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The concrete question concerns a very common kind of map in
algebraic geometry, and it answers it with one concrete computation. The map
takes a space $X$ down to a smaller space $Z$; the fibres have trivial
canonical class in a suitable sense; and the question is how badly the base
piece of the canonical bundle formula can be written when you are allowed to
fiddle with the representative. The answer says: not badly at all, as long as
the total dimension is at most four and the coefficients at the boundary are
drawn from a fixed finite set. Every quantity that the formula produces then
has a denominator bounded by a constant that knows only the dimension and the
coefficient set, never the individual map.
There are three intended readers. The first is a graduate student in
algebraic geometry who has met the minimal model program and the canonical
bundle formula but who has not yet seen this uniformity argument. The second
is a strong undergraduate who wants the shape of the argument and is willing
to take some of the deep input theorems on faith. The third is a curious
high-school reader who wants to understand what is being claimed and why it is
hard. Sections~sec:question and~sec:vocab, and the opening words of
every later section, are written for the third audience: they say in plain
language what is about to be proved before any formula appears. Everything
after those opening words is written at the first audience's level.
Two registers run throughout. There is a plain form, one to
five sentences, that a reader can take in without stopping. There is a
technical form, which is the actual statement or proof. The two
describe the same object. If the technical form is out of reach, the plain
form still tells you what the object is and what it is for.
Two features shape the result: it is a uniformity (not existence) theorem, and
it uses that the total dimension is at most four. Sections~sec:question
through~sec:residue prove the main theorem; Sections~sec:effective
and~sec:torsion apply it.
\medskip
\medskip\noindentIn words. Take a compact space $X$ that maps onto a smaller space
$Z$, and suppose the fibres have ``trivial canonical class''. Then a classical
theorem says that the canonical class of $X$ is the pullback of a class on
$Z$ plus a correction that measures how singular the fibres are. That
correction has two pieces: a boundary divisor that records which fibres are
badly singular, and a ``moduli'' part that records something finer, the
one-parameter family of fibres through each point of $Z$. The moduli part is
only defined up to adding a principal divisor from the base, and adding a
principal divisor can change its coefficients by nasty fractions. The question
of the argument is whether, for a fixed dimension bound and a fixed set of
allowed boundary coefficients, one can always choose the representative so
that those fractions have a common, bounded denominator. The answer is yes,
and the bound knows nothing about the map.
\medskip\noindentTechnical setting. Let $(X,B)$ be a projective log canonical pair
over $\C$ of dimension $d$, with $B$ an effective boundary whose coefficients
lie in a fixed finite set $\Phi\subset[0,1]\cap\Q$. Let
\[
f\colon X→ Z
\]
be a contraction: a surjective projective morphism of normal projective
varieties with $f_*\OO_X=\OO_Z$. Assume $\dim Z>0$ and that
\[
K_X+B\sim_{\Q} f^*D
\]
for a rational Cartier divisor $D$ on $Z$. Then the adjoint of the generic
fibre is rationally linearly trivial, and $(X,B)$ is a log Calabi--Yau
fibration over $Z$.
\medskip\noindentThe canonical bundle formula. On a suitable smooth base model
$q\colon W\to Z$ one writes
\[
K_X+B\sim_{\Q} f^*(K_Z+B_Z+M_Z),
\]
where $B_Z$ is the discriminant: at a prime $P\subset Z$ its coefficient
is $1-t_P$, with $t_P$ the log canonical threshold of $f^*P$ over the generic
point of $P$. The remaining piece $\mathbf M$ is the moduli b-divisor.
It is a b-divisor rather than a divisor because it only descends to a divisor
$M_W$ on some high birational model $W\to Z$, and its pullbacks to higher
models are declared. When $M_W$ is nef one says $\mathbf M$ is b-nef;
when $pM_W$ is Cartier one says $p\mathbf M$ is b-Cartier.
The defect is the phrase ``on a suitable model''. Change $D$ to a different
divisor $D_Z$ in the same rational linear class, and $M_W$ changes by a
principal divisor that comes from the base. That can change denominators
arbitrarily. The point of an exact presentation is to freeze the choice
of $D_Z$ so that the moduli part has a denominator controlled from outside.
\medskip\noindentPlain form of the main theorem. Fix the dimension $d\le 4$ and the
coefficient set $\Phi$. There is a single integer $p$, and a DCC set
$\mathcal B$ of allowed discriminant coefficients, both depending only on $d$
and $\Phi$, with the following property. For every log Calabi--Yau fibration
of this dimension and coefficient set there is a corrected principal divisor
$(1/p_0)\Div(\psi)$ and a representative $D_Z$ of the base class such that
\[
K_X+B+\frac1{p_0}\Div(\psi)=f^*D_Z
\]
holds as an equality of actual rational divisors, with
$D_Z=K_Z+B_Z+M_Z$, with $B_Z$ effective and coefficients in $\mathcal B$, with
$(Z,B_Z+M_Z)$ generalized log canonical, and with $p\mathbf M$ b-Cartier. The
integer $p_0$ divides $p$ and clears the denominators of $\Phi$. The bound $p$
does not depend on the fibration, only on $d$ and $\Phi$.
\medskip\noindentTechnical form.
[Uniform relative denominators]
Fix a finite set $\Phi\subset[0,1]\cap\Q$ and an integer $1\le d\le 4$.
There are positive integers $p_0\mid p$, with $p_0$ clearing $\Phi$, and a
rational DCC set $\mathcal B\subset[0,1]$, both depending only on $d$ and
$\Phi$, with the following property. Suppose $(X,B)$ is a projective log
canonical pair of dimension $d$ with coefficients in $\Phi$, and
\[
f\colon X→ Z,\qquad \dim Z>0,\qquad K_X+B\sim_{\Q} f^*D,
\]
where $f$ is a contraction and $D$ is a rational Cartier divisor. Then there
exist $\psi\in\C(X)^*$ and $D_Z\sim_{\Q}D$ such that
K_X+B+\frac1{p_0}\Div(\psi)=f^*D_Z,\qquad D_Z=K_Z+B_Z+M_Z,
where $B_Z$ is effective with coefficients in $\mathcal B$, the pair
$(Z,B_Z+M_Z)$ is generalized log canonical, and $\mathbf M$ is b-nef with
$p\mathbf M$ b-Cartier. If $(X,B)$ is klt then the generalized pair is
generalized klt.
\medskip\noindentWhy the two integers are separate. The integer $p_0$ does two
jobs: it lets the principal correction $\psi$ produce an exact pullback, and on
the generic fibre it is a trivializing degree for the log Calabi--Yau pair
there. The integer $p$ is larger: it clears the moduli trace on a model that
determines $\mathbf M$. The discriminant is controlled separately, by the DCC
set $\mathcal B$.
\medskip\noindentWhat is genuinely new. Qualitative versions of the formula put the
moduli part on a model and make it nef, but the index may depend on the
fibration (Ambro, Fujino--Gongyo). The curve case of Floris even shows that,
when the generic fibre is a rational curve and the fibre index is allowed to
vary, no polynomial bound on the clearing integer can exist. The theorem says
that when the total dimension is at most four and the boundary
coefficients are from a fixed finite set, the index can be chosen uniformly.
The source of the uniformity is the index theorem for the reduced special
fibre, whose dimension is one less than the total.
\noindent{\smallhttps://arxiv.org/pdf/math/0301180}
\medskip
A short glossary follows. Each item is one or two sentences; a reader who
already knows the term can skip it. Everything later refers back by item
number. Where a term has a long standard definition only the shape is recorded,
not the full definition, and point the reader to Koll\'ar--Mori or the Stacks
Project.
\item[(V1)] A variety is an integral separated scheme of finite type over
$\C$. Unless stated otherwise all varieties here are normal and, in the global
statements, projective. Normal means the local rings are normal domains;
a normal variety is smooth outside a closed set of codimension at least two.
A resolution is a proper birational morphism from a smooth variety.
\item[(V2)] A contraction $f\colon X\to Z$ is a surjective projective
morphism of normal projective varieties with $f_*\OO_X=\OO_Z$. Equivalently, the
function field $\C(Z)$ is algebraically closed in $\C(X)$, so the generic fibre
is geometrically integral and the fibres are connected.
\item[(V3)] A divisor on a normal variety is a formal finite $\Z$-linear
combination of prime divisors. A $\Q$-divisor allows rational
coefficients. A divisor is effective when all coefficients are
nonnegative. A boundary is an effective $\Q$-divisor; the boundary
coefficients are its coefficients.
\item[(V4)] Two $\Q$-divisors are linearly equivalent, written $\sim$,
when their difference is the divisor of a rational function. They are
rationally linearly equivalent, written $\sim_{\Q}$, when a positive
integer multiple of the difference is. They are numerically equivalent,
written $\equiv$, when they meet every complete curve the same way. A divisor
is big, nef, or pseudoeffective when its class has the
corresponding positivity: big means it grows like an ample class on high
multiples, nef means nonnegative degree on every curve, pseudoeffective means it
lies in the closure of the cone of effective classes.
\item[(V5)] The canonical divisor $K_X$ of a normal variety is the
divisor of a rational top form; on a smooth variety it is the determinant of
the cotangent bundle. It is defined up to linear equivalence. A log pair
$(X,B)$ is a normal variety together with a boundary.
