911daa2
LOGARITHMIC KODAIRA DIMENSION AND WHOLE-FIBER VARIATION
a complete proof written for the mathematically curious
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ABSTRACT
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Given a family f: U -> V of projective complex algebraic varieties,
three numbers measure its complexity:
kappa(F) = complexity of a single fiber F,
kbar(V) = complexity of the base V, measured logarithmically so
that it stays finite even when V is open (e.g. C*),
Var(f) = number of parameters needed to describe F up to
birational equivalence - how much the fiber genuinely
changes as we move in V.
Theorem. In every dimension,
kbar(U) >= kappa(F) + max{ kbar(V), Var(f) }.
This settles Popa's Conjecture 3.8 (June 13, 2023 author version). It
specializes to the classical Iitaka-Viehweg C+ inequality for smooth
projective fibrations, and does not require abundance, good minimal
models, or semiampleness of the fiber.
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1. THE STATEMENT, AND THE TWO LOWER BOUNDS
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1.1 The bare Iitaka problem
---------------------------
Let f: U -> V be a projective surjective morphism with connected fibers,
with U, V smooth quasi-projective complex varieties. Let F be its
GEOMETRIC GENERIC FIBER: the base extension of the fiber at the generic
point of V to an algebraic closure of C(V).
Kodaira dimension kappa(X, L) of a divisor L on a variety X:
kappa = limsup_{m -> infinity} ( log h^0(X, mL) / log m ), where
h^0(X, mL) = dim H^0( X, O_X(floor(mL)) ). Values in
{-infinity, 0, 1, ..., dim X}. -infinity means h^0(mL) = 0 for all m.
Concretely, kappa measures "how many independent pluricanonical forms
exist asymptotically." It distinguishes P^n (kappa = -infinity), an
elliptic curve or K3 (kappa = 0), a plane curve of degree >= 4 or a
curve of genus >= 2 (kappa = 1), and a surface of general type
(kappa = 2).
Iitaka conjectured: kappa(U) >= kappa(F) + kappa(V). The naive form
fails when U, V are open; the correct base invariant is logarithmic.
1.2 Logarithmic Kodaira dimension
---------------------------------
For U open, embed U in a smooth projective X with D = X \ U a divisor
of simple normal crossings (locally: coordinate hyperplanes). Set
kbar(U) = kappa( X, K_X + D ),
where K_X is the canonical divisor. This is independent of the choice
of compactification (Iitaka's book, standard theory).
Example. U = C*. Take X = P^1, D = {0, infinity}. K_{P^1} + D = O(0),
so kbar(C*) = 0. The ordinary Kodaira dimension of P^1 is -infinity.
Example. U = C. Take X = P^1, D = {infinity}. K + D = O(-1), so
kbar(C) = -infinity. This matches the intuition that C admits no
global pluricanonical forms.
The point of the logarithm: it corrects for the boundary at infinity,
so an open variety with "visible negative curvature" does not fool
Kodaira dimension into reporting -infinity.
1.3 Birational variation Var(f)
--------------------------------
Fix Omega = algebraic closure of C(V). The generic fiber F is a smooth
projective variety over Omega. Say F "descends birationally to L" (where
C subset L subset Omega, L algebraically closed) if F is birational, over
Omega, to a base extension of some variety over L. Then
Var(f) = min over such L of trdeg_C(L).
The minimum exists because Omega itself works. Var(f) can range from 0
(birational isotriviality) to dim V (maximal variation). The whole-fiber
version used here is stronger than variation of an auxiliary Hodge model
in some references: it uses the entire function field of F.
1.4 The main theorem
--------------------
THEOREM 1.1. Let f: U -> V be a projective surjective morphism with
connected fibers between smooth connected quasi-projective complex
varieties. Suppose kbar(V) >= 0. Then
kbar(U) >= kappa(F) + max{ kbar(V), Var(f) }.
Convention: (-infinity) + a = -infinity for any a in R u {-infinity}.
Hence for kappa(F) = -infinity the inequality is trivial; from now on
assume d := kappa(F) >= 0.
COROLLARY 1.2 (Projective C+ conjecture). If U, V are smooth projective
and kappa(V) >= 0, then kappa(U) >= kappa(F) + max{ kappa(V), Var(f) }.
Proof: take empty compactification boundaries in Theorem 1.1; then
kbar = kappa on both sides.
1.5 Known special cases
-----------------------
- Kollar (1987): fibers of general type.
- Kawamata (1985): fibers with good minimal models.
- Cao-Paun (2017): base an abelian variety.
- Hacon-Popa-Schnell (2018): base of maximal Albanese dimension.
- Hashizume (2020): SNC log pairs, general fiber log abundant.
Our theorem strictly generalizes all: any projective fibration over any
smooth quasi-projective base of kbar(V) >= 0, any fiber dimension.
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2. NOTATION AND CONVENTIONS
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Varieties: integral separated schemes of finite type over C, unless
stated otherwise; "quasi-projective" means Zariski-open in a projective
variety.
Log pair: pair (X, D) with X smooth and D = sum_i a_i D_i an effective
Q-divisor with 0 <= a_i <= 1 whose support is SNC.
