positive-dimensional linear algebraic group cannot act birationally
on a smooth complete variety carrying a nonzero pluricanonical
system. Thus each class contributes only finitely many maps, so
Bir(V) is a countable union of finite sets, hence countable.
Step C: every moving effective divisor is big. Suppose J is an
effective Weil divisor on V with Iitaka dimension kappa(V, J) in
(0, n). We use MMP and the index bound. Since V is Q-factorial,
small delta > 0 gives a klt pair (V, delta J). LA gives a good
model; call it V', and let f: V' -> Z be the semiample contraction
of the adjoint. The canonical class is preserved crepantly, so V'
is still klt with K_{V'} ~_Q 0 and unchanged canonical index. Apply
the moduli-denominator bound (Prop 4.1 of paper) to V' -> Z; its
generalized adjoint D_Z is rationally linearly trivial. Since Z is
a positive-dimensional rational image of V, its smooth resolution is
rationally connected by Section 9. So Proposition 4.6 (rc-torsion)
applies and gives a uniform principal multiple of D_Z, hence of
K_{V'}. The bound depends only on n and not on J or delta. Discarding
finitely many terms of the sequence makes this impossible. So every
positive-Iitaka divisor is big.
Step D: general-type fibres. Combined with Step A (terminality) and
Step C (moving divisors are big), we get (3) and (4) via:
LEMMA (General-type fibres, paper Lemma 5.5). Let V be projective
and Q-factorial and terminal, with K_V ~_Q 0, and suppose every
effective Weil divisor of positive Iitaka dimension is big. Let
pi: Y -> V be the cyclic index cover with deck group G. Then every
intermediate G-equivariant rational fibration of Y has general-
type smooth geometric generic fibre. The same holds on V with no
group action.
Proof sketch:
- The cover is connected (minimality of the index) and quasi-etale.
Y is terminal: for an exceptional valuation over Y, its image
under the finite cover to V is exceptional over V (since the
cover is quasi-etale), and Y's log discrepancy is positive because
V's is. Also K_Y ~ 0.
- 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) at the same transcendence degree, and is relatively
algebraically closed. So the fibration descends (birational
fibration of V).
- Resolve maps, get u: U -> Y smooth and f: U -> T', g: U -> Z'
with f constant on generic fibres of g. Take a general very ample
hyperplane H on Z' and pull it back via the rational map from V;
closure gives a divisor D on V. Properness ensures this is
defined in codimension one. The section ratios of the pullbacks
vary with the hyperplane, so kappa(V, D) > 0, hence D is big
(this uses Step C).
- Pull back to U: q^*D = g^*H + E, with E >= 0 exceptional over
Y. Fix the resolution, then choose H general so it contains no
generic image of any exceptional prime divisor: E's coefficients
are then effective and q^*D = g^*H + E holds with strict
effectivity.
- Let F be the smooth geometric generic fibre of f. The class
q^*D|_F is big (D is big and F is not contained in the support of
the negative part). Since g^*H|_F ~ 0, the class E|_F is big.
Terminality gives K_U ~_Q sum d_a E_a with d_a > 0 over a
u-exceptional sum. Some positive rational multiple of E is
coefficientwise dominated by this sum. So K_F = K_U|_F is big.
The proof for V is the same without the cover.
This completes the proof of Proposition 5.1.
============================================================
PART IV. SCALAR ESTIMATES
(paper Section 5, second half)
============================================================
11. TWO MEASUREMENTS: GAMMA AND EPSILON
Two scalar invariants measure how big a polarization P is on a
variety V, from the "point" direction (order of vanishing) and the
"curve" direction (degree of curves through a point).
Order. For a big, nef, semiample rational Cartier divisor P on V,
and a smooth point x in V, consider nonzero sections s of O_V(kP)
in divisible degrees k = m * k_0. The order of vanishing ord_x(s) of
s at x grows linearly in k for fixed s. Its normalization ord_x(s)/k
is scale-free. The invariant gamma(P; V, x) is the supremum of this
normalized order over all nonzero sections in all sufficiently
divisible degrees. Its value at a very general point of V is
denoted gamma(P; V). Intuitively, gamma(P) records: how many jets at
a general point can pluricanonical sections encode, per unit of P.
Seshadri constant. The very general Seshadri constant
epsilon(P; x) is the infimum of P.C / mult_x(C) over all integral
curves C through x, where P.C is the degree of P restricted to C and
mult_x(C) is the multiplicity of C at x. It is again scale-free
after rescaling P. Taking the very general value over x gives
epsilon(P). Intuitively, epsilon(P) records: how "expensive" is a
curve through a general point of V, in units of P.
Both invariants are homogeneous under positive rational rescaling
of P. Moreover they are bounded above/below in a first-nontrivial
way by the top self-intersection P^n:
gamma(P; V) >= (P^n)^{1/n},
because asymptotic pluricanonical section counts at degree l give a
Riemann-Roch estimate proportional to P^n, while vanishing to order
l at a smooth point imposes at most binom(n+l-1, n) linear
conditions.
The scalar-estimate proposition (paper Prop 5.3) puts a ceiling on
gamma and a floor on epsilon, both proportional to (P^n)^{1/n},
provided the variety satisfies the general-type-fibre hypothesis
from Part III:
PROPOSITION (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 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},
epsilon(P) >= c_n (P^n)^{1/n}.
The constants are independent of V, of the group G, and of the
denominator of P.
Notice what this proposition says. The upper bound says: order of
vanishing at a very general point is at most C_n times the natural
scale (P^n)^{1/n}. The lower bound says: curves through very general
points have degree at least c_n times the same scale. Both bounds
are uniform in the whole family described. Without the general-type
hypothesis both would fail: a fibration whose fibres are varieties
of very large canonical degree and very small height would break
them.
The proof has two independent parts. First an upper order bound on
gamma, then a lower curve bound on epsilon that uses the upper bound.
We take them in order.
12. THE UPPER ORDER BOUND
If the upper bound fails, a rescaling argument produces a sequence
of (V, P) with P^n -> 0 and gamma(P; V) > 1. We will follow that
sequence and derive a contradiction against the general-type-fibre
hypothesis.