\item[(V6)] For a prime divisor $E$ over $X$ (that is, on some resolution
$g\colon Y\to X$ with $E$ exceptional or strict transform), the
discrepancy of $E$ with respect to $(X,B)$ is the coefficient of $E$ in
$K_Y-g^*(K_X+B)$ when $g$ is chosen so that this pullback makes sense; it is the
discrepancy when $K_Y$ and $K_X$ are fixed compatibly. The pair is
log canonical (lc) when every discrepancy is at least $-1$, and
klt when every discrepancy is strictly bigger than $-1$. The pair is
dlt when it is lc and the coefficients are at most one and the crossings
are simple normal; dlt is a convenient regular representative of an lc pair.
\item[(V7)] The pair is slc, semi-log-canonical, when it is defined on a
possibly nonnormal variety with pure-dimension components, is $S_2$, is nodal in
codimension one, and its normalization with the conductor included is lc. This
is the right singular class on a reduced fibre that may be reducible and
nonnormal.
\item[(V8)] The log canonical threshold $\lct(X,B;P)$ of a prime $P$
over a point of $Z$ is the largest $t\ge0$ such that the pair
$(X,B+t\,f^*P)$ is lc over that point. In the canonical bundle formula the
discriminant coefficient at $P$ is $1-t_P$. A threshold is a single rational
number, so it is the natural quantity to try to control uniformly.
\item[(V9)] A b-divisor $\mathbf M$ on a normal variety $Z$ is a rule that
assigns a $\Q$-divisor $M_{Z'}$ to every birational model $Z'\to Z$, compatibly
with pushforward. It is b-nef when $M_{Z'}$ is nef on some model and
stays nef under pullback, and b-Cartier when some integer $pM_{Z'}$ is
Cartier on a determining model. Here $\mathbf M$ denotes the moduli
b-divisor and $p\mathbf M$ b-Cartier means $pM_W$ is an honest Cartier divisor
on a model $W$ that determines $\mathbf M$.
\item[(V10)] A generalized pair $(Z,B_Z+M_Z)$ consists of a boundary
$B_Z$ and the trace $M_Z$ of a b-divisor on a model. Its adjoint is
$K_Z+B_Z+M_Z$, and its discrepancies are read off the crepant boundaries on
higher models. It is generalized lc (or klt) when those discrepancies are
at least $-1$ (or bigger than $-1$). The crepant boundaries on higher models
may have negative coefficients; that is part of the definition, not an error.
\item[(V11)] A DCC set is a subset of $\R$ that contains no infinite
strictly decreasing sequence; an ACC set is the reverse. The relevance is
that a DCC set of coefficients cannot contain smaller and smaller rationals
approaching a limit from above, and an ACC theorem says that a collection of
coefficients produced by a geometric process must lie in some DCC set or even a
finite set. The argument uses the ascending chain condition for log canonical
thresholds, quoted from Hacon--McKernan--Xu.
\noindent{\smallhttps://arxiv.org/pdf/1208.4150}
\item[(V12)] The index of a log Calabi--Yau pair $(T,B_T)$ with
$K_T+B_T\sim_{\Q}0$ is the smallest positive integer $n$ with $n(K_T+B_T)\sim0$.
An index theorem bounds $n$ in terms of the dimension and a finite set of
allowed boundary coefficients. The argument imports the slc index theorem of
Jiang--Liu in dimension at most three.
\noindent{\smallhttps://arxiv.org/pdf/2002.11928}
\item[(V13)] For an integral Weil divisor $G$ on a normal variety, the
reflexive divisorial sheaf $\OO_X(G)$ has sections, on an open $U$,
the nonzero rational functions $v$ with $\Div(v)|_U+G|_U\ge0$. For a rational
divisor $D$, the rounded system is the complete system of
$\OO_X(\lfloor D\rfloor)$: nonzero $v$ with $\Div(v)+D\ge0$. This needs no
Cartier condition. The field generated by a system is the subfield of
$\C(X)$ generated by ratios of nonzero sections.
\item[(V14)] A cyclic cover of a smooth curve $C$ at a point $c$ is the
normalization of $C$ in $\C(C)(w)$ with $w^m$ a local uniformizer at $c$. It is
totally ramified of degree $m$ over $c$ and \'etale elsewhere. Pulling back a
fibration to a cyclic cover and comparing a rational generator with its Galois
translates is the mechanism by which a denominator is read off a
character.
\item[(V15)] The residue of a rational top form with a logarithmic pole
along a smooth divisor is the form obtained by contracting with a local
generator of the normal bundle. Iterated residues give a form on a reduced
normal crossing fibre, and the residues of two generators differ by a scalar
determined by the local node. This is the one analytic-flavoured operation the
paper uses.
\item[(V16)] Rational connectedness of a smooth projective variety $Y$
means that any two points are joined by a chain of rational curves. It implies
$\Pic^0(Y)=0$ and, with it, that the N\'eron--Severi group is finitely
generated. The divisor class group $\Cl(Z)$ of a normal variety with a
rationally connected resolution is then finitely generated, and its torsion
is bounded using Kummer theory.
\medskip
\medskip\noindentIn words. The proof of the main theorem has two halves. The first
half is conceptual: it produces the exact presentation and
turns the moduli bound into a statement about one prime at a time. The second
half, Sections~sec:curve and~sec:residue, is a single hard
computation for that one prime. The whole strategy is: reduce a global claim
to a local number, then compute that number once.
\medskip\noindentHalf one, step by step. The argument starts with the generic fibre of $f$. It
is a log Calabi--Yau pair of dimension $d-\dim Z\le d-1\le 3$, so the known
index theorem in low dimension applies to it. That theorem provides a degree
$p_0$ in which the adjoint of the generic fibre is principal, with an explicit
rational function. Because the function field extension is regular, a Galois
descent (Hilbert's theorem 90) moves that function down to the actual generic
fibre, not just its algebraic closure. The result is a function $\psi$ on $X$
whose $(1/p_0)$-divisor corrects $K_X+B$ to be vertical over $Z$. A short
argument in the same spirit then chooses $D_Z$ in the rational class of $D$ so
that the corrected adjoint is literally $f^*D_Z$, with no vertical error left.
That is the exact presentation~(eq:relative-exact-presentation).
Once the presentation is exact, the qualitative lc-trivial fibration theorem
gives the structural facts for free: the discriminant $B_Z$ is
effective and the moduli b-divisor $\mathbf M$ is b-nef and descends on a
smooth model. What the qualitative theorem does not give for free is a
uniform Cartier index for $\mathbf M$. That is the content of the argument, and
it is what remains to be reduced.
\medskip\noindentTwo independent boundings. The discriminant coefficients $1-t_P$
must fall into a fixed DCC set; this comes from ACC for log canonical
thresholds, applied prime by prime, and is purely qualitative and already
uniform. The moduli coefficients must have bounded denominators; this is the
hard part. On a smooth model $W\to Z$ with $M_W=D_W-K_W-B_W$, a coefficient of
$M_W$ at a prime $P\subset W$ equals an integer (from $K_W$) plus
$coeff_P D_W$ plus $coeff_P(-B_W)$. If
$\alpha=coeff_P D_W$ and $B_W$ has coefficient $1-t_P$ at $P$
over the generic point, that coefficient is $\alpha+t_P$, up to an integer.
So it suffices to show that one fixed integer $p$, independent of the
fibration and of $P$, has
\[
p(\alpha+t_P)\in\Z
\qquadfor every prime P\subset W.
\]
This is the target~(eq:relative-coefficient-target) below. Everything in
Sections~sec:curve and~sec:residue is aimed at it.
\medskip
\medskip\noindentTechnical form. Put $k=\dim Z$ and let $K=\C(Z)$. The contraction
condition makes $K$ algebraically closed in $\C(X)$, so the extension
$\C(X)/K$ is regular; in characteristic zero it is separably generated, and
the generic fibre $X_\eta$ is geometrically integral. It is normal by
localization from $X$, and normality is geometric for a finite-type variety
over the perfect field $K$.
Choose a rational top form on $Z$, and use it together with the form defining
$K_X$ to fix a relative top form on $X_\eta$. Its divisor is the canonical
divisor $K_{X_\eta}$. At codimension-one points of $X_\eta$, which are smooth
over $K$, this divisor is the coefficientwise localization of $K_X$, so
vertical primes disappear; when $d=k$, $X_\eta=\Spec K$ and the
relative form is a nonzero $0$-form. Restricting a log resolution and its
crepant sub-boundary to the geometric generic fibre exhibits the original
geometric generic pair as lc: generic smoothness makes the horizontal strata
simple normal crossing there. That pair is projective of dimension
$d-k\le 3$, its boundary coefficients lie in $\Phi$, and its adjoint is
rationally linearly trivial.
[generic-fibre trivialization]
There is a positive integer $p_0$, depending only on $d$ and $\Phi$ and
clearing the denominators of $\Phi$, and a rational function
$\psi\in\C(X)^*$ such that
\[
p_0(K_X+B)+\Div(\psi)
\]
is vertical over $Z$.