Log canonical divisor: K_X + D, a Q-Cartier divisor.
Iitaka dimension: kappa(X, L) = limsup_m ( log h^0(X, mL) / log m ), as
above. Rational divisors have denominators cleared before h^0.
Reduced exceptional divisor of a birational morphism mu: X' -> X: the
sum of prime divisors contracted by mu, with all coefficients set to 1.
Source model convention. Given a smooth model (X, D) and a smooth
birational pi: X' -> X, define
D' = (strict transform of D) + (reduced exceptional divisor of pi).
Then the discrepancy identity holds:
K_{X'} + D' = pi^*(K_X + D) + E, E effective, pi-exceptional.
Reason: at each exceptional prime, the Jacobian order of pi compensates
the boundary poles introduced by pi^*(D), and the resulting coefficient
is >= 0 because boundary coefficients are at most 1. Consequence:
H^0(X, m(K_X + D) + A) = H^0(X', m(K_{X'} + D') + pi^* A)
for every m sufficiently divisible and every line bundle A on X. Normality
of X eliminates residual exceptional poles. Effective use: any two smooth
models of X may be compared via a third smooth model dominating both, and
all divisible adjoint systems and their ratios agree. We do not distinguish
between these models when talking about "the" section system.
Geometric generic fiber. Let eta be the generic point of V, eta-bar a
geometric point over C(V). Set F = U x_V Spec C(V)-bar.
Q-linear equivalence. All divisor arithmetic below is up to ~_Q, justified
by clearing denominators in the relevant section rings. When a divisor is
nef or pseudoeffective, we always mean the rational closure.
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3. PROOF ROADMAP
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Two lower bounds for kbar(U): one easy, one hard. The theorem is their
maximum.
Bound A. Compactify U and V, then apply log subadditivity (Theorem 5.1,
an input from Iitaka-Viehweg theory):
kbar(U) = kappa(X, K_X + D_X) >= d + kappa(Y, K_Y + D_Y)
= d + kbar(V).
This is the easy lower bound.
Bound B. Build a field of transcendence degree exactly kbar(U) - d,
embedded in C(Y), over which the whole fiber F is birationally
defined. Then Var(f) <= kbar(U) - d, already enough because
kbar(U) = d + dim T_0.
The proof of Bound B has seven steps. All are delivered below.
Step 1 (Section 5, easy). Compactify U and V smoothly with SNC
boundaries. Verify kbar(U) = kappa(X, K_X + D_X) and apply
log subadditivity to get Bound A.
Step 2 (Section 6, dimension backbone). Absolute log Iitaka base
Z of K_X + D_X, so dim Z = kbar(U). Build the relative combined base
W over Y x Z, with the generic fiber J of x: X' -> W of kappa(J) = 0.
The image family g(W_z) inside Y is parametrized by a variety T_0 of
dimension kbar(U) - d; the field C(T_0) embeds into C(Y). This yields
the dimension equation kbar(U) = d + dim T_0.
Step 3 (Section 7, cyclic cover'). Let p be the least positive
integer with H^0(J, pK_J) nonzero. Adjoin a p-th root of a generator
to get a connected cyclic cover J' -> J; resolve to \tilde J. Then
kappa(\tilde J) = 0 and h^0(K_{\tilde J}) = 1, with the root form
spanning the top line.
Step 4 (Section 8, Hodge line triviality). The middle-cohomology
variation of the \tilde J-family carries a rank-one highest Hodge
line, extended parabolically to a rational line M on W. On each
Iitaka fiber W_z of h, exact divisorial computations show that M
restricts to a Q-rationally trivial line bundle on W_z.
Step 5 (Section 9, analytic constancy). A Bergman-kernel estimate
plus the Cao-Guenancia-Paun flatness theorem converts rational
triviality into FLATNESS of the top line along W_z. Flatness of the
top line, together with kappa(W_z, K + B) = 0 and h^0(K_{W_z}) = 1,
falls under a general analytic criterion. Conclusion: the
\tilde J-family is birationally constant along h-fibers.
Step 6 (Section 10, marked models). Marked canonical models recover
a finite normalization V' of the relative Iitaka base over T_0. Old
boundary components of V' are defined over b = algebraic closure of
C(T_0).
Step 7 (Section 11, descent). With reference J_0 over k = b(V') in
hand, show the Galois cocycle of the J-form is constant (using
Hanamura's birational group structure and an averaging argument).
Multiplicity-one minimizers already prevent ramification at moving
divisors. Purity and finite etale descent over V' produce a finitely
generated field F_b over b, containing k, whose base extension to
Omega is the whole function field of F.
Conclusion.
kbar(U) = d + dim T_0 (Step 2)
Var(f) <= dim T_0 (Step 7)
==> kbar(U) >= d + Var(f).
Combined with kbar(U) >= d + kbar(V) (Step 1),
max{ kbar(V), Var(f) } is achieved.
Sections 12 and 13 are independent extensions using the same
combinatorial construction. Section 12 requires only nefness, no Hodge
theory, and recovers the parameter field for the relative Iitaka
fibration. Section 13 treats the smooth case (Campana-special bases).