Step 1: setup. Take small projective Q-factorializations of the V in
the sequence and pull back P. After passing to a subsequence, we have
fixed dimension d and data (Z, N) such that:
(A) Z is projective, Q-factorial and klt; N is big, nef, and
semiample; and N^d -> 0.
(B) A smooth model of Z has nonnegative Kodaira dimension.
(C) K_Z is rationally dominated by a_0 N for a fixed positive
rational a_0.
(D) gamma(N; Z) has a positive lower bound.
These conditions hold initially by the canonicality of V and
crepantness of the Q-factorialization. They cannot hold in
dimension one, because the order of a section on a curve is bounded
above by its degree against P. So if we can show that conditions
(A)-(D) persist to a strictly lower-dimensional family when the
order is high, then the initial assumption is false.
Step 2: shrink dimension. Choose d+1 fixed, equally spaced, positive
rational levels strictly below the order lower bound in (D). For
late sequence terms, all these levels lie above (N^d)^{1/d}. The
tracking lemma (paper Lemma 5.8, taken from U4) then supplies a
covering family of integral subvarieties of dimension e strictly
between 0 and d, such that the smooth projective model W of a
geometric generic member satisfies
(i) N^e restricted to W is at most (2/Delta)^{d-e} N^d.
(ii) K_W * (N|_W)^{e-1} is at most
(K_Z + (2d/Delta) N)|_W * (N|_W)^{e-1}.
Here Delta is the spacing of the levels chosen. Combining (ii) with
K_Z <= a_0 N and N^d -> 0 gives
(N|_W)^e -> 0,
K_W * (N|_W)^{e-1} <= C' * (N|_W)^e
for a fixed C', independent of the sequence. The tracking lemma also
supplies kappa(W) >= 0.
Step 3: Iitaka fibres. Apply the small-volume lemma (Lemma 5.3,
below). The Iitaka fibre U of W (possibly all of W if kappa(W) = 0,
but always positive-dimensional) has kappa(U) = 0 and
vol(K_U + L_U) -> 0,
where L_U = N|_W restricted to U.
Step 4: pass to a good model. Represent L_U by a klt boundary Gamma
from a divided general free member. LA gives a Q-factorial good
model Z' of (U, Gamma); let N' be its semiample big adjoint. Then
(N')^{dim Z'} -> 0, and K_{Z'} is dominated by N'. Additionally, on
a common resolution, one gets a polarization comparison
x^*(L_U) <= y^* N'
obtained from an explicit fixed-part argument: write
x^*(K_U + Gamma) = y^* N' + F with F >= 0 exceptional over Z', then
show that F is dominated by x^* D for a divisor D rationally
linearly equivalent to K_U (which exists because kappa(U) = 0).
Step 5: iteration and termination. If gamma(N'; Z') -> 0 on a
subsequence, stop. Otherwise, pass to a subsequence with a positive
lower bound on gamma(N'; Z') and repeat steps 2-4. All conditions
(A)-(D) persist at the new dimension. Dimension one is impossible.
So the iteration terminates at some dimension, having strictly
decreased each time.
Step 6: trace back to V. Going backward through the dimension-
decreasing steps produces covering members A subset V of positive
dimension, whose smooth geometric generic models A^* satisfy
kappa(A^*) = 0,
gamma(P|_A; A^*) -> 0.
Careful: each polarization comparison in the iteration is an
injection from the complete space of the restricted preceding
polarization into that of the next one, in divisible degrees,
preserving order at a general point (multiplication by a fixed
effective difference preserves the vanishing order at a generic
point of a sweeping member). Composition of these injections
across all stages converts the early-to-zero order bound into the
single-to-zero bound above.
Step 7: chain contradiction. Mark these covering members at generic
smooth points and include all G-translates. The order bounds survive
because g^*P ~_Q P. Apply the chain lemma (paper Lemma 5.7, from
U4): chains of generic movements produce generic leaves of a
rational fibration. If the leaf had full dimension n, the chain
lemma's order estimate at the leaf would force gamma(P; V) -> 0,
contradicting the lower bound from the initial assumption. So the
chain leaf has positive but strictly smaller dimension: it is the
generic fibre of a G-equivariant intermediate rational fibration of V.
That geometric generic leaf is covered by the kappa(A^*) = 0 members
from Step 6: the chain lemma ensures each generic marked member is
contained in its leaf. The covering assertion of the tracking lemma
then forces the leaf to have nonnegative Kodaira dimension. It
cannot be of general type. This contradicts the general-type-fibre
hypothesis in the scalar-estimate proposition.
So the upper order bound holds.
13. THE SMALL-VOLUME LEMMA
We used above a lemma (paper Lemma 5.3) that says: a sequence of
small-volume e-folds with controlled canonical gives Iitaka fibres
whose own adjoint volume also goes to zero. Precisely:
LEMMA (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, kappa(U_i) = 0, and
vol(K_{U_i} + L_i|_{U_i}) -> 0.
The interesting case is 0 < kappa(W_i) < e. One first extracts a
flat proper model f: W -> T with W birational to W_0 and T birational
to the Iitaka base; flattening and normalization do not change the
smooth generic fibre. Then one exploits weakly positive direct
images f_* O_W(q l (K_{W/T} + L)), using Fujino's twisted weak
positivity theorem. Sections over a big open of T multiply
fibrewise into sections of an enlarged adjoint on W. Section counts
on base and fibre multiply, giving the lower bound
vol(a K_{W_0} + L_0) >= ((e!)/(g! j!)) * H^j * vol(K_U + L|_U)
with g = e - j and H^j >= 1. On the other hand, the left side goes to
zero along the sequence (via the adjoint-volume upper bound
vol(a K_W + L) <= (a C' + 1)^e L^e derived from the initial small-
volume hypotheses). So the right side goes to zero, forcing
vol(K_U + L|_U) -> 0.