Proof. [Plain form]
Apply the slc index theorem to the geometric generic fibre in dimension at most
three, getting a trivializing degree. Descend the trivialization across the
Galois action by taking ratios of conjugates, whose divisors are vertical and
hence constant, and clearing the resulting cocycle with Hilbert 90.
Proof. [Technical form]
The geometric generic fibre is a projective lc pair of dimension at most three
with boundary coefficients in $\Phi$ and rationally trivial adjoint. The slc
index theorem (Jiang--Liu, quoted as input) gives a common integer $p_0$,
depending only on $d$ and the finite coefficient set, such that
$p_0(K_{X_{\overline\eta}}+B|_{X_{\overline\eta}})$ is principal on the
geometric generic fibre. Choosing $p_0$ also to clear $\Phi$ gives an
integral divisor.
Descend to $K$. Choose a finite Galois extension $L/K$ over which a
principalizing rational function is defined. The quotient of two Galois
conjugates of it has zero divisor on the proper geometrically integral normal
fibre over $L$, hence is a nonzero constant. These quotients form a
multiplicative Galois cocycle, and Hilbert's theorem 90 rescales the function
to be Galois-invariant; being defined over $K$, it descends to $\C(X)^*$.
Thus there is $\psi\in\C(X)^*$ making $p_0(K_X+B)+\Div(\psi)$ vertical.
\medskip
\medskip\noindentTechnical form. Choose $a$ divisible by $p_0$ and
$\phi\in\C(X)^*$ with
\[
a(K_X+B)+\Div(\phi)=f^*(aD).
\]
Such a pair exists because $K_X+B\sim_{\Q}f^*D$ and $a$ clears denominators.
The divisor of $\psi^{a/p_0}/\phi$ is vertical. On the proper normal generic
fibre this function has neither zeros nor poles, so it belongs to $K^*$.
Writing it as $h$, set
\[
D_Z=D+\frac1a\Div(h).
\]
Substituting gives the first equality of~(eq:relative-exact-presentation)
as an equality of actual rational divisors. In particular $D_Z$ is rational
Cartier and $D_Z\sim_{\Q}D$.
\medskip
\medskip\noindentTechnical form. Apply the lc-trivial-fibration theorem of
Fujino--Gongyo. Besides generic log canonicity and the rational pullback
relation it needs a rank-one condition: on a log resolution the modified
discrepancy divisor drops the discrepancy $-1$ terms, and because
$0\le B\le 1$ rounding up leaves, over the generic point of the base, only an
effective exceptional divisor; its divisorial sheaf pushes forward to $\OO_X$
by normality, and pushing on to $Z$ has rank one. The theorem then makes
$\mathbf M$ b-nef and gives its descent to a nef rational Cartier divisor on a
sufficiently high smooth projective model $W\to Z$. The statement is stable
under changing the rational trivialization, since that adds a rational
principal b-divisor from the base.
Write
\[
D_W=(W\to Z)^*D_Z,\qquad M_W=D_W-K_W-B_W.
\]
For a prime $P\subset W$, set $\alpha=coeff_P D_W$. Since
$coeff_P B_W=1-t_P$ and the coefficient of $K_W$ is an integer,
\[
coeff_P M_W
=\alpha+t_P-1-coeff_P K_W.
\]
Hence it is enough to prove, uniformly in the choice of $P$ and the fibration,
p(\alpha+t_P)\in\Z.
Once~(eq:relative-coefficient-target) holds for a single $p$, the divisor
$pM_W$ is an integral divisor on the smooth variety $W$, hence Cartier, and
$p\mathbf M$ is b-Cartier.
\medskip
\medskip\noindentTechnical form. For primes $P\subset Z$, shrink about the generic
point of $P$ so that $P$ is Cartier. In a log resolution computing the
threshold, only divisors with positive order along the pullback of $P$
matter; discard those whose images are proper subsets of $P$. Then $t_P$ is an
ordinary log canonical threshold of an lc pair with coefficients in $\Phi$
against an effective Cartier divisor with positive integral coefficients. The
threshold ACC theorem (Hacon--McKernan--Xu) applies, and since thresholds are
rational and lie in $[0,1]$, their complements $1-t_P$ form the required
rational DCC set $\mathcal B\subset[0,1]$. The singularities of
$(Z,B_Z+M_Z)$ are generalized lc (respectively generalized klt) because the
crepant sub-pair over $Z$ is sub-lc (respectively sub-klt): $t_P\ge0$ always,
and $t_P>0$ in the klt case. On the original base $Z$ the boundary upstairs is
effective with positive integral multiplicity over $P$, which forces
$t_P\le1$ and hence $B_Z\ge0$.
\medskip
After these considerations the main theorem is reduced to a single statement: there is
an integer $p$, depending only on $d$ and $\Phi$, such that the coarse identity
p(\alpha+t_P)\in\Z
\qquadfor every prime P of every determining model W
holds. The threshold $t_P$ is disguised: it was computed as an lc threshold over
the generic point of $P$, but the required uniformity is insensitive to the
individual $P$. The next two sections compute~(eq:problem) by slicing to
a curve through the generic point of $P$, running a relative minimal model
program there, and reading the answer off a residue on the resulting reduced
fibre.
\medskip
\medskip\noindentIn words. The goal is to prove one congruence, $p(\alpha+t_P)\in\Z$, for
one prime $P$ of one determining model $W$. The trick is to make the situation
one-dimensional by cutting with general divisors through a general point of
$P$. A curve has no room for complications: the "fibre" over a point of the
curve is a divisor, the singularities are controlled, and a residue
computation can see the denominator. So slice and run the relative minimal
model program, and reduce the congruence to a statement about a single reduced
special fibre.
\medskip\noindentThe plan. Fix $P\subset W$. Build a smooth model $V\to X$ that maps
to $W$ and is marked along everything relevant. Cut $V$ by $k-1=\dim W-1$
general very ample divisors to get a curve $C$ and a total space $V_C\to C$.
On $V_C$ write an exact relative identity whose base coefficient at the chosen
point $c\in C$ is exactly $\alpha$. Construct a dlt pair by adding the reduced
fibre and a horizontal boundary. Run a minimal model program over $C$. The
program does not change the function field and does not change the coefficient
of interest; it makes the total space terminal enough that the reduced fibre
is well behaved. The output is a dlt model $N\to C$ on which an exact divisor
relation holds:
\[
L+\Div(\psi_C)=p_0(\alpha+t)f_N^*[c],\qquad L=p_0(K_N+T+H),
\]
with $T$ the reduced fibre. That relation is handed directly to the residue
lemma of Section~sec:residue.
\medskip
\medskip\noindentTechnical form. Fix a prime $P\subset W$. Take a smooth projective
model $V\to X$ mapping to $W$ by $g\colon V\to W$, with simple normal crossing
markings for the crepant boundary $B_V^c$, the exceptional divisors over $X$,
and the support of $g^*P$. Because the function field extension $\C(W)\subset
\C(V)$ is regular (the same argument as for $f$), the Stein factorisation of $g$
is $W$ itself, so $g$ has connected fibres. Let $\theta_V$ be the rational top
form compatible with the chosen canonical divisor on $X$, and put
$K_V=\Div(\theta_V)$. Crepant pullback gives
K_V+B_V^c+\frac1{p_0}\Div(\psi)=g^*D_W.
Put $t=t_P$. At the generic point of $P$, the boundary $B_V^c+t\,g^*P$ has
coefficients at most one, with equality on at least one component dominating
$P$; this is the threshold computation.
Choose general very ample divisors $A_1,...,A_{k-1}$ on $W$ (taking
$C=W$ when $k=1$) and let $C=A_1\cap...\cap A_{k-1}$. Bertini applied
successively to the pulled-back systems and the finitely many marked strata
makes $C$ and $V_C=g^{-1}C$ smooth, with the markings still SNC. The curve $C$
is connected; the restriction of $g$ has connected fibres, so $V_C$ is
connected and hence integral. Also required are $C\not\subset\Supp D_W$ and
$V_C\not\subset\Supp K_V\cup\Supp B_V^c\cup\Supp\Div(\psi)$. The dimension of
$V_C$ is $d-k+1\le4$. Choose an intersection point $c\in C\cap P$ transverse and
general on $P$, avoiding at $c$ all other components of $\Supp D_W$, the bad
strata on $P$, and the images of all nondominating strata not containing $P$.
\medskip
\medskip\noindentTechnical form. Let $h:V_C\to C$, $F=h^*[c]$, and $B^c=B_V^c|_{V_C}$.
To keep the coefficient of $P$ the canonical divisor is specified using
complete-intersection adjunction. Choose general representatives $A_j'\sim A_j$
avoiding $c$ and rational functions $\chi_j\in\C(W)^*$ with
$\Div(\chi_j)=A_j'-A_j$. Define $\theta_C$ as the iterated Poincar\'e residue,
in the order of the cuts, of
\[
\theta_V\prod_{j=1}^{k-1}g^*\chi_j
\]
along $g^{-1}A_1,...,g^{-1}A_{k-1}$. Generality makes this a nonzero rational
top form on $V_C$; for $k=1$ the product is empty and the residue is the
identity. Each $\chi_j$ supplies a simple pole along its divisor and a zero along
its chosen representative, so, cancelling the cutting divisors before
restriction, the residue formula gives an equality of actual divisors
\[
K_{V_C}:=\Div(\theta_C)
&=(\Div(\theta_V\prod_jg^*\chi_j)+\sum_jg^*A_j)|_{V_C}\\
&=(K_V+\sum_jg^*A_j')|_{V_C}.