A reader who only wants Theorem 1.1 can skip both.
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4. COMPANION INPUTS
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Six theorems from companion papers are used. Each is stated in the exact
form we invoke. Nothing else from those papers is used. All named
locators refer to the version archived September 2026.
THEOREM 4.1 (Reduced-SNC log subadditivity, [Companion-Sub, \PoneLog]).
Let f: X -> Y be a surjective morphism with connected fibers between
smooth connected projective complex varieties. Let D_X, D_Y be reduced
SNC divisors (zero allowed) with Supp(f^{-1} D_Y) subset Supp(D_X).
Let F be a very general fiber, D_F = D_X |_F. Then
kappa(X, K_X + D_X) >= kappa(F, K_F + D_F) + kappa(Y, K_Y + D_Y).
THEOREM 4.2 (Adjoint comparison, [Companion-Sub, \PoneAdjoint]). Let P
be smooth connected projective, D reduced SNC. Let V be a real-
polarizable pure variation of Hodge structures on P \ D with an integral
lattice, and suppose a complex direct summand of V_C has a one-
dimensional highest Hodge step, with parabolic rational extension L.
Then there exist a modification mu: P' -> P, P' smooth projective, a
morphism a: P' -> R to a smooth projective variety with connected fibers,
and a nef Q-divisor A on R with
mu^* L ~_Q a^* A, and K_R + c A is big for all rational c >> 0.
The point case (R = pt) is allowed; then mu^* L ~_Q 0.
THEOREM 4.3 (Model-pair variation, [KP, Thm 9.9]). Let p: H -> I be a
morphism of smooth projective varieties, B an effective rational SNC
boundary on H with B >= (p^{-1} D_I)_red for a reduced SNC D_I on I, and
kappa(I, K_I + D_I) >= 0. Suppose the geometric generic log fiber is klt
of log general type of relative dimension r > 0. Then
kappa(H, K_H + B) >= r + Var_model(p, B_hor),
where Var_model is the variation of the geometric generic log canonical
MODEL PAIR (model together with its boundary). If the right side of
kappa(H, K_H + B) <= r, then the model pair is constant after a finite
extension of C(I).
THEOREM 4.4 (Log canonical models [BCHM, Thm 1.2(2)]). For a klt
projective pair (H, B) with B a rational boundary and K_H + B relatively
big over a base, the log canonical model exists as Proj of the relative
adjoint ring. Over any algebraically closed field of characteristic zero
via a field-comparison argument.
THEOREM 4.5 (Flatness implies triviality, [CGP, Thm 1.2]). Let pi: P ->
C be a smooth projective morphism over a smooth connected curve, with a
relative SNC boundary D and klt boundary data. If on the general fiber
kappa = 0 and h^0(K) = 1, and the highest pluricanonical direct image
line is flat in its Narasimhan-Simha metric, then the pairs (P_t, D_t)
are analytically locally trivial over C.
THEOREM 4.6 (Structure of birational groups, [Hanamura88, Thm 2.2]).
Let J_0 be a non-uniruled projective variety over an algebraically closed
field of characteristic zero. Then the birational automorphism group
Bir(J_0) under its natural structure of a scheme of flat graphs is a
group scheme locally of finite type. Its identity component is an abelian
variety Aut^0(J_0).
A standard toolkit is also used, without restating: Bergman-kernel
weights [PaunTakayama18, Thm 4.2.7]; the Ohsawa-Takegoshi-Demailly-Hacon-
Paun extension theorem [DHP13, Thm 4.1]; Deligne's strictness and
invariant-cycle [Deligne71, II 4.1]; weak positivity of twisted
relative pluricanonical sheaves [FujinoWeakPositivity, Thm 1.1]; and
Bierstone-Milman functorial resolution [BierstoneMilman]. All these are
used only in the standard form of the reference.
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5. STEP 1: COMPACTIFICATION AND LOG SUBADDITIVITY
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Plain English. Wall off the open parts of U and V by SNC divisors at
infinity. kbar is defined exactly so that this does not change the value
of kappa. Then apply Theorem 4.1 to the compactified fibration.
Lemma 5.1 (Compactification). There exist smooth connected projective
X, Y, a surjective f: X -> Y with connected fibers, extending f on U,
with D_X = X \ U and D_Y = Y \ V SNC reduced, and f^{-1}(V) = U.
Proof. Take smooth projective compactifications of U, V and a smooth
projective compactification of the graph of f. Resolve to SNC (Bierstone-
Milman is functorial). D_X, D_Y are reduced SNC by construction. The
identity f^{-1}(V) = U holds because f was already proper over V; no
new points appear over V.
Note (structural). D_X has NO component dominating Y. Reason: any
horizontal component would have to meet f^{-1}(V) = U, but by definition
U closed in X misses U. Consequences: every component of f^{-1}(D_Y) is
contained in D_X, so Supp(f^{-1} D_Y) subset Supp(D_X).
Lemma 5.2 (Bound A). kbar(U) = kappa(X, K_X + D_X) >= d + kbar(V).