14. THE LOWER CURVE BOUND
Once the upper order bound is known, the lower bound on epsilon
follows by a thickening argument. Rescale P so 1 <= P^n <= 2, and
let gamma(P; V) <= D_0 be the upper order bound. Suppose epsilon(P)
has no positive lower bound. Then there is a covering marked curve
family on V whose generic member C_x through a very general point x
satisfies P.C_x / mult_x(C_x) -> 0. Include G-translates. The chain
lemma gives chain leaves of a fixed dimension h on a subsequence,
with (P^h H) -> 0.
Since P^n is between 1 and 2, we must have 0 < h < n. Resolve the
chain fibration and choose a very general smooth fibre F of it, of
dimension h. Because F is a fibre of a smooth morphism, its normal
bundle is trivial. Let s = n - h.
The ideal sheaf I_F of F in V has powers I_F^u that filter the
structure sheaf O_V / I_F^u. The associated graded pieces are
Sym^j (I_F / I_F^2) for 0 <= j < u, of total rank binom(u + s - 1, s).
Consequently the rank of the map
"restrict global sections of l P to the u-th order neighbourhood
of F" is at most
binom(u + s - 1, s) * h^0(F, l P|_F).
Take u = ceil(D l) with a fixed D > D_0. In large divisible degrees
l, for a fixed chain fibration, the leading coefficient of that
rank is
(D^s / (s! h!)) (P^h F) l^n,
which tends to zero along the sequence, while the ambient section
count has leading coefficient P^n / n! >= 1/n!. Thus, first picking
a sufficiently late sequence term and then a sufficiently large
divisible l, there is a nonzero ambient section vanishing to order
u along F.
At any smooth point x of F, this section has order >= u = ceil(D l)
at x, so its normalized order is at least D > D_0, contradicting the
upper bound gamma(P; V) <= D_0. Hence epsilon(P) is bounded below by
c_n (P^n)^{1/n} for a positive constant c_n.
This completes the scalar-estimate proposition. Both inequalities
hold uniformly in the family, and both constants depend only on n.
15. WHERE THIS FEEDS IN
The scalar estimates are needed in two places in the remaining
arguments:
- Bounding Seshadri constants from below gives a lower bound on
epsilon(pi^*L) for the pullback of a fixed polarization to the
cyclic cover pi: Y -> V of Part III, in terms of the covering
degree r. This is the "small curves exist everywhere" fact that
makes the transport argument of Part VI bite.
- Bounding order of vanishing from above controls how many curve
degrees can be small, feeding into the diagonal-product estimate
of Part V.
Both numbers are what makes the entire low-index contradiction
argument work smoothly in every dimension, not just up to four.
============================================================
PART V. FROBENIUS JETS AND COMPATIBLE LEAVES
(paper Section 6)
============================================================
16. THE DIAGONAL PRODUCT AND WHAT IT IS FOR
Part III reduced the missing bound K_n to a hypothetical sequence of
projective Q-factorial terminal n-folds V with growing canonical
index r, satisfying four structural properties (countable
birational group, rationally connected rational images, moving
divisors are big, general-type fibres). Part IV then gave uniform
control on two scalar invariants of any ample polarization on such
a V: gamma from above and epsilon from below. Part V now uses a
construction on many copies of the cyclic cover to derive a
contradiction, disposing of the hypothetical sequence and thus
proving K_n.
The construction: fix V, take the cyclic cover pi: Y -> V of degree
r, and for each integer t >= 2 consider the diagonal quotient
Z_t = Y^t / mu_{r,diag},
where mu_r acts on every slot by the same deck transformation. The
natural map
theta_t: Z_t -> V^t
is finite of degree r. Pull the product polarization L^{boxplus t} =
sum_i pr_i^* L back to Z_t to get
P_t = theta_t^* L^{boxplus t}.
We want upper bounds on the Seshadri constant epsilon(P_t), because
by the theory of very general Seshadri constants, a bounded
epsilon(P_t) is equivalent (roughly) to the existence of curves
through very general points with small P_t-degree. Those small
curves are what the transport argument of Section 18 uses as raw
material.
Two facts interact. The lower bound epsilon(pi^*L) >= c r^{1/n}
from Part IV means the pullback has large Seshadri constant. But
somehow the diagonal P_t has bounded Seshadri constant. The tension
between Y and Z_t is what generates the contradiction.
The precise statement of the diagonal jet bound is Proposition 6.1
of the paper, and we state it and describe its proof in Section
17. Then Section 18 gives the transport contradiction
(Proposition 6.11). Section 19 closes the loop back to K_n.
17. THE DIAGONAL JET BOUND (PROPOSITION 6.1)
PROPOSITION (Diagonal jet bound). Let V be a normal projective
canonical complex n-fold (n >= 2) with K_V ~_Q 0 and canonical
index r. Let pi: Y -> V be its canonical cyclic cover (of degree
r). Suppose L is ample rational Cartier and C_1 > 0 is rational,
with
1 <= L^n <= 2, gamma(L; V) <= C_1.
Choose a positive rational b_0 such that
3 (2b_0)^n < 1/4, b_0 (C_1 + 1) < 3/4.
For every integer t >= 2, set Z_t = Y^t / mu_{r,diag}, theta_t:
Z_t -> V^t the quotient map, P_t = theta_t^* L^{boxplus t}. Then
P_t is ample and
epsilon(P_t) <= (n+1) t^2 / b_0.
The constant C_2 := (n+1)/b_0 does not depend on r.
The proof is a determinant argument comparing two different upper
bounds on the order of vanishing of a single section at a very
general point. If the diagonal bound fails, ordinary jets (in the
fixed torsor Z_t) produce a full-rank matrix; we specialize to
large positive characteristic, use Frobenius jets to show that
matrix has rank drop at the diagonal; this forces a maximal minor
to vanish to very high order at a specified point. But a movable
curve test produces a much lower bound for the same order.
Contradiction.
Three small technical lemmas support the proof.
(1) Truncated-power slopes (paper Lemma 6.4). Take E a rank-rho
vector bundle on a smooth projective curve C of genus g over a
field of characteristic p > 0, with mu_min(E) >= 0. Let A_j(E)
be the symmetric algebra modulo p-th powers of degree-one
elements from 0 to p-1, so that A_j(E) is the "truncated j-th
power". Then with G = max(0, 2g-2), u = rho(p-1), and
c_2 = rho(rho-1)G, every nonzero graded piece satisfies
mu_max(A_j(E)) <= (p-1) deg(E) + c_2, 0 <= j <= u.