\]
Set $\psi_C=\psi|_{V_C}$ and $D_C'=(D_W+\sum_jA_j')|_C$. Restricting
(eq:relative-resolution-presentation) gives
K_{V_C}+B^c+\frac1{p_0}\Div(\psi_C)=h^*D'_C,
\qquadcoeff_c D'_C=\alpha.
The coefficient assertion uses that $C$ meets $P$ transversely at $c$ while the
$A_j'$ and all other components of $\Supp D_W$ avoid $c$.
\medskip
\medskip\noindentTechnical form. Along $F$, the coefficients of $B^c+tF$ are at most
one, with equality on a component of $F$: the divisors dominating $P$ restrict
transversely and reducedly, and near $c$, $g^*P|_{V_C}=F$. Over the
generic point of $C$ there is still a projective birational comparison with the
normal fibre of the original model $X$, and the restricted exceptional divisors
are exceptional there; this is because the generic point of $C$ lies where
$W\to Z$ is an isomorphism, and generic flatness and openness of geometric
normality give normal integral original fibres on an open base with an open
birational isomorphism locus whose exceptional images have fibre codimension at
least two.
On $V_C$ define an effective boundary $\Gamma$ as $F_{red}$, plus the
horizontal strict transforms of the original boundary with their original
coefficients, plus all horizontal restricted exceptional divisors with
coefficient one. Then $(V_C,\Gamma)$ is log smooth, its horizontal coefficients
lie in the fixed set $\Phi\cup\{1\}$, and
E=\Gamma-B^c-tF
is effective near $c$ and has coefficient zero on a component of $F$. On the
generic fibre, $E$ is effective: its horizontal nonexceptional terms cancel,
while every horizontal exceptional term has coefficient $1-coeff
B^c\ge0$, and its support there is exceptional over the original normal fibre.
Equation~(eq:relative-curve-presentation) gives the exact identity
\[
K_{V_C}+\Gamma+\frac1{p_0}\Div(\psi_C)=h^*(D'_C+t[c])+E.
\]
Run a minimal model program over $C$. All negative coefficients of $E$ are
vertical and occur over finitely many points, so there is an effective rational
divisor $A$ on $C$ with $E^+:=E+h^*A\ge0$. The identity becomes
\[
K_{V_C}+\Gamma+\frac1{p_0}\Div(\psi_C)=h^*(D'_C+t[c]-A)+E^+.
\]
In particular $K_{V_C}+\Gamma\sim_{\Q,C}E^+$, whose restriction to a very general
fibre is rationally linearly equivalent to an effective divisor, hence
pseudo-effective over $C$.
\medskip
\medskip\noindentTechnical form. The existence-and-preservation result of
Chen--Tsakanikas permits a minimal model program with scaling of an ample over
the base for an lc generalized pair whose underlying variety is $\Q$-factorial
klt; applied to the $\Q$-factorial dlt pair $(V_C,\Gamma)$ over $C$, it keeps the
stages $\Q$-factorial and dlt, and each step is negative for the transformed
adjoint. The zero nef datum satisfies the NQC hypothesis. In dimension four,
their theorem terminates every flip sequence over $C$ for an NQC lc generalized
pair whose adjoint is pseudo-effective over $C$; in dimension three their
statement has no pseudo-effectivity hypothesis. These give the program and
terminate the flip tail.
Fix canonical divisors on every birational stage using $\theta_C$, and push
forward $\Gamma,E^+$; every step is over $C$, so the identity pushes forward to
\[
K_{V_i}+\Gamma_i+\frac1{p_0}\Div(\psi_C)=h_i^*(D'_C+t[c]-A)+E_i^+,
\qquad E_i^+\ge0.
\]
A fibre-type negative extremal contraction is impossible: a general
positive-dimensional contracted fibre has a curve $\ell$ not contained in
$\Supp E_i^+$, and $\Q$-factoriality plus effectivity makes the intersection
$E_i^+\cdot\ell\ge0$, while the contraction makes it negative, a contradiction.
Divisorial contractions strictly decrease the relative Picard number, so an
infinite program would eventually consist only of flips, which the termination
statements exclude; in dimensions at most two no flips occur. The endpoint is a
projective morphism $f_N:N\to C$ with $N$ $\Q$-factorial, $(N,\Gamma_N)$ dlt, and
$K_N+\Gamma_N$ nef over $C$. The unchanged function field and Stein factorisation
give connected fibres, so $f_N$ is a contraction.
\medskip
\medskip\noindentTechnical form. Let $E_N^+$ be the transform of $E^+$ on $N$. The
pushforward of $h^*A$ is $f_N^*A$, so $E_N:=E_N^+-f_N^*A$ is the pushforward of
the original $E$, and the transformed identity reads
K_N+\Gamma_N+\frac1{p_0}\Div(\psi_C)=f_N^*(D'_C+t[c])+E_N.
Take a common resolution $r:\widetilde V\to V_C$, $r':\widetilde V\to N$.
Discrepancy comparison for a $K+\Gamma$ program gives
\[
r^*(K_{V_C}+\Gamma)-r'^*(K_N+\Gamma_N)\ge0,
\]
and cancelling common base pullbacks and principal divisors in the input and
output identities gives
r'^*E_N\le r^*E.
Let $\eta$ be the generic point of $C$ and $q:\widetilde V_\eta\to X_\eta$ the
map to the normal original fibre. The restriction of $E_N$ to $N_\eta$ is
effective, being the pushforward of the effective restriction of $E$; the
restriction of $r^*E$ is effective and $q$-exceptional; so by
(eq:relative-error-monotonicity) the restriction of $r'^*E_N$ is
$q$-exceptional. But (eq:relative-output-error) gives
$E_N\equiv_C K_N+\Gamma_N$, which is nef over $C$, and hence $q$-nef. The
negativity lemma makes it zero. Thus $E_N$ has no horizontal component; after
shrinking about $c$ it is effective, supported on the special fibre, and nef
over $C$.
Write $F_N=f_N^*[c]=\sum_i e_iS_i$ and $E_N=\sum_i q_iS_i$ with $e_i>0$,
$q_i\ge0$, and set $\lambda=\max_i(q_i/e_i)$. Then
$\Delta=\lambda F_N-E_N$ is effective, anti-nef over $C$, and misses at least one
component of the fibre. If the fibre has positive dimension and $\Delta\ne0$,
connectedness gives a component $S$ outside $\Supp\Delta$ meeting it; since $N$
is $\Q$-factorial a positive multiple $b\Delta$ is Cartier, and its restriction
to the integral projective $S$ is a nonzero effective Cartier divisor, so
$((b\Delta)|_S)\cdot A_S^{n-1}>0$ for very ample $A_S$, contradicting anti-nefness.
Hence $\Delta=0$, i.e. $E_N=\lambda F_N$; for a zero-dimensional fibre, $f_N$ is
an isomorphism of normal curves and the proportionality is immediate. Since
$r'^*F_N=r^*F$ and $E$ has coefficient zero on a component of $F$, comparing that
coefficient forces $\lambda=0$. So $E_N=0$ near $c$.
\medskip
\medskip\noindentTechnical form. Shrink $C$ about $c$ and set
$T=(f_N^*[c])_{red}$. No divisor was extracted in the program, so every
surviving component of this fibre has coefficient one in $\Gamma_N$; hence
$\Gamma_N=T+H$, with $H$ horizontal and coefficients in $\Phi\cup\{1\}$. The
reduced fibre $T$ is connected and projective, and its components have dimension
$d-k\le3$. Removing the boundary from the $\Q$-factorial dlt pair shows that $N$
is klt. Shrink again so that $D'_C$ is supported at most at $c$. For the
integral Weil divisor $L=p_0(K_N+T+H)$ the exact relation is now
L+\Div(\psi_C)=p_0(\alpha+t)f_N^*[c].
This is exactly the hypothesis of the residue lemma, with
$\beta=\alpha+t$. The remaining work is to show that one uniform $p$, depending
only on $d$ and $\Phi$, has $p\beta\in\Z$ in every such configuration.
\medskip
\medskip\noindentIn words. This is the one genuinely delicate computation. The setup provides
handed a normal total space $N$ over a curve, a reduced fibre $T$, and a
horizontal boundary $H$, with an exact relation
\[
p_0(K_N+T+H)+\Div(\psi)=p_0\beta\,F,
\qquad F=f^*[c],\quad T=F_{red}.
\]
The question is whether $p\beta$ is an integer for one fixed $p$ that does not
depend on the configuration. The strategy is: restrict the relation to the
reduced fibre, where it becomes a statement that a certain log Calabi--Yau pair
$(T,B_T)$ has a trivial $p$-th pluricanonical class; take a global generator $u$
of that trivialization; pull back to a cyclic cover of the curve that opens up
the denominator; and show, by comparing residues across the nodes of the cover,
that the would-be extra character is forced to be trivial. The denominator must
therefore divide $p$.