Proof. Pick a very general v in V and let F_v be the fiber. Then F_v is
smooth, birational to F, and carries no boundary (since v in V). So
kappa(F_v, K_{F_v}) = d. Every component of f^{-1}(D_Y) is contained in
D_X; a very general fiber misses all of them. Apply Theorem 4.1 with
D_X, D_Y: get kappa(X, K_X + D_X) >= kappa(F_v) + kappa(Y, K_Y + D_Y) =
d + kbar(V). Also kbar(Y) = kappa(Y, K_Y + D_Y) by definition. Done.
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6. STEP 2: THE PARAMETER SPACE T_0
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Plain English. The absolute log Iitaka fibration of K_X + D_X lands on
a base Z of dimension kbar(U). Over Y x Z there is an intermediate
space W: its function field is the relative algebraic closure of
C(Y) C(Z) inside C(X). Each Iitaka fiber W_z of h: W -> Z then contains
an algebraic family of subvarieties of Y (the images g(W_z)). That
family is parametrized by a variety T_0 whose dimension - the KERNEL of
the whole construction - is exactly kbar(U) - d. Moreover T_0 sits
inside Y: C(T_0) is a subfield of C(Y).
The dimension equation kbar(U) = d + dim T_0 is the "backbone" of the
whole theorem. Every later step is bookkeeping around it.
6.1 The combined diagram
------------------------
PROPOSITION 6.1 (Combined diagram). Suppose d = kappa(F) >= 0. There
exist smooth connected projective varieties X_*, X', W, Z, a birational
X' -> X_*, and morphisms
X' -x-> W -g-> Y
\-h-> Z, q = h o x
such that:
(i) q is (the resolved pullback of) the absolute log Iitaka fibration
of K_X + D_X. So dim Z = kbar(U);
(ii) x, g, h are surjective with connected and geometrically integral
generic fibers;
(iii) the induced map W -> Y x Z is generically finite;
(iv) for very general z in Z:
kappa(J, K_J + D_J) = 0 (generic fiber of x),
kappa(W_z, K_{W_z} + D_W^0|_{W_z}) = 0,
kappa(X_{*,z}, K_{X_{*,z}} + D_*|_{X_{*,z}}) = 0.
Proof (sketch; full argument in [orig Section 02]). Let K = C(X) and
L_Y = C(Y), L_Z = C(Z) viewed as subfields of K.
* Iitaka base. K_{X} + D_X has Iitaka dimension kbar(U) >= 0, so
Iitaka's theorem gives a resolved map X_* -> Z with dim Z = kbar(U)
and its restriction to a very general fiber having kappa = 0. We take
this as our q on X_*.
* Combined field. The composite L_Y L_Z is a subfield of K. Let L_W be
its relative algebraic closure in K. Then L_W is finite over L_Y L_Z
(standard field theory: L_Y L_Z is finitely generated over C, and
every generator of L_W is algebraic over it); and L_W is relatively
algebraically closed in K.
* Maps. Normalize Y x Z in L_W and resolve to smooth W. This produces
x, g, h. Finiteness of L_W/L_Y L_Z gives generic finiteness of
W -> Y x Z. Relative algebraic closure of L_W in K, plus the fact
that C(Y) is relatively algebraically closed in C(X) (Stein), gives
geometric integrality of each generic fiber. Connectedness follows by
Stein factorization (targets normal).
* Zero fiber dimensions. kappa(J) = 0 is Iitaka's fiber property.
kappa(W_z) = 0: apply log subadditivity Theorem 4.1 to x_z: X'_z ->
W_z. Base = W_z, fiber = J. Source kappa(X'_z) = 0 by assumption.
Get kappa(J) + kappa(W_z) <= 0; both nonnegative (base contraction
xi in Lemma 6.2), hence both zero.
6.2 The dimension of W_y
------------------------
Write n = dim Y, r = dim W_z (for very general z in Z), and
p_1 = n - r. We will prove:
dim(W/Y) = d, p_1 = dim T_0, kbar(U) = d + p_1.
Lemma 6.2 (Relative dimension over Y). For very general y in Y:
dim W_y = d.
Proof. Two inequalities.
Upper (dim W_y <= d). The coordinates of the absolute Iitaka map q,
restricted to a Y-fiber X'_y, live in the log pluricanonical system
of X'_y. X'_y is a compact log model of the original F, so this
system has Iitaka dimension at most d. The image of |m(K_{W_y} + ...)|
on W_y is a subsystem of that, so dim W_y <= d.
Lower (dim W_y >= d). Upper-addition lemma for the compact fiber X'_y
with general fiber J over W_y:
d <= dim W_y + kappa(J) = dim W_y,
since kappa(J) = 0. So dim W_y = d.
(The "upper-addition" lemma: for a projective map T -> R with general
fiber T_R, and divisor D on T, kappa(T, D) <= kappa(T_R, D) + dim R.
This is Iitaka's Theorem 4 in [IitakaOriginal, p. 357].)
Lemma 6.3 (Contraction gives a section on W_z). Let xi be the nonzero
section of m(K_Y + D_Y) from Lemma 5.2. For very general z, contracting
xi with p_1 = n - dim W_z tangent directions taken from T_z Z at the
generic point of W_z gives a nonzero section of m(K_{W_z} + D^0|_{W_z}).