The argument uses the canonical connection of absolute Frobenius
(Langer 2004). Frobenius pullback of a stable bundle has slope
p times the original, minus a bounded error; iterating gives a
bound on the minimal slope of iterated-Frobenius pullback; and a
quotient argument then controls all pieces of A_j(E). The
sum of ranks of all A_j(E) is p^rho.
(2) Sections from a fixed flag (paper Lemma 6.6). Given a fixed
complete-intersection flag in an n-fold W cut by very ample
divisors in |l_i H|, ending in a smooth curve C, and a bundle E
of rank e with mu_max(E|_C) <= M and M >= 0,
h^0(W, E) <= e (M+1) prod_i (1 + M/d_i),
where d_i = l_i H . C. This is a successive-divisor-sequence
bound.
(3) Orders in an external product (paper Lemma 6.7). If Q_i are
line bundles on integral varieties W_i with nonzero sections,
and if a_i is the maximum order of a section of Q_i at a smooth
point x_i, then the maximum order of a section of
boxplus_i pr_i^* Q_i at the point (x_1, ..., x_t) is at most
sum_i a_i. This is Kunneth adapted to the local order
filtration.
Now the argument.
Stage 1 (ordinary jets). Choose a smooth very general z in Z_t
with local coordinates above V_reg, and a rational c with
(n+1) t < c < b epsilon(P_t; z).
On the blow-up of z, the -c E divisor (E the exceptional) below
b P_t produces an ample class; Serre vanishing on the blow-up gives
a k_0 with surjectivity of H^0 onto ordinary jets through degree
(n+1)t k_0. Choose k_0 also so that Q = k_0 b L is Cartier. The
result (paper 6.7):
H^0(Z_t, k_0 b P_t) -> O_{Z_t, z} / m_z^{(n+1)t k_0 + 1}
is surjective.
The index cover gives explicit coset-tensor decompositions: a
section in the fixed jets can be written as sum of terms
tau^{-m_i} s_i with s_i in H^0(V_reg, Q + m_i K_V) for 0 <= m_i < r,
sum m_i ≡ 0 mod r. Extend each s_i to the resolution W. Only
finitely many such tensors are needed. Retain only the finite jet
data.
Stage 2 (movable curve test). Choose an ample rational divisor
H = L + delta H_amp with small delta > 0, so that
H^n <= 3, L H^{n-1} <= H^n,
(q + t - 1) K_W H^{n-1} <= b_0 H^n.
Here q = 2tr. Generic semipositivity of Omega^1_W (Campana-Paun)
combined with Mehta-Ramanathan's restriction theorem gives a fixed
complete-intersection flag whose ending curve C satisfies
mu_min(Omega^1_W|_C) >= 0.
Choose a very general x in W, isolating it by blowing up with
sigma: W^ -> W, exceptional J. Define D^+ = b L + (q+1) K_W and
d_0 = b (C_1 + 1). Sections of D^+ in divisible degrees are just
pluricanonical sections of L multiplied by a fixed exceptional
factor (using rK_V ~ 0), and the fixed factor has order zero at x.
So the normalized order of a section of D^+ at x is at most b C_1.
If sigma^* D^+ - d_0 J were pseudo-effective, interpolation with the
big divisor sigma^*D^+ would make sigma^* D^+ - d J big for some
rational d strictly between bC_1 and d_0, producing a section
contradicting the order bound. So sigma^* D^+ - d_0 J is not
pseudo-effective. BDPP's duality theorem gives a strongly movable
curve beta on W^ with
D^+ . beta < d_0 (J . beta),
J . beta > 0,
L . beta >= 0,
K_W . beta >= 0.
The last two inequalities select the "right" generator of the
movable cone.
Stage 3 (specialize to char p). Spread the finite data over an
integral finitely generated Z-algebra inside C. After shrinking,
smoothness, geometric integrality, ampleness, dominance of the test
morphism, and the intersection numbers survive. Invert r and the
finitely many denominators. For every remaining geometric fibre in
characteristic p, the movable test beta preserves its nonnegative
pairing with effective divisors.
Stage 4 (Frobenius full rank). Let F: W -> W' be relative Frobenius
in characteristic p. Set
N = floor(p / k_0), B = N Q,
E_e = F_* O_W(B), s = p^n, R = s^t.
E_e has rank s. Products of N of the fixed jet sections span
Frobenius-thick jets through degree n t (p-1); this is the
Mustata-Schwede comparison of ordinary and Frobenius jets.
Etaleness identifies the completed local rings at the torsor lift
and its tuple. For an N-fold product of pure tensors with slot
exponents m'_i, choose a_i in [r, 2r-1] with p a_i ≡ m'_i mod r
and sum a_i ≡ 0 mod r; then the multiplicative factor
eta^{(p a_i - m'_i)/r} is a p-th power of a unit on the thick
Frobenius point. The result is a column map
⊕_{a_i >= 0, sum a_i = q} (⊗_i H^0(W, B + p a_i K_W))
⊗ O_(W')^t ( - sum_i a_i K_{W',i} )
-> E_e^{boxplus t}
of full rank R at a very general point, hence at the generic
point.
Stage 5 (rank drop at the diagonal). On the diagonal W' sub (W')^t,
each source line bundle restricts to a fixed negative twist of
O_{W'}. Filter by the ideal of the reduced diagonal Delta_F (the
t-fold fiber product over Frobenius) and push forward. The graded
pieces are
G_j = O_W(t B + p q K_W) ⊗ A_j((Omega^1_W)^{oplus(t-1)}),
0 <= j <= n(t-1)(p-1).
Apply truncated-power slopes on the fixed flag curve C. Using
B.C <= p b L.C, H^{n-1} control from Stage 2, and K_W.C >= 0:
mu_max(G_j|_C) <= 2 b_0 p D + c_2 =: M_p,
where D = l_1 ... l_{n-1} H^n and c_2 is independent of p.