\medskip\noindentTechnical form.
[The reduced fibre and the cyclic residue test]
Fix a finite set $\Phi\subset[0,1]\cap\Q$ and a positive integer $p_0$ clearing
its denominators. There is a positive integer $p$, depending only on $\Phi$ and
$p_0$, divisible by $p_0$, with the following property. Let
\[
f\colon N→ C
\]
be a projective contraction from a normal integral variety to a smooth complex
curve, let $c\in C$, and suppose $1\le\dim N\le4$. Set $F=f^*[c]$ and
$T=F_{red}$. Assume that $N$ is $\Q$-factorial, that $(N,T+H)$ is dlt,
and that $H$ is an effective horizontal $\Q$-divisor with coefficients in
$\Phi$. Suppose, on a neighbourhood of $F$, that there are $\beta\in\Q$ and
$\psi\in\C(N)^*$ such that, for a fixed canonical divisor,
p_0(K_N+T+H)+\Div(\psi)=p_0\beta F
as an equality of actual $\Q$-divisors. Then $p\beta\in\Z$.
Proof.
The proof has four movements. Adjoin on the whole reduced fibre to get an
slc pair $(T,B_T)$ with a uniform index. Take a global generator $u$ and a
cyclic cover that unwinds the denominator. Compare residues on the components
of the covering fibre and glue the scalars. Kill the remaining character.
\medskip
\medskip\noindentTechnical form. Shrink $C$ about $c$ throughout; fix a uniformizer
$z\in\C(C)$ with $\Div_C z=[c]$ there. The fibre $T$ is connected and
projective, and its irreducible components have dimension $\dim N-1$, because
$F$ is an effective Cartier divisor on the integral variety $N$ with support the
whole fibre. When $\dim N=1$, $f$ is an isomorphism of normal curves and
comparing the coefficient at $c$ already gives $p_0\beta\in\Z$; so assume
$1\le\dim T\le3$.
Write $T=∪_iT_i$. By dlt adjunction and the structure of dlt strata, each
$T_i$ is normal and
(K_N+T+H)|_{T_i}\sim_{\Q}K_{T_i}+Θ_i,
where $(T_i,Θ_i)$ is lc and $Θ_i$ is the effective different. If $n$
is sufficiently divisible, the $n$-th tensor power of the ordinary residue at
the generic point of $T_i$ extends to an isomorphism
\[
\OO_N(n(K_N+T+H))|_{T_i}\simeq\OO_{T_i}(n(K_{T_i}+Θ_i)),
\]
identifying actual rational pluricanonical forms.
Intersections of distinct components are unions of lc strata. At the generic
point of a codimension-two intersection in $N$ the pair is simple normal
crossing, exactly two components of $T$ pass through the point, and $H$ is
absent; the conductor prime on either branch has coefficient one in $Θ_i$.
The remaining coefficients of the differents lie in the set
\mathcal D(\Phi)={(r-1+b)/(r)\ \middle|\
r\in\N_{>0},\ b=\sum_{\phi\in\Phi\cup\{1\}}n_\phi\phi\le1}\cap[0,1],
which is a fixed rational DCC set: the possible sums $b\le1$ form a finite set,
and for each such sum the displayed coefficients increase in $r$ toward one.
To apply the index theorem to $T$ itself and not just to its components, $T$ must be the right kind of scheme. Apply the depth theorem to the dlt pair
$(N,T+H)$ with the effective integral divisor $D=T\le\lfloor T+H\rfloor$ and
$L=0$; it makes $\OO_N$, $\OO_N(-T)$, and $\OO_T$ Cohen--Macaulay, so $T$
satisfies $S_2$. The normality of the $T_i$ and the generic crossing description
make $T$ smooth away from ordinary double curves, hence seminormal (the
square--cube criterion, checked at codimension-one points, plus the $S_2$
extension property). Thus $T$ is demi-normal, its normalization is the disjoint
union of the $T_i$, and its conductor is the union of the double curves.
Remove the coefficient-one conductor divisors from the $Θ_i$ and push the
remaining boundary cycles to $T$; call the result $B_T$. Off the conductor the
normalization is an isomorphism in codimension one, so nothing is double
counted. The boundary $B_T$ is effective, has coefficients in $\mathcal D(\Phi)$,
and contains no conductor component. Choosing an open $j:T^\circ↪ T$
whose complement has codimension at least two and where $T^\circ$ has only
smooth points and ordinary double crossings, one defines
$\OO_T(n(K_T+B_T))$ first on $T^\circ$ and then extends by $S_2$. For every
sufficiently divisible even $n$ the residue maps give a natural isomorphism
\OO_N(n(K_N+T+H))|_T\simeq\OO_T(n(K_T+B_T)).
At a node with normal-crossing coordinates $x,y$, hypersurface adjunction for
$xy=0$ matches the ambient log canonical line with the dualizing line of the
node; the two iterated residues of $dx/x\wedge dy/y$ differ by a sign whose
$n$-th power is $+1$ for even $n$, and their next residues agree. It follows
that $K_T+B_T$ is $\Q$-Cartier and its pullback to the normalization is
$K_{T_i}+Θ_i$ with the conductor included, so $(T,B_T)$ is slc. Combining
(eq:slc-local-relation) with $F=\Div(f^*z)$ and restricting a sufficiently
divisible trivialization in~(eq:slc-ambient-restriction) gives
K_T+B_T\sim_{\Q}0.
\medskip
\medskip\noindentTechnical form. Apply global ACC to the projective normalized
component pairs $(T_i,Θ_i)$, whose adjoints are numerically trivial by
(eq:slc-fibre-trivial). Their dimensions are at most three and their
coefficients lie in the DCC set $\mathcal D(\Phi)$, so their nonzero coefficients
lie in a fixed finite rational set; likewise for $B_T$. The slc index theorem in
dimension at most three then gives a common integer annihilating $K_T+B_T$.
Enlarging it, choose once and for all an even multiple $p$ of $p_0$, depending
only on $\Phi$ and $p_0$, with
pB_T integral,\qquad p(K_T+B_T)\sim0.
Choose a nowhere-vanishing generator $u$ of this trivial log pluricanonical line;
on the normalization it is a tuple of rational $p$-pluricanonical forms $u_i$
with the prescribed boundary poles and the conductor gluing.
\medskip
\medskip\noindentTechnical form. Write
p_0\beta=(a)/(m),\qquad a\in\Z,\ m\in\N_{>0},\ \gcd(a,m)=1.
The case $m=1$ is immediate; assume $m>1$. Let $C'\to C$ be the normalization
in $\C(C)(w)$ with $w^m=z$, and let $\pi:Y\to N$ be the normalization of the
dominating component of $N\times_CC'$. The polynomial $w^m-z$ is Eisenstein at
$c$, so $C'$ has a unique point $c'$ over $c$ with total ramification of degree
$m$; and since $\C(C)$ is algebraically closed in $\C(N)$ and the extension is
regular, $\C(C')/\C(C)$ is linearly disjoint from it. The map $\pi$ has degree
$m$ and cyclic Galois group $\mu_m$ acting by $w\mapsto\zeta w$. Write
$f_Y:Y\to C'$ and $T_Y=(\pi^{-1}T)_{red}$. Figure~fig:root-cover
records the maps; $Y$ is the normalization of the base change.
[ht]
\centering
[row sep=large,column sep=huge]
T_Y \arrow[r] \arrow[d,hook] & T \arrow[d,hook] \\
Y \arrow[r,"\pi"] \arrow[d,"f_Y"'] & N \arrow[d,"f"] \\
C' \arrow[r,"w^m=z"'] & C
Caption: The normalized cyclic base change and its whole reduced special
fibres over $c'$ and $c$. The covering group may permute components of
$T_Y$, so the residue comparison must use their conductor gluing.
The morphism $f_Y$ is projective and its function field is regular over
$\C(C')$, so Stein factorisation gives $(f_Y)_*\OO_Y=\OO_{C'}$; in particular
$T_Y$ is connected and projective. The divisor $p_0(K_N+T+H)$ in
(eq:slc-local-relation) is integral, and comparing coefficients with
(eq:slc-denominator) shows $m$ divides every coefficient $e_i$ of
$F=\sum_ie_iT_i$. At the generic point of $T_i$ write $f^*z=\tau^{e_i}v$ with
$v$ a unit; after dividing $w$ by $\tau^{e_i/m}$ the normalized extension is
obtained by extracting an $m$-th root of a unit and is \'etale. Away from $T$
the function $f^*z$ is a unit and the same holds, so $\pi$ is \'etale in
codimension one.
Put $H_Y=\pi^*H$ and choose canonical divisors compatibly. Since $\pi$ is
quasi-\'etale and the coefficient-one divisor pulls back without divisorial
ramification,
K_Y+T_Y+H_Y=\pi^*(K_N+T+H).
The pair on the left is dlt. Finite crepant discrepancy comparison multiplies a
downstairs log discrepancy by the ramification index of a divisorial
prolongation, so the upstairs pair is lc and every upstairs lc centre maps onto
a downstairs lc centre; a finite quasi-\'etale map is \'etale over the smooth
locus of its target by purity of the branch locus, so the upstairs pair is
simple normal crossing at the generic points of all its lc centres, which is the
dlt condition. In particular each component $T_{Y,j}$ of $T_Y$ is normal.