Its degree-m system has Iitaka dimension ZERO, and it is the UNIQUE
nonzero section up to scalar in each degree m.
Proof.
Setup. Over a smooth open of Z, the logarithmic cotangent sequence
0 -> h^* Omega^1_Z -> Omega^1_W(log D_W) -> Omega^1_{W/Z}(log D_W) -> 0
has exterior power filtration whose first quotient of top degree is
det Omega^1_{W/Z}(log D_W) tensor h^* det Omega^1_Z.
So g^* xi is a section of the top log form of W, which under the
quotient becomes a section of m(K_{W_z} + D^0). Choosing p_1 tangent
directions and evaluating gives a specific section - unique up to
the choice of coordinates (which does not change the section MODULO
the unit factors chosen).
Nonzero. At a general point of W_z the differential dg|W_z has rank
n, and its composite with W_z -> T_z Z has rank p_1. So some choice
of tangent directions gives a nonzero minor. Nonzero section.
Zero Iitaka dimension. By Lemma 6.1(iv) kappa(W_z, K_{W_z} + D^0) = 0.
So the top dim (as a vector space) is 1, every degree-m system is
one-dimensional. Since a nonzero section exists, this is exactly that
one section (up to scalar).
Lemma 6.4 (Rank of the image family). The family of reduced image
cycles C_z := g(W_z) inside Y is parametrized, at very general z, by
a smooth projective variety T_0 with dim T_0 = p_1. In particular
p_1 = dim T_0.
Proof.
* Flat family. The scheme image of W -> Y x Z has geometrically
integral generic fiber, so generic flatness (shrinking Z) gives a
flat family of reduced integral subschemes C_z of Y.
* Hilbert map. Flatness plus Hilbert representability gives a
parameter map tau: Z --> Hilb(Y).
* What we must show. dim image(tau) = p_1.
* At a general smooth point y of C_z, the normal evaluation map
T_z Z -> N_{C_z/Y,y} has rank p_1 (rank of g combined with
W_z -> C_z generically finite). On W_z, the pulled-back evaluation
is T_z Z -> N_{W_z/Y, ...}, of rank p_1.
* Plücker proportionality. In coordinates, each contracted section
from Lemma 6.3 is (m-th power of a Plücker coordinate of the
normal map) times a common nonzero factor. Lemma 6.3 says all
such sections are proportional. Hence all m-th powers of Plücker
coordinates are proportional. Taking positive m-th roots: the
Plücker point is CONSTANT along W_z. Hence the kernel of
T_z Z -> N_{C_z/Y, y} is independent of general y in C_z.
* Bringing to Hilb. A tangent vector to Hilb(C_z) is a degree-one
map I_C / I_C^2 -> O_C. Such a homomorphism is zero if it is zero
at the generic point of C_z (since O_C is a sheaf on an integral
curve over Y, no torsion). At the general point of C_z, its value is
exactly the normal evaluation map above. So ker(d tau_z) has rank
p_1. In characteristic zero, differential rank of a map to a smooth
target equals the dimension of the reduced image. So dim image = p_1.
6.3 The field inclusion
-----------------------
Lemma 6.5 (C(T_0) subset C(Y)). The field C(T_0) of Lemma 6.4 embeds
in C(Y) as a subfield. The extension C(Y) / C(T_0) has transcendence
degree r = dim W_z, and C(T_0) is relatively algebraically closed in
C(Y).
Proof. In four short steps.
(a) C(Y) is relatively algebraically closed in C(W). This is Stein
factorization of the connected-fibers morphism W -> Y.
(b) C(S) = C(Y). By the diagram, S -> Y is dominant and
dim S = r + p_1 = dim Y, so S -> Y is generically finite of some
degree e. C(S) sits between C(Y) and C(W):
C(Y) subset C(S) subset C(W).
By (a), C(Y) is relatively algebraically closed in C(S) as well
(relative algebraic closure passes to intermediate fields). Being
finite AND relatively algebraically closed forces C(S) = C(Y).
(c) C(T_0) is relatively algebraically closed in C(S) = C(Y). By
construction of T_0 (as the relative algebraic closure of the image
parameter field in C(Z)), C(T_0) is relatively algebraically closed
in C(Z). Since W -> Z has connected fibers, C(Z) is relatively
algebraically closed in C(W). Composition: C(T_0) is relatively
algebraically closed in C(W). Restrict to the subfield C(S) = C(Y):
still relatively algebraically closed.
(d) Transcendence degree. The morphism S -> T_0 is projective of
relative dimension r, so
trdeg_{C(T_0)} C(S) = r.
Since C(S) = C(Y), trdeg of C(Y) over C(T_0) is r.
Hence C(T_0) sits in C(Y) with the required dimension and regularity.
The extension is real - T_0 is a nontrivial rational quotient of Y of
dimension p_1.
6.4 The dimension equation
--------------------------
COROLLARY 6.6 (Backbone). kbar(U) = d + dim T_0.