The flag bound (technical lemma (2)) gives
h^0(W, G_j) <= e_j p^n ((2b_0)^n H^n + O(p^{-1})).
For large p, the leading coefficient is controlled by
3(2b_0)^n < 1/4. Summing over j: the rank of the column map at any
diagonal point is at most
p^{nt} ((2b_0)^n H^n + O(p^{-1})) <= R/4.
So on the diagonal, the column map has rank at most R/4, whereas
generically it has full rank R.
Stage 6 (determinant contradiction). Pick R columns of the column
map with nonzero determinant. The determinant is a section of an
external product of line bundles Q_i whose divisors we can compute:
the normalized class of each Q_i paired with the movable test beta
is at most D^+.beta, and hence any nonzero slot section of Q_i has
order at most R d_0 at the chosen point. Lemma (3) (external-product
orders) says the determinant has order at most
R t d_0 = R b_0 (C_1 + 1) < 3R/4.
Meanwhile, the rank-drop bound at the diagonal point means at least
R - floor(R/4) rows of the matrix vanish there after constant
invertible row operations. Every determinant then vanishes to
order at least 3R/4 at that point. Contradiction.
This proves Proposition 6.1. The diagonal product P_t has small
Seshadri constant, with an explicit quadratic control in t.
18. THE HIGH-INDEX CONTRADICTION (PROPOSITION 6.11)
Now we use the diagonal bound. The full statement is:
PROPOSITION. 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 -> oo, satisfying:
(a) Bir(V_nu) is countable.
(b) There are ample rational Cartier divisors L_nu and
constants c, C_2 > 0, independent of nu, such that
epsilon(L_nu) >= c,
epsilon(pi_nu^* L_nu) >= c r_nu^{1/n}.
(c) For every t >= 2, on a tail allowed to depend on t,
epsilon(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 builds, for each fixed integer T and sufficiently large
nu, sequences of curves in Z_{nu,t} (t <= T) of bounded
cost P_t-degree divided by marked multiplicity. It arranges those
curves into chains, obtains positive-dimensional leaves of a
rational fibration on V, uses single-slot finiteness to control
projection degrees, gets a degree-one forgetting of one slot from
the others, and turns the tangent distribution of the leaf into
uncountably many birational self-maps of V.
Here is the structure of the argument, slowly.
Step A: curves from small Seshadri. For each t <= T, the strict
inequality epsilon(P_t) < B := C_2 T^2 + 1 gives a covering family
of integral curves on Z_t with bounded ratio P_t . D / mult_z(D).
The relevant theorem is the strict-inequality form of the definition
of Seshadri constant as an infimum over curves.
Step B: closing under projections and deck translates. Take deck
translates of the curve family, and image families under every
ordered coordinate projection Z_t -> Z_{t'} (2 <= t' <= T). Closure
is finite. The bounds are preserved: for a curve map D -> D' of
degree e with marked point z over z', the inequality
e mult_{z'}(D') >= mult_z(D)
(resulting from a general local hyperplane at z') combines with
the fact that P_t minus the pullback of the lower polarization is
nef to give the cost comparison.
Step C: chain leaves. Apply the chain lemma (Lemma 5.7) to the
resulting list of marked covering families. The geometric generic
leaf H_t of the resulting rational fibration xi_t: V^t - -> S_t has
positive dimension h_t and satisfies
P_t^{h_t} . H_t <= (h_t B)^{h_t}.
The finite quotient Z_t -> V^t commutes with the rational quotient,
giving a corresponding leaf on Z_t.
Step D: finiteness of the single-slot projection. Each H_t projects
generically finitely onto its image in every V-slot. Suppose not.
Then for some slot, the image I has dimension j < h_t and
positive-dimensional generic fibre J of dimension a = h_t - j.
The marked point is ambient very general on I, giving the
Seshadri-type bound
L^j . I >= c^j.
Over a geometric generic point of that slot, the fibre of Z_t -> V
identifies with Y^{t-1} (fix one lift in the torsor). The
polarization P_t restricts to the external sum of pi^* L on the
remaining copies. Seshadri bound on pi^* L in restriction:
epsilon(pi^* L) >= c r^{1/n}. So
P_t^a . J >= (c r^{1/n})^a.
Combining and using P_t - pr^* L nef gives
P_t^{h_t} . H_t >= c^j (c r^{1/n})^a,
which contradicts Step C as r -> oo, since h_t <= n T and T is
fixed. So each H_t is generically finite over its image in every
slot, in particular 1 <= h_t <= n.
Step E: degree of a single slot projection. Let d_t be the degree of
H_t over its first-slot image. Then d_t <= K T^{2n}, where
K = max_{1 <= h <= n} ((h (C_2+1)) / c)^h.
Step F: a degree-one forgetting. Projection compatibility gives
h_t <= h_{t-1} for t >= 3 (the leaf of the t-fold chain projects
into the leaf of the (t-1)-fold chain). When equality holds,
W_t -> W_{t-1} is an honest covering of integral varieties, so
the ratio d_t / d_{t-1} = deg(W_t -> W_{t-1}) is a positive
integer.
In the T-1 dimensions h_t, some stretch of length at least (T-1)/n
has h constant. Choose T large so that
(T-1)/n - 1 > log_2 (K T^{2n}).
If all adjacent degrees in that stretch are >= 2, the product of
them is >= 2^{(T-1)/n}, which contradicts the degree bound d_t <=
K T^{2n}. So for some k >= 3,
h_k = h_{k-1} = h, W_k -> W_{k-1} is birational (degree one).
Permutation invariance gives the same conclusion for every
single-slot omission.
Step G: rational recovery of a missing coordinate. Birationality of
W_k -> W_{k-1} says, in function fields, that the coordinate
xi_k determines the last slot from the other k-1 slots. Make this
explicit: define
Lambda = (pr_{<k}, xi_k): V^k - -> D,
the closure of the image of (u, xi_k(u)) in V^{k-1} x S_k, where
S_k is the base of the leaf fibration. The map Lambda is
birational (degree-one forgetting of the last coordinate). Let
Lambda^{-1} be its rational inverse. Then
ev(u, s) := last-coordinate of Lambda^{-1}(u, s)
is a rational evaluation map, giving the last coordinate from the
first k-1 slots and the quotient value.