\medskip
\medskip\noindentTechnical form. Let $\theta$ be the rational top form defining
$K_N$, and define
s=(\pi^*\theta)^{ ⊗ p_0}\,\pi^*\psi\,w^{-a}.
Since $\pi^*F=m\Div_Yw$, equations~(eq:slc-local-relation) and
(eq:slc-cover-crepant) give $\Div(s)+p_0(T_Y+H_Y)=0$. Thus $s$ generates
the degree-$p_0$ divisorial log pluricanonical sheaf on $Y$, and its
sufficiently divisible powers generate the corresponding invertible sheaves.
\medskip
\medskip\noindentTechnical form. Let $T_i$ be the component of $T$ dominated by
$T_{Y,j}$, and let $\rho_j:T_{Y,j}\to T_i$ be the induced finite \'etale map
at their generic points. The generic residue
\[
v_j=\res_{T_{Y,j}}(s^{p/p_0})
\]
is a nonzero rational $p$-pluricanonical form, as is $\rho_j^*u_i$, and the
quotient
r_j=(v_j)/(\rho_j^*u_i)\in\C(T_{Y,j})^*
is a rational function. Choose a multiple $n=bp$ sufficiently divisible for
(eq:slc-ambient-restriction) and for component adjunction upstairs. Then
$v_j^b$ is a nowhere-vanishing generator of the degree-$n$ adjunction sheaf on
$T_{Y,j}$. The same is true of $\rho_j^*u_i^b$: near any point of $T$, the
section $u^b$ is a generator of the restriction of the invertible sheaf
$\OO_N(n(K_N+T+H))$, hence is a unit relative to a local ambient generator; such
a unit lifts after restricting the neighbourhood, so $u^b$ is the residue
restriction of a local ambient log generator. Pull that generator back to $Y$;
it remains a log generator by (eq:slc-cover-crepant), and its residue on
$T_{Y,j}$ generates the upstairs adjunction sheaf and equals $\rho_j^*u_i^b$
because residue commutes with \'etale pullback. Therefore $r_j^b$, and hence
$r_j$, has zero divisor; a rational function with zero divisor on a normal
projective integral variety is a nonzero constant. So
v_j=r_j\,\rho_j^*u_i,\qquad r_j\in\C^*.
\medskip
\medskip\noindentTechnical form. Suppose two distinct components $T_{Y,j}$ and
$T_{Y,j'}$ meet. Their intersection contains a stratum of codimension two in $Y$
along which the pair is generically simple normal crossing, and its image is a
codimension-two lc centre downstairs where $\pi$ is \'etale and the downstairs
pair is simple normal crossing. The two downstairs vertical branches are
distinct, and $H$ is absent at this generic crossing. At the generic point $\xi$
of the upstairs crossing choose the two normal coordinates $x,y$ and a regular
ambient $(\dim Y-2)$-form $\eta$. Since $s^{p/p_0}$ is a log generator, it has
the local form
\[
s^{p/p_0}=ε((dx)/(x)\wedge(dy)/(y)\wedge\eta)^{ ⊗ p},
\qquadε\in\OO_{Y,\xi}^*.
\]
The two next residues differ by $(-1)^p$ and are nonzero, because $ε$
is a unit; as $p$ is even they agree. The components of $u$ have the same
matching, nonzero next residues downstairs, since $u$ generates the slc log
pluricanonical line whose local node description is the even tensor power of the
dualizing line, and \'etale pullback preserves this. Taking next residues in
(eq:slc-component-scalar) gives $r_j=r_{j'}$.
\medskip
\medskip\noindentTechnical form. Since $T_Y$ is connected, its component-incidence
graph is connected, so all $r_j$ equal one scalar $r\in\C^*$. In tuple notation
on the normalized components of $T_Y$,
(\res_{T_{Y,j}}(s^{p/p_0}))_j=r\,\pi^*u,
and $\pi^*u$ is invariant under the covering group, with the natural action also
permuting components. On the other hand, for $\zeta\in\mu_m$,
(eq:slc-cover-generator) gives $\zeta^*(s^{p/p_0})=\zeta^{-ap/p_0}s^{p/p_0}$,
and residue is natural under this action including its permutation of
components. The nonzero scalar comparison~(eq:slc-global-scalar) forces
this character to be trivial. So $m\mid ap/p_0$; since $\gcd(a,m)=1$, obtain
$m\mid p/p_0$, and hence
\[
p\beta=(p)/(p_0)\cdot(a)/(m)\in\Z.
\]
This proves the lemma.
\medskip
Apply Lemma~lem:slc-residue-test to (eq:relative-local-cyclic) with
$\beta=\alpha+t$ and the fixed coefficient set $\Phi\cup\{1\}$. It gives a single
multiple $p$ of $p_0$, depending only on $d$ and $\Phi$, for which
(eq:relative-coefficient-target) holds at every prime $P$ of the
determination $W$. Consequently $pM_W$ is integral and Cartier, and since the
moduli b-divisor descends on $W$ the b-divisor $p\mathbf M$ is b-Cartier. Every
construction used only the dimension bound $d\le4$ and the fixed coefficient
set $\Phi$. Together with the discriminant bound of Section~sec:generic
this completes the proof of Theorem~thm:relative-denominators.
\medskip
\medskip\noindentIn words. The main theorem was about writing the base divisor with
bounded denominators. A closely related question is: can pluricanonical
sections upstairs actually see the base downstairs? When the base
divisor is big, the answer is yes in one uniform degree: for every large enough
multiple $l$, the complete rounded system $|\lfloor l(K_X+B)\rfloor|$ is
nonempty, and the field generated by ratios of its sections is exactly the base
function field $\C(Z)$, no more and no less. ``No more'' is the substantive
half: the sections must not accidentally generate a larger subfield of
$\C(X)$.
\medskip\noindentTechnical form.
[Effective systems recovering the full base field]
Under the hypotheses of Theorem~thm:relative-denominators, suppose in
addition that $D$ is big. There is a positive integer $m=m(d,\Phi)$ such that,
for every positive multiple $l$ of $m$, the complete divisorial system
\[
|\left\lfloor l(K_X+B)\right\rfloor|
\]
is nonempty and generates precisely the subfield $\C(Z)\subset\C(X)$. Its
rational map is therefore birationally equivalent to $f$ and to the Iitaka
fibration of $K_X+B$.
Proof. [Plain form]
First identify the complete section space upstairs with the pullback of a
complete section space on the base, using the exact presentation and a valuation
argument valid for reflexive sheaves. Then produce birational sections on the
base by the polarized effective-birationality theorem applied to a big adjoint
on a smooth model. To identify the Iitaka fibration, observe that multiples of
$m$ form a cofinal divisible subsequence of the Cartier section series, which
has the same field of ratios.
Proof. [Technical form]
Put $L=K_X+B$ and $K=\C(Z)$, and use the exact
presentation~(eq:relative-exact-presentation). For every positive integer
$l$ divisible by $p_0$, the equality of subspaces of $\C(X)$ is first proved
H^0\!(X,\OO_X(\lfloor lL\rfloor))
=\psi^{l/p_0}f^*
H^0\!(Z,\OO_Z(\lfloor lD_Z\rfloor)),
where $f^*$ on the right means pullback of rational functions. For
$v\in K^*$, the exact divisor identity gives
lL+\Div_X\!(\psi^{l/p_0}f^*v)
=f^*\!(lD_Z+\Div_Z(v)).
If $v$ is a section of the rounded system on $Z$, then
$lD_Z+\Div_Z(v)\ge0$ by the section convention of Section~sec:vocab. This
divisor is rational Cartier, so local effective Cartier multiples show that its
pullback is effective; (eq:relative-section-divisors) gives the inclusion
from right to left in~(eq:relative-section-spaces).
Conversely, let $u\ne0$ be a section on the left and let $X_\eta$ be the
generic fibre over $K$. The exact presentation rewrites the section inequality
as $\Div_X(u/\psi^{l/p_0})+lf^*D_Z\ge0$, whose horizontal valuations give
\[
\Div_{X_\eta}\!(u/\psi^{l/p_0})\ge0.
\]
The generic fibre is normal, so this quotient is regular there; and the
contraction condition $f_*\OO_X=\OO_Z$ gives
$H^0(X_\eta,\OO_{X_\eta})=K$. Hence $u=\psi^{l/p_0}f^*v$ for some $v\in K^*$. By
(eq:relative-section-divisors) the source inequality is $f^*G\ge0$ where
$G=lD_Z+\Div_Z(v)$ is rational Cartier. Effectivity of $G$ descends to $Z$:
near the generic point of any prime $P\subset Z$, write a Cartier multiple of
$G$ as $aP$ with $P$ locally principal; the local Cartier pullback of $P$ has a
prime component dominating $P$ with positive multiplicity $e$ because $f$ is
surjective, and its coefficient in the pullback of the chosen multiple is $ae$.
Thus $f^*G\ge0$ forces $a\ge0$. Doing this for every $P$ proves $G\ge0$, so $v$
is a section on $Z$, and (eq:relative-section-spaces) holds. Since $p_0$
clears $\Phi$, the divisor $lL$ is integral Weil, and the argument applies to its
reflexive sheaf even when $lL$ is not Cartier.