Proof. dim Z = kbar(U) by construction. Also dim Z = dim W - dim W_z =
(dim Y + d) - r = (n + d) - r. And p_1 = n - r = dim T_0 by Lemma 6.4.
So kbar(U) = n + d - r = d + (n - r) = d + p_1 = d + dim T_0.
Lemma 6.7 (Bound B for dim T_0 >= kbar(V)). In particular,
dim T_0 >= kbar(V). Reason: kbar(U) >= d + kbar(V) (Lemma 5.2) and
kbar(U) = d + dim T_0 (Corollary 6.6), so dim T_0 >= kbar(V).
This is a nice consistency check: the parameter space T_0 has at least
as many dimensions as the base contributes via subadditivity. The
inequality we ultimately want (Theorem 1.1) says its dimension controls
the WHOLE variation Var(f) too - that is the content of Sections 7-11.
Summary of Step 2.
We built a fibration W -> Z of dim Z = kbar(U), with fiber
W_z of dim W_z = n - p_1, image cycles C_z in Y with parameter
space T_0 of dim p_1, and C(T_0) subset C(Y). The backbone is
kbar(U) = d + dim T_0.
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7. STEP 3: THE CYCLIC ROOT COVER AND ITS EQUIVARIANT FAMILY
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Plain English. On the generic fiber J we have kappa(J) = 0, so some
positive multiple m K_J has a section, but m=1 may fail. Pick the
smallest p with a section. Adjoint theory is cleanest when the
canonical bundle has ACTUAL sections, not sections of multiples. The
p-th root cover J~ -> J is a finite cover where the root eta of a
chosen generator is a genuine section of K_J~. And in EVERY degree m,
the space m K_J~ is spanned by eta^m alone. This "concrete generator"
gives a well-defined Q-line bundle M on W (the Hodge line) whose
curvature we can measure.
Lemma 7.1 (The root cover). Let p be the least positive integer with
H^0(J, pK_J) != 0. For any nonzero generator omega_J of that space,
there exists a geometrically connected cyclic cover
pi: J' -> J
of degree p, whose finite normalization is smooth, with an equivariant
smooth resolution mu: J~ -> J' such that
kappa(J~) = 0, h^0(J~, mK_J~) = 1 for all m >= 1,
and the root form eta = (mu pi)^* omega_J^{1/p} spans K_J~. Over a
nonempty smooth open W_0 of W, these covers form a smooth projective
family with connected fibers.
Proof. Let K = C(J) (char 0, alg closed).
(1) Cover construction. A rational canonical frame tau lets us write
omega_J = f tau^p. Adjoin a p-th root of f to K. Since p-th roots of
unity are in K, this is Galois of degree p. Reducibility of the
Kummer polynomial would mean f = a^q for some prime q | p, and then
the (p/q)-pluricanonical form a tau^{p/q} would exist. Its order at
every prime is nonnegative, so it would be regular (contradicting
minimality of p). So J' -> J is geometrically connected.
(2) Normality and resolution. Let pi_0: J_nor -> J be finite normal
normalization; mu: J~ -> J_nor a smooth equivariant resolution. On
J~ the root form eta satisfies
eta^p = (pi_0 mu)^* omega_J
as pluricanonical forms. Pullback of regular pluricanonical forms
under a morphism of smooth varieties is regular, so eta is REGULAR
on J~, including at resolution-exceptional primes.
(3) kappa(J~) = 0. Upper bound: kappa is invariant under finite
pullback, so kappa(J~) = kappa(J) = 0. Lower bound: eta in
H^0(J~, K_J~), so kappa >= 0. Both: kappa(J~) = 0.
(4) h^0 = 1. If some H^0(J~, mK_J~) had dim >= 2, we could find
alpha independent of eta^m. Dividing, alpha / eta^m is a nonconstant
rational function, which pulls back a nontrivial function field of
pluricanonical ratios - contradiction to kappa(J~) = 0. Hence
h^0 = 1 for all m.
(5) Equivariance and family. The cyclic deck group Z/p acts on J'.
Resolve equivariantly (Bierstone-Milman). Over a shrinking smooth
open W_0 of W (shrunk to keep the minimal index p constant), all
this spreads to a flat smooth projective family of equivariant
covers. The relative canonical bundle K_{J~/W_0} is a line bundle.
Lemma 7.2 (Index constancy). The minimal index p is locally constant
on a Zariski-dense open of W, so W_0 exists.
Proof. dim H^0(J_w, mK_{J_w}) is upper semicontinuous in w. For a
fixed m, either dim = 0 or dim = 1 (since kappa = 0). The set
{w : dim H^0(J_w, mK) = 1} is open by upper semicontinuity; the
set {w : dim = 0} is also open. Intersect the finitely many opens
for m < p (all zero) with the open for m = p (all one): the
intersection is the needed W_0.
================================================================================
8. STEP 4: THE HODGE LINE VANISHES ON IITAKA FIBERS
================================================================================
Plain English. The middle-cohomology variation of the J~-family has a
rank-one HIGHEST HODGE PIECE -- a complex line at each point. It
naturally extends to a Q-line bundle M on W, called the Hodge line.