Step H: transport of tangent directions. The tangent distribution
D_k = ker(d xi_k) at a generic tuple has rank h. By single-slot
finiteness of Step D, it projects injectively to each slot's tangent
space, isomorphically to the lower distribution under every
omission. Let A_i sub T_{V,x_i} be the image in slot i: this is a
distribution of rank h on V that depends only on x_i (a
"field-intersection" argument: in a Grassmann chart, its
coordinates belong to the field of every (k-1)-tuple containing
slot i, so they belong to C(V) in slot i alone).
The transport R(x,y): A(x) -> A(y) obtained by lifting to D_k and
projecting to slot j from slot i 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) span D_k.
Step I: local flows and uncountable birational maps. Choose a
general pair (u, y_0) in V^{k-1} x V, in open sets where
everything is regular. For small z in C^h, let E_z be the ordered
composition of the local holomorphic flows of v_1, ..., v_h for
times z_1, ..., z_h. The simultaneous flows preserve xi_k. So for
y near y_0:
E_z(y) = ev( E_z(u), xi_k(u, y) ).
For fixed z, the right side is rational in y. Equality on an
analytic open shows it extends to a rational self-map of V. The
reversed composition E_z^- of negative-time flows gives a rational
inverse on a smaller analytic open. Each E_z is therefore
birational. The image z -> E_z(y_0) has nonzero derivative and is
uncountable. Thus Bir(V) is uncountable.
This contradicts countability of Bir(V). So no such sequence
exists. The high-index contradiction is complete.
19. CLOSING K_n
The pieces are all in place. Take the high-index sequence from
Proposition 5.1 (Part III). Rescale each L_i so that
1 <= L_i^n <= 2. Part IV's scalar estimates give uniform C_1 and c:
gamma(L_i) <= C_1 and epsilon(L_i) >= c, with a matching lower
bound epsilon(pi_i^* L_i) >= c r_i^{1/n}. Section 17's diagonal bound
gives the C_2 t^2 upper bound for P_t. Part III guaranteed
Bir(V_i) is countable.
So the sequence satisfies every hypothesis of Proposition 18. But
Proposition 18 forbids the existence of such a sequence. Contradiction.
Therefore the canonical indices of projective klt n-folds with
K_X ~_Q 0 are uniformly bounded. This proves K_n in all dimensions.
============================================================
PART VI. THE INDEX STEP AND THE COMPLETION
(paper Sections 3 and 7)
============================================================
20. FROM K_n TO L_n: BOUNDARY LOG CANONICAL INDEX
We have proved K_n. The next step is the pair-index statement L_n,
for log canonical pairs with boundary.
PROPOSITION (Reduction to the zero-boundary klt index, paper
3.1). Assume L_j for every j < n and K_n. Then L_n holds: for
every finite set Gamma contained in [0,1] intersect Q there is an
integer c(n, Gamma) > 0 such that, for every normal projective
lc n-fold pair (X, B) with coefficients in Gamma and
K_X + B ~_Q 0,
c(n, Gamma) (K_X + B) ~ 0.
The proof has two structural halves and one technical descent.
Reduction to a Mori fibre space. Take a crepant Q-factorial dlt
modification. Then T = floor(B) is reduced. Enlarge Gamma to include
1. Pick a small rational epsilon > 0 so that (X, B - epsilon T) is
klt and
K_X + B - epsilon T ~_Q - epsilon T
is not pseudo-effective: its intersection with the (n-1)-power of an
ample divisor is negative. So the klt MMP applied to this pair
terminates in a Mori fibre space f: X' -> Z.
Case (i): dim Z = 0. Then X is Fano type. The bounded complement
theorem (Birkar 2019) gives a bounded c and a complement B^+ >= B
with c(K_X + B^+) ~ 0. The effective divisor B^+-B is numerically
trivial, hence zero by intersection with an ample. Done.
Case (ii): dim Z > 0. On the Mori fibre space the surviving
transform of T is relatively ample (it satisfies epsilon T ~_Q
-(K_X + B - epsilon T) with the right-hand side relatively ample).
So some prime S of T dominates Z. Normalize S: nu: S^nu -> S.
Divisorial adjunction gives
(K_X + B)|_{S^nu} = K_{S^nu} + Theta,
with (S^nu, Theta) a normal lc pair, Theta effective, coefficients
in the fixed rational DCC set of differents (Kollar-Mori). Its
adjoint is Q-linearly trivial. So L_{n-1} bounds a trivializing
multiple. Each term of the arrangement is now within reach.
The generic fibre of f is a normal projective lc pair of dimension
< n with coefficients in Gamma. Applying L_{<n} bounds a trivializing
multiple there; and linear triviality descends in that same degree
to C(Z) by a faithful-flatness argument (Lemma 3.2 part 1).
Now a chain of reductions.
Lemma 3.2 (vertical descent). Let f: X -> Z be a contraction of
normal projective varieties over a field of characteristic zero.
A rational function on X with divisor having no horizontal component
belongs to C(Z) subset C(X). Moreover, if (X, B) is a rational pair
with K_X + B ~_Q 0 and q(K_{X_eta} + B_eta) ~ 0 on the generic
fibre, then there is a rational q-canonical form s and rational
Cartier D ~_Q 0 on Z such that
Div(s) + q B = f^* D.
Reduction 1 (finding the form). Choose q clearing both the
fibre-adjoint trivialization and K_{S^nu} + Theta's. Lemma 3.2
applied to the q-canonical form on X yields
Div(s) + q B = f^* D.
Rewrite as
Div(s) + q B - q T = f^* D (modulo an integral boundary
correction).
The residue of s along S gives a rational q-canonical form v on S^nu
with, by adjunction,
Div(v) + q Theta = (f o nu)^* D.
So bounding the RHS becomes a question of bounding this principal
divisor.