Now produce birational sections on the base. Choose a smooth projective
determination $q:W\to Z$ of the nef data that also resolves the boundary, and
write
\[
K_W+B_W^{cr}+M_W=q^*D_Z.
\]
Let $A$ be the strict transform of $B_Z$ plus the reduced exceptional divisor.
On nonexceptional primes $A$ agrees with $B_W^{cr}$; on exceptional
primes its coefficient is one, at least the coefficient of $B_W^{cr}$ by
generalized log canonicity. Thus
K_W+A+M_W=q^*D_Z+E_W,\qquad E_W=A-B_W^{cr}\ge0
exceptional over Z.
Put $Λ=\mathcal B\cup\{1\}$. The pair $(W,A)$ is an ordinary log smooth lc
pair with coefficients in the fixed DCC set $Λ$, and $pM_W$ is nef Cartier.
Since $D$ is big and $D_Z\sim_{\Q}D$, the adjoint on the left of
(eq:relative-big-model) is big.
For each $1\le k\le d$ let $b_k=b(Λ,k,p)$ be the integer in the
polarized effective-birationality theorem of Birkar--Zhang. Its conclusion is
that $|\lfloor l(K_W+A+M_W)\rfloor|$ is birational for every positive $l$
divisible by $b_k$ when $\dim W=k$, the floor taken on the whole adjoint. Set
\[
m=lcm(p_0,b_1,...,b_d),
\]
an integer depending only on $d$ and $\Phi$. Fix a positive multiple $l$ of $m$
and take $k=\dim Z$. For a rational section $v\in\C(W)=K$ of the displayed
birational system, pushing its divisor inequality forward using
(eq:relative-big-model) gives $\Div_Z(v)+lD_Z\ge0$. Thus these sections
belong to the complete rounded system on $Z$, and their ratios generate $K$
because the system on $W$ is birational. The complete system on $Z$ is therefore
nonempty and generates $K$, and (eq:relative-section-spaces) transfers
exactly these ratios to $X$: the common factor $\psi^{l/p_0}$ cancels. The
complete source system is nonempty and generates the embedded field $K\subset
\C(X)$ for every such $l$.
\noindent{\smallhttps://arxiv.org/pdf/1411.2118}
The function field of the image of a system's rational map is generated by its
section ratios, so each of these maps is birationally equivalent to $f$. To
compare with the Iitaka fibration, choose for the individual pair an integer
$r>0$ such that $rL$ is Cartier. Degrees divisible by $lcm(m,r)$
form a cofinal divisible subsequence of the Cartier section series of $L$, and
such a subsequence has the same field of section ratios: given $u/v$ in one
degree, choose $a\ge1$ with the multiplied degree in the subsequence and write
$u/v=(uv^{a-1})/v^a$. The field defining the Iitaka fibration is therefore $K$,
and $\kappa(L)=\trdeg_{\C}K=\dim Z$. This identifies $f$ with the Iitaka
fibration up to birational equivalence. The individual index $r$ is used only
for this comparison and does not enter $m$.
\medskip
\medskip\noindentIn words. The moduli denominator theorem says the adjoint on the
base can be written with bounded denominators. When the base is reasonably
simple, one can do better: a uniform multiple of the adjoint is not merely
integral, it is the divisor of an actual rational function. ``Reasonably
simple'' means rationally connected, the algebraic-geometric analogue of
connected-by-rational-curves, which forces the linear-algebraic obstruction (the
Picard group, and its torsion) to be bounded. The plan is: bound the boundary
coefficients and the singularities by global ACC, bound the torsion of the
divisor class group by boundedness, then clear denominators and apply the
torsion bound.
\medskip\noindentTechnical form.
[Uniform principal multiple over a rationally connected base]
Fix an integer $1\le d\le4$, a DCC set $I\subset[0,1]\cap\Q$, and a positive
integer $p$. There exists a positive integer $\ell=\ell(d,I,p)$ with the
following property. Let $Z$ be a normal projective variety of dimension $d$
with a rationally connected smooth projective resolution. Suppose
$(Z,B+M_Z)$ is generalized klt, $B\ge0$ has coefficients in $I$, and the
b-nef rational b-divisor $\mathbf M$ has $p\mathbf M$ b-Cartier. If
$D=K_Z+B+M_Z\sim_{\Q}0$, then $\ell D$ is an integral principal divisor.
Consequently $a\ell D$ is principal and $h^0(Z,\OO_Z(a\ell D))=1$ for every
integer $a\ge1$.
Proof.
There are three steps: global ACC gives a finite coefficient set and a uniform
positive lower bound for generalized log discrepancies; boundedness up to
isomorphism in codimension one bounds torsion in the divisor class group; and
clearing the coefficients of the actual adjoint applies that torsion bound.
\medskip
\medskip\noindentTechnical form. Suppose an exceptional prime $E$ over $Z$ is marked
with generalized log discrepancy $a\in(0,1)$. Choose a high projective log
resolution $g:W\to Z$ on which the nef data descend and $E$ appears, and write
$K_W+B_W+M_W=g^*D$ with $\Supp B_W$ SNC and all coefficients less than one.
Arrange also for an effective exceptional Cartier divisor $F$ on $W$ with $-F$
$g$-ample and $\Supp(B_W)\cup\Supp F$ SNC, by resolving the singular locus and
then the marked valuation with centres of codimension at least two so that the
negative of each new exceptional divisor is relatively ample.
Choose a small positive rational $\delta$ with all coefficients of
$B_W+\delta F$ still below one and, when $E$ is marked,
$\delta\mult_E F<a$. For a positive ample Cartier $A$ on $Z$, the divisor
$M_W+g^*A-\delta F$ is ample; a general member $Λ$ of a divisible multiple
makes $(W,B_W+\delta F+Λ)$ sub-klt. There is a principal $P$ on $W$ with
$\delta F+Λ=M_W+g^*A+P$, and pushing forward gives $K_Z+B+g_*Λ=
D+A+g_*P$ with $g^*(K_Z+B+g_*Λ)=K_W+B_W+\delta F+Λ$. So, with
$\Xi=B+g_*Λ$, the pair $(Z,\Xi)$ is an ordinary effective rational klt
pair, and a marked $E$ has ordinary log discrepancy $a-\delta\mult_E F\in(0,1)$.
Apply the extraction corollary of BCHM to the klt pair $(Z,\Xi)$ with the
prescribed set $\{E\}$ or the empty set. This gives a projective birational
$h:Z^+\to Z$ with $Z^+$ $\Q$-factorial and with exactly the prescribed
exceptional primes (a small $\Q$-factorial modification in the empty case).
Restore the generalized data by $K_{Z^+}+B^++M_{Z^+}=h^*D$ with the same nef
b-divisor; the pair is generalized klt with rationally trivial adjoint, and
$B^+$ is effective, being the strict transform of $B$ together with $(1-a)E$ in
the extraction case.
Now apply global ACC. If the nef trace is not numerically trivial on a model
determining $\mathbf M$, write $M_W=\frac1p(pM_W)$ with $pM_W$ nef Cartier, and
use the generalized global ACC theorem: the nef weight $1/p$ is a single fixed
weight adjoined to the DCC set, so the boundary coefficients lie in a finite
set. If instead $M_W\equiv0$, take a common model $\widetilde W\to Z^+$; the
trace $M_{Z^+}$ is $\Q$-Cartier and the difference $M_{\widetilde W}-\pi^*M_{Z^+}$
is exceptional and numerically trivial over $Z^+$, so the negativity lemma makes
it zero, hence $M_{Z^+}\equiv0$ and the generalized discrepancies are the ordinary
discrepancies of $(Z^+,B^+)$; ordinary global ACC again gives finitely many
coefficients. Applying this first to the small modification with no marked
divisor shows the coefficients of $B$ lie in a fixed finite set $I_0\subset I$
and the pairs $Z$ are uniformly generalized $ε$-lc for some
$ε=ε(d,I,p)>0$: otherwise a sequence of marked valuations
with discrepancies decreasing to zero would, after extraction, produce boundary
coefficients $1-a_i$ in a DCC set that global ACC would force to be finite, a
contradiction.
\medskip
\medskip\noindentTechnical form. Birkar's boundedness theorem applies to the
varieties $Z$: projectivity, fixed dimension, generalized $ε$-lc
singularities, effective boundary, rationally trivial adjoint, and rational
connectedness. It gives, for each $Z$, a projective $V_Z$, from one bounded set,
isomorphic to $Z$ in codimension one; replacing by finite normalization keeps
this and stays bounded. Hence a projective morphism $\mathcal V\to S$ over a
finite-type base whose normal fibres are isomorphic to the $Z$ in codimension
one.