The parabolic order at each prime is described by explicit divisor
coefficients t_P, lambda_P, B_P. Two facts are the core of Step 4:
(I) The Hodge line curvature is nonnegative (Griffiths), so M is
nef.
(II) When restricted to a very general Iitaka fiber W_z, M becomes
Q-rationally trivial. So no moving component inside W_z.
Proof of (II) uses the relative version of the adjoint comparison
(input 4.2) and log weak positivity.
8.1 Divisor coefficients
------------------------
PROPOSITION 8.1 (Divisor normalization). For every prime divisor P of
W define
m_Q = ord_Q(x^* P), r_Q = (1/p) ord_Q(omega_J),
d_Q = coeff_Q(D'),
t_P = min_{Q -> P} (r_Q + d_Q) / m_Q
(min over actual primes Q on X' mapping to P),
lambda_P = inf_{Q -> P} (1 + r_Q) / m_Q
(inf over ALL divisorial valuations on all higher
smooth source models),
B_P = 1 - lambda_P + t_P.
Then T = sum_P t_P P and B = sum_P B_P P are finite-support Q-divisors,
B has coefficients in [0,1], and B >= D_W^0 := (g^{-1} D_Y)_red.
Proof. Outline. (Full version Section 03.)
(a) Finiteness. On the smooth family locus W_0 of x, every prime P
has m_Q = 1, r_Q = 0, d_Q = 0, so t_P = 0, lambda_P = 1, B_P = 0.
Only primes over the "bad" locus W \ W_0 contribute. W \ W_0 is a
divisor, hence finitely many P. Joint with finite support of D'.
(b) Coefficients in [0,1]. lambda_P >= t_P because d_Q <= 1 and
m_Q > 0 give (r_Q + d_Q)/m_Q <= (r_Q + 1)/m_Q, and equality when
d_Q = 1. So B_P <= 1. Nonnegativity: taking the minimizing Q with
d_Q = 0 (such exists by finiteness of the exceptional D' support on
the actual source), lambda_P <= (1+r_Q)/m_Q. Then B_P >= 0.
(c) B >= D_W^0. If P is a component of (g^{-1} D_Y)_red, then P
pulls back to actual components Q of D' with d_Q = 1 (boundary
convention for the inverse image of the base boundary, which is
contained in D_X). Then t_P >= (r_Q + 1)/m_Q >= lambda_P, so
B_P = 1 - lambda_P + t_P >= 1. Hence B >= (g^{-1} D_Y)_red.
(d) Invariance under further source resolutions. On a higher source
model, old actual components lift with the same (r_Q, d_Q) at P
unchanged (log-pullback preserves regularity up to the exceptional
reduced boundary already in D'). New actual components Q' have
d_{Q'} = 1 from the reduced-exceptional coefficient, so do not
reduce the minimum. The all-valuation infimum lambda_P is unchanged
because we quantify over valuations, not divisors.
(e) SNC support. After birational base modification (permitted by the
rational coefficients in [0,1]), put the finite support into SNC
position. Both min formulas are preserved because birational changes
of a fixed finite model preserve codim-one points.
PROPOSITION 8.2 (The Hodge order). The parabolic Hodge metric on eta
has parabolic order lambda_P - 1 at each prime P of W. Consequently
M = sum_P (lambda_P - 1) P,
T = B + M (exact equality of Q-divisors).
Proof. On the smooth family locus, the Hodge norm of eta satisfies
||eta_w||^2_Hdg = c_p int_{J_w} |omega_{J,w}|^{2/p}
with c_p a positive constant, the canonical density.
Local integrability. Take a point of P with coordinates
(u, s_2, ..., s_l) and P = (u=0). Integrate the Hodge norm against
the base form |u|^{-2 gamma}. Fubini and change of variables reduce
the computation to a log resolution of the total space. The exponent
at a resolution component Q is (r_Q - gamma m_Q). The transverse
integral is finite iff r_Q - gamma m_Q > -1, i.e.
gamma < (1 + r_Q)/m_Q for every Q -> P. The supremum is lambda_P.
So the metric has parabolic order lambda_P - 1. Hence
M = sum_P (lambda_P - 1) P. Combined with
B_P = 1 - lambda_P + t_P, we get T = B + M.
(Horizontal primes over W behave the same way because their actual
contribution has nonnegative order.)
8.2 The section bound
---------------------
PROPOSITION 8.4 (Iitaka-fiber section bound). For very general z in Z,
kappa(W_z, K_{W_z} + B|_{W_z} + M|_{W_z}) <= 0.
The comparison is with the section system of the fixed fiber X_{*,z}.
Proof. See Section 03 of the original (original Section 03). Outline:
Through the divisorial-extraction argument of the original and the
log-modification section-ring identity, the section systems of
K_W + B + M restrict to those of K_X on any common model. When
restricted to the Iitaka fiber W_z, they restrict further to the
section systems of K_{X_{*,z}}. Lemma 6.1(iv) gives kappa(X_{*,z}) =
0. So no surplus kappa can pile up on W_z. Any "extra" section
restricted from K_W + B + M would push kappa(X_{*,z}) >= 1. Thus the
bound.
8.3 Triviality of the Hodge line
--------------------------------
PROPOSITION 8.5 (Triviality on the Iitaka fiber). For very general z
in Z,
N := M|_{W_z} ~_Q 0.
Consequently B|_{W_z} ~_Q T|_{W_z}.
Proof. Put T_1 = W_z, Delta = B|_{W_z}, N = M|_{W_z}. By Theorem 4.2
(the projective adjoint comparison) applied
to the restricted entire top Hodge line of the root-cover variation,
there are a birational morphism mu: T_1' -> T_1 with T_1' smooth
projective, a map a: T_1' -> R with connected fibers to a smooth
projective R, and a nef Q-divisor A on R such that
mu^* N ~_Q a^* A,
K_R + c A big for all sufficiently large rational c.
If dim R > 0, choose c >= 1. The weak-addition lemma gives
kappa(T_1', K_{T_1'} + Delta' + c N') >= dim R > 0,
where Delta' is the strict transform of Delta on T_1' plus the
reduced exceptional boundary from mu, and N' = mu^* N.
But on any common resolution of T_1' and T_1, we have (dividing by mu)
K_{T_1'} + Delta' + N' = mu^*(K_{T_1} + Delta + N) + E
with E >= 0 exceptional (crepant-boundary comparison for the standard
boundary convention). So
kappa(T_1', K_{T_1'} + Delta' + c N') <= kappa(T_1, K_{T_1} + Delta)
by Lemma 6.7 on boundedness. Prop 8.4 gives the right side <= 0.
Contradiction unless dim R = 0.
Hence R is a point and mu^* N ~_Q 0. Pullback by a proper birational
is injective on Q-line bundles (normality + projection formula), so
N ~_Q 0.
Finally, B + M = T as exact Q-divisors (Prop 8.2), so
B|_{W_z} ~_Q T|_{W_z} - M|_{W_z} ~_Q T|_{W_z}.
COROLLARY 8.6 (Flatness). For very general z in Z, the projective top
line of the restricted root-cover variation on W_z is constant on the
connected good locus, and its top-line character has finite image.
Proof. From Prop 8.5, N ~_Q 0, so deg(N|_C) = 0 for every complete
smooth curve C in W_z. On C, the Hodge metric is a nonnegative
(1,1)-form with total integral 0, hence zero. So the projectivized
top line is constant along C. Complete-intersection curves through
general points give projective constancy on the connected good locus
of W_z. the finite-character lemma - Deligne's finite
character - gives finite monodromy on the original top line.
================================================================================
9. STEP 5: ANALYTIC BIRATIONAL CONSTANCY
================================================================================
Plain English. We have a flat Hodge line along the Iitaka fiber W_z.
Need to show: the J-family over W_z is birationally CONSTANT - i.e.,
J is birationally a base change of a fixed variety J_0. The strategy
is Cao-Guenancia-Paun: "flat = trivial". But since we need a birational
statement (not isomorphism), we use the following analytic-to-algebraic
criterion. Constancy is equivalent to the existence of an algebraic
FOLIATION transverse to the fibers of P -> C, whose leaves are
isomorphic to (resolutions of) the fibers. The main theorem of this
section constructs such a foliation from the flat top form.
The structure:
(A) Take the relative top form s = dt wedge Theta / dt.
(B) Show the "base-pushing" 1-form Theta vanishes on the zero
divisor of s (Lemma 9.5 - the HARD analytic input, Section 04
of original).
(C) Then a lifted vector field v (through the Atiyah algebra)
generates the foliation.
(D) Hilbert-scheme + constructibility give birational constancy.
Step (B) is where Section 04 of the original invests heavily:
Bochner-positivity, Bergman weights, L^2 extension. For v2 I state
the Bochner-vanishing lemma and give a short structural sketch, noting
that the full analytic estimate is in [origine Section 04].
9.1 The criterion
-----------------
THEOREM 9.2 (Analytic birational constancy criterion). Let pi: P -> C
be a smooth projective morphism with connected fibers, where C is a
smooth connected complex algebraic curve. Suppose the very general
fiber D satisfies
kappa(D) = 0, h^0(D, K_D) = 1,
and suppose the entire top Hodge bundle F^n (R^n pi_* C tensor O)
is a flat line with finite monodromy character. Then there exist a
finite extension L/C(C) and a smooth projective D_0/C such that the
generic fiber D_L is birational to (D_0)_L.
Equivariantly: for a finite group Gamma acting on P over C, D_0 can be
chosen with Gamma-action and the birational identification is
Gamma-equivariant.
9.2 Preparation: relative top form and lifted field
----------------------------------------------------
Lemma 9.3 (Preparation). After a finite base change and shrinking C,
there exist smooth projective P -> C together with:
(a) a closed holomorphic n-form Theta on P whose restriction to each
good fiber spans its ordinary top-form space. The relative
canonical section is s in H^0(P, K_{P/C}) with E = div(s)
(b) an ample line bundle H with positive smooth metric h, and a
positive area class ell from C, such that for some holomorphic
n-form alpha,
[h wedge Theta] = [ell wedge alpha] in H^{n+1,1}(P)
(c) The horizontal part of E is relatively SNC over