Reduction 2 (Stein factorization). Choose a trivializing q-log form
v_0 on (S^nu, Theta). Then a = v/v_0 in C(S)^* satisfies
Div(a) = (f o nu)^* D. Factor the structure map as
S^nu --g--> Y --h--> Z
with g a contraction and h a finite morphism. Lemma 3.2 part 1
gives a in C(Y). Injectivity of pullback for Cartier divisors
under a surjective map gives h^* D = Div_Y(a).
Reduction 3 (Norm). The norm identity for finite extensions of
function fields gives
(deg h) D = Div_Z ( N_{C(Y)/C(Z)}(a) ).
So the bound on the Cartier divisor D on Z reduces to the degree of
the finite morphism h: Y -> Z. This is the Stein degree.
The Stein degree bound (paper SD, the arithmetic Stein-degree
bound). Applied to the geometric generic
fibre (X_eta, B_eta) over C(Z) with its prime coefficient-one
divisor S_eta (parameter t = 1), this bounds the algebraic closure
of C(Z) inside C(S_eta) -- which is exactly C(Y). Hence deg h is
bounded in terms of the fibre dimension and the base field, hence
in terms of n.
With deg h bounded, D is bounded by the norm calculation. And q is
bounded above by the assumed inductions and Gamma. Taking a common
multiple, we get a uniform multiple c(n, Gamma) of K_X + B. The
result follows.
Two further results are proved alongside and used in Part IV.
Lemma (Volume characters, paper 3.8). 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 with E
reduced, and let omega be a rational top form satisfying
Div(omega) + E = 0. Let gamma be a finite-order automorphism of
the pair, with gamma^* omega = lambda omega. Then
lambda^{c_j} = 1.
Proof. Quotient by the cyclic group <gamma>. The boundary of the
quotient is a normal lc pair with DCC coefficients; its adjoint is
rationally trivial. L_j bounds a trivializing multiple c_j. Pulling
back gives a bound on lambda.
Lemma (Cover comparison, paper 3.6). Under a finite quasi-etale
cover R -> P of a product of blocks, there is a bounded-index
subgroup of the deck group that acts regularly on the individual
factor fields. This is the natural input to Section 9's "product
varieties have bounded index" step.
Combining the lemmas closes the induction: K_n implies L_n, given
L_{<n}.
21. THE COMPLETION OF THEOREM 1.1
Now we reach the theorem we started with. Two more ingredients are
needed. The first is the comparison of pluricanonical sections
between X and its minimal model.
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 "Good minimal models" (LA, Thm 11.1) gives the good
minimal model V. Using a common resolution W, write K_W = q^* K_V + A
with 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, which is
also effective. Conversely, a form regular on W is regular at every
prime divisor of V, hence is a divisorial section there by
normality. Same argument for W -> X. So all degrees agree.
Now the two theorem proofs are short.
PROOF OF THEOREM 1.2 (lc-index). Assume I_j, L_j, K_j for all
j < n. Part V proved K_n. Section 20 proves L_n from K_n and
L_{<n}. Global ACC gives the rational-DCC version from the
finite-coefficient version. So L_n holds for all n.
PROOF OF THEOREM 1.1 (main). Induct on n = dim X. Let X be a
smooth projective n-fold with kappa(X) >= 0. Take V a good
minimal model of X (Lemma 7.1); the semiample canonical divisor
defines a contraction
f: V -> Z, K_V ~_Q f^* D,
where D is a rational ample divisor on Z. This contraction is
the Iitaka fibration. Its base Z has dimension kappa(X).
Case kappa(X) = 0: Z is a point, K_V ~_Q 0. L_n (just proved)
gives a common pluricanonical degree m(n) whose image is a point.
Done.
Case kappa(X) > 0: Section 4's Proposition 4.1 applies. Its
base adjoint K_Z + B_Z + M_Z is rationally linearly equivalent
to D, hence big. Corollary 4.7 (big-base) supplies m(n) whose
section ratios give exactly C(Z) from C(X). And m(n) is
divisible by p_0, which was needed for the section comparison
formula. Done.
Either way, for each fixed base dimension we get a single m.
Take the common multiple over dim Z = 0, 1, ..., n. That single
m works for all cases.
Ratios in a larger multiple: if m is divisible and we pass to qm,
the ratio s'/s_0 of two degree-m sections is also the ratio
(s' s_0^{q-1})/(s_0^q) of two degree-qm sections. So the field
is preserved. This closes the simultaneous induction starting
from dimension zero.
Theorems 1.1 and 1.2 hold in every dimension.
22. OVER OTHER FIELDS
Finally we extend from C to any algebraically closed field of
characteristic zero.
Given a variety (or pair) with all its finite defining data, we can
descend to a finitely generated subfield k_0 over which everything
makes sense. Then embed k_0 in C. Smoothness, integrality,
singularity conditions, dims of section spaces are preserved by
algebraically closed extension (spreading arguments + flat base
change + faithful flatness).
For Theorem 1.2 (index). Include a rational principalization of a
sufficiently divisible multiple of K_V in the descended data. The
uniform trivialization over C descends by the one-dimensional-
section argument (Lemma 3.2 part 1 of the paper, also stated in
Section 20) and extends to k.
For Theorem 1.1 (Iitaka). Include a nonzero pluricanonical section
in the descended data, so that kappa(X) >= 0 survives the descent
and extension. The same m(n) gives a nonempty system after every
extension. To check the field, choose a sufficiently large degree
r divisible by m(n) that also defines the Iitaka fibration. The
products constructed earlier give a dominant rational map from the
degree-r image to the degree-m(n) image. It is birational iff it is
birational after an algebraically closed extension: equivalently,
the corresponding function-field extension has degree one, a
property detected by faithful flatness. Since it is birational
after comparison with C, it is birational over k.
Thus the same m(n) works over every algebraically closed field of
characteristic zero. No field-specific argument is needed beyond
this standard descent-and-compare.
23. THE ARCHITECTURE, SUMMARIZED
Here is the paper's dependency structure, one more time, in the
form of a summary that the reader can compare with the section
headings of the original.
Black-box inputs (external to the paper):
* Log abundance in characteristic zero (LA, Thm 11.1).
* BCHM existence of good minimal models / MMP termination.