Let $U=Z_{reg}$. First, $H_1(U(\C),\Z)$ is invariant under isomorphism
in codimension one of normal varieties: the common open subset, intersected with
the regular loci, is each regular locus with a closed subset of complex
codimension at least two removed, and removing such a subset from a smooth
complex variety does not change the fundamental group (perturb paths and
homotopies, relative to boundary, transverse to a smooth stratification whose
strata have real codimension at least four). Second, the groups take only
finitely many types. Replace $S$ by its reduction and cover by finitely many
flat affine strata with $\mathcal U=Sm(\mathcal V/S)$ the
relative smooth locus; embed $\mathcal V\subset\PP^N_{\C}\times S$ and identify
$\PP^N(\C)$ with Hermitian matrices $H^2=H$, $\tr H=1$. The map
$\mathcal U(\C)\to S(\C)$ is continuous semialgebraic, so the Delfs--Knebusch
semialgebraic triviality theorem partitions $S(\C)$ into finitely many pieces
over which the whole inverse image is a product; each semialgebraic fibre has the
homotopy type of a finite complex. Thus the $H_1$ take only finitely many
finitely generated isomorphism types.
For each $Z$, $\Cl(Z)$ is finitely generated: a smooth projective rationally
connected resolution $r:Y\to Z$ gives a surjection $\Pic(Y)\to\Cl(Z)$ (strict
transforms), and since the Albanese map of $Y$ is constant,
$\Pic^0(Y)=0$, so $\Pic(Y)$ is finitely generated by N\'eron--Severi finiteness.
Normality and projectivity give $\Gamma(U,\OO_U^*)=\C^*$, and
$\Pic(U)=\Cl(Z)$. For every $n\ge1$ the \'etale Kummer sequence and
classification of finite abelian torsors give
\Cl(Z)[n]\simeq H^1_{\'et}(U,\mu_n)\simeq
\Hom(H_1(U(\C),\Z),\mu_n(\C)).
For a fixed $Z$ the orders on the left are bounded because $\Cl(Z)$ is finitely
generated; if $H_1(U(\C),\Z)$ had a free summand of positive rank the orders on
the right would grow without bound, so it is finite. Finitely many finite groups
occur, so there is a uniform $T$ divisible by all their exponents, and every
torsion class of $\Cl(Z)$ is killed by $T$: take $n$ its order in
(eq:rc-kummer), whose right side has exponent dividing $T$.
\medskip
\medskip\noindentTechnical form. Since $pM_Z$ is integral Weil, being the pushforward
of a Cartier divisor on a determining model, choose $q$ divisible by $p$ and by
all denominators in the finite set $I_0$; then $qD$ is integral Weil. The
relation $D\sim_{\Q}0$ makes $[qD]\in\Cl(Z)$ torsion: if $nD=\Div(v)$ then
$n(qD)=\Div(v^q)$. Hence $TqD$ is principal, and $\ell=Tq$ proves the
proposition in dimension $d$; a common multiple over the finite dimensional range
handles the stated subset. All positive multiples of $\ell D$ are principal, and
projectivity with integrality gives the one-dimensional section space.
\medskip\noindentFinally, the corollary pulling the two theorems together.
[Uniform principal multiple from a fibration]
Fix a finite rational coefficient set $\Phi\subset[0,1]\cap\Q$. There is a
positive integer $r=r(\Phi)$ such that the following holds. Let $(X,B)$ be a
projective klt fourfold pair with effective boundary having coefficients in
$\Phi$, and suppose $K_X+B\sim_{\Q}0$. If there is a contraction $f:X\to Z$ with
$\dim Z>0$ and with a rationally connected smooth projective resolution of $Z$,
then $r(K_X+B)$ is an integral principal divisor. This includes $B=0$, and every
positive multiple is also principal.
Proof. [Plain form]
Apply the uniform denominator theorem with zero base class. The base adjoint is
rationally trivial, so the rationally connected torsion proposition applies to
it, giving a principal multiple. Multiplying the exact presentation by that
multiple writes the total multiple as the divisor of an actual function.
Proof. [Technical form]
Apply Theorem~thm:relative-denominators to $f$ with the zero divisor as
the rational pullback class. It gives uniformly bounded integers $p_0\mid p$, a
function $\psi\in\C(X)^*$, and an exact identity
K_X+B+\frac1{p_0}\Div(\psi)=f^*D_Z,\qquad D_Z=K_Z+B_Z+M_Z,
with $(Z,B_Z+M_Z)$ generalized klt, $B_Z$ effective with coefficients in a fixed
rational DCC set, and $p\mathbf M$ b-Cartier with nef data. Since
$f^*D_Z\sim_{\Q}0$ and $f_*\OO_X=\OO_Z$, the projection formula makes
$D_Z\sim_{\Q}0$.
Proposition~prop:rc-torsion gives a uniform $\ell$ and $v\in\C(Z)^*$ with
$\ell D_Z=\Div(v)$. Choose $r$ divisible by both $\ell$ and $p$, so $p_0\mid r$.
Multiplying (eq:rc-exact-relative) by $r$ gives
\[
r(K_X+B)=\Div\!((v\circ f)^{r/\ell}\,\psi^{-r/p_0}).
\]
Both exponents are integers, so this is the divisor of an actual rational
function and includes the integrality of the adjoint multiple. All constants
depend only on $\Phi$, since the possible dimensions of $Z$ form a finite set and
the denominator theorem supplies uniform $p_0,p$ and a fixed DCC set.
\appendix
\medskip
The following table collects symbols used in more than one place. A section
reference points to the introduction of the symbol.
\small
{@{}p{.16\textwidth}p{.52\textwidth}p{.22\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
$(X,B)$ & projective lc pair of dimension $d\le4$, boundary $B$ & sec:question \\
$\Phi$ & fixed finite set of allowed boundary coefficients in $[0,1]\cap\Q$ & sec:question \\
$f:X\to Z$ & contraction with log Calabi--Yau adjoint & sec:question \\
$K_X+B\sim_{\Q}f^*D$ & rational pullback hypothesis & sec:question \\
$B_Z$ & discriminant divisor, coefficient $1-t_P$ & sec:question \\
$t_P$ & log canonical threshold of $f^*P$ over the generic point & sec:question \\
$\mathbf M$, $M_W$ & moduli b-divisor and its trace on a determining model $W$ & sec:generic \\
$p_0$ & generic-fibre trivializing degree, clears $\Phi$ & sec:generic \\
$p$ & uniform multiple making $\mathbf M$ b-Cartier & sec:generic \\
$\psi$ & principal correction with vertical divisor & sec:generic \\
$D_Z$ & exact base representative, $D_Z=K_Z+B_Z+M_Z$ & sec:generic \\
$\mathcal B$ & DCC set of discriminant coefficients & sec:generic \\
$\alpha$ & coefficient of $D_W$ at a prime $P\subset W$ & sec:generic \\
$V_C\to C$ & curve slice of a smooth model & sec:curve \\
$F=h^*[c]$ & special fibre over the sliced point $c$ & sec:curve \\
$\Gamma$, $E$ & dlt boundary and the error $\Gamma-B^c-tF$ & sec:curve \\
$N\to C$ & endpoint of the relative minimal model program & sec:curve \\
$T=F_{red}$ & reduced special fibre, connected & sec:residue \\
$H$ & effective horizontal boundary with coefficients in $\Phi$ & sec:residue \\
$(T,B_T)$ & slc log Calabi--Yau pair obtained by adjunction on $T$ & sec:residue \\
$\mathcal D(\Phi)$ & DCC set of different coefficients & sec:residue \\
$a/m$ & $p_0\beta$ in lowest terms, the denominator to kill & sec:residue \\
$C'\to C$ & cyclic cover $w^m=z$ & sec:residue \\
$\pi:Y\to N$ & normalized base change, Galois group $\mu_m$ & sec:residue \\
$s$ & ambient generator $(\pi^*\theta)^{ ⊗ p_0}\pi^*\psi\,w^{-a}$ & sec:residue \\
$u$ & global generator of $p(K_T+B_T)$ & sec:residue \\
$r_j$ & residue ratio on a covering component & sec:residue \\
$m=m(d,\Phi)$ & degree generating $\C(Z)$ from sections & sec:effective \\
$\ell=\ell(d,I,p)$ & uniform principal multiple on the base & sec:torsion \\
$r=r(\Phi)$ & uniform principal multiple in the fourfold corollary & sec:torsion \\
\bottomrule
\normalsize
\medskip\noindentConventions. The symbols $\sim$, $\sim_{\Q}$, and $\equiv$ denote linear,
rational linear, and numerical equivalence. Tensor products of line bundles are
written additively: $A+B$ means $A ⊗ B$ and $mA$ means $A^{ ⊗ m}$.
Equalities such as~(eq:relative-exact-presentation) are equalities of
actual rational divisors for compatible choices of canonical divisors and the
displayed rational function; the canonical divisors are fixed using rational top
forms and are not rescaled silently. An integral principal multiple $rD$ means
$rD=\Div(v)$ for an actual rational function $v$, which includes integrality of
every coefficient of $rD$. The rounded system of a rational divisor $D$ is the
complete system of $\OO_X(\lfloor D\rfloor)$; a nonzero $v$ is a section exactly
when $\Div(v)+D\ge0$, and this needs no Cartier condition. The field generated by
a system is the subfield of $\C(X)$ generated by ratios of its nonzero sections.
The two integers $p_0$ and $p$ satisfy $p_0\mid p$; all bounds depend only on the
dimension $d\le4$ and the coefficient set $\Phi$, never on the individual
fibration.