* Polarized effective birationality (Birkar-Zhang 2016, Thm 1.3).
* Arithmetic Stein-degree bounds (SD).
* Equivariant chain and tracking lemmas (U4, Lemmas 5.7, 5.8).
* Beauville-Bogomolov decomposition (Druel 2018; GGK 2019;
Horing-Peternell 2019).
* Birkar's boundedness for generalized eta-lc pairs (2025).
* ACC for log canonical thresholds (Hacon-McKernan-Xu 2014) and
generalized discrepancies.
New contributions of the paper:
* Relative denominator bound (Section 4, Part IV of tour).
* Constancy of rc-torsion on rationally connected bases
(Proposition 4.6, Corollary 4.7).
* Arbitrary-dimensional extension of the scalar estimates
(Section 5, Part IV of tour).
* Arbitrary-dimensional extension of the diagonal jet bound and
the compatible-leaf transport argument (Section 6, Part V).
Induction architecture:
To prove I_n and L_n, assume I_j and L_j for j < n.
1. K_n (zero-boundary klt index): Parts III, IV, V.
2. L_n (boundary lc index): K_n and L_{<n} and SD.
3. I_n (Iitaka theorem): L_n, Corollary 4.7, and
polarized effective birationality.
Then induction proceeds.
The dependencies are strict and short. The deepest black box is LA
-- the good minimal model / abundance theorem -- and the deepest
new contribution is the diagonal-transport argument of Part V.
Everything else is either standard MMP or an arithmetic/topological
uniformity statement.
A reading guide for the original paper:
Sections 1-2: Setup and conventions. Part I of the tour.
Section 3: Boundary lc from zero-boundary klt; volume
characters; cover comparison. Section 20 of tour.
Section 4: Moduli denominator; rc-torsion; big-base.
Sections 4-7 of tour.
Section 5: Structural reductions; scalar estimates.
Sections 8-15 of tour.
Section 6: Frobenius diagonal; compatible leaves and
transport. Sections 16-19 of tour.
Section 7: Completion and characteristic-zero field
descent. Sections 21-22 of tour.
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GLOSSARY
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algebraic variety -- the zero set of polynomial equations.
birational -- two varieties are birational if they have
isomorphic dense open subsets.
birational group Bir(X) -- the group of birational self-maps.
blow-up -- a birational map that replaces a closed subvariety
by its projectivized normal bundle.
b-divisor -- a divisor on a higher birational model, pushed
forward to the variety along the map.
Calabi-Yau block -- irreducible simply-connected variety with
trivial canonical class and no intermediate holomorphic forms.
canonical divisor K_X -- the divisor of any meromorphic top
form on X; independent of the choice up to linear equivalence.
canonical index -- the smallest r > 0 such that r K_X is an
integral principal divisor. Measures the "denominator" of the
canonical class.
complete-intersection curve -- the common zero locus of
codimension-minus-one number of very ample hypersurfaces.
DCC set -- descending chain condition set, subset of [0,1] with
no strictly decreasing infinite chain.
dlt -- divisorially log terminal. See Part I.
Du Bois -- a singularity class for which cohomology and base
change behave well; slc varieties are Du Bois.
effective divisor -- a nonnegative formal sum of codim-one
subvarieties.
flip -- a small birational MMP step, replacing a K-negative
curve locus with a K-positive one.
Frobenius -- the p-th power map in positive characteristic.
generic fibre -- the scheme-theoretic fibre over the generic
point of the base.
geometric generic fibre -- the generic fibre, base-changed to an
algebraic closure.
good minimal model -- a minimal model on which the adjoint is
semiample.
Iitaka dimension kappa(X) -- the growth rate of pluricanonical
sections.
Iitaka fibration -- the fibration defined by all pluricanonical
sections in all sufficiently large divisible degrees.
klt -- Kawamata log terminal. See Part I.
lc -- log canonical. See Part I.
minimal model -- a Q-factorial terminal model on which the
canonical divisor is nef.
MMP -- minimal model program; the sequence of K-negative
contractions and flips that simplifies a variety.
moduli b-divisor -- the M_Z part of the canonical bundle
formula.
multiplicity mult_x(C) -- the order of vanishing of the local
equation of a curve at a point.
nef -- numerically effective; nonnegative intersection with
every curve.
order of vanishing ord_x(s) -- the order to which a section s
vanishes at a point x.
pluricanonical -- referring to the bundle or divisor m K_X for
m > 0.
polarized effective birationality -- a uniform bound (in a
bounded family) on how many pluricanonical sections suffice
for birationality.
Q-factorial -- a variety on which every Weil divisor is
Q-Cartier.
rational Cartier divisor -- a Q-Cartier Q-divisor.
rational map -- a morphism defined on a dense open subset.
section ratio -- s'/s for two sections of the same bundle.
semiample -- the pullback of a nef divisor under a contraction.
Seshadri constant epsilon(P; x) -- infimum of P.C / mult_x(C)
over integral curves C through x.
semistable reduction -- a base change and blowup procedure that
puts all fibres of a family into SNC form.
SNC -- simple normal crossing.
symplectic block -- an irreducible holomorphic symplectic
variety, carrying a generically nondegenerate two-form.
terminal -- an MMP endpoint whose exceptional divisors all
have strictly positive discrepancy.
Weil divisor -- a formal integral combination of codim-one
subvarieties; not necessarily Cartier.
weak positivity -- a positivity property of direct image
sheaves under a fibration, weaker than nefness.
Zariski dense -- a subset whose Zariski closure equals X; in
the analytic context, a subset containing a dense analytic
open in X.
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END OF TOUR
============================================================
The reader has now seen the plot of a research paper. The argument
has been broken into six parts:
Part I: the question and the two theorems.
Part II: uniform denominators, via Beauville-Bogomolov.
Part III: structural reductions for the canonical index.
Part IV: scalar order and Seshadri estimates.
Part V: Frobenius jets and compatible leaves.
Part VI: the boundary lc step and the completion.
A patient reader who has made it this far should be able to open
the original paper, look up a specific estimate by name, and
recognize why the estimate has to be there.
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END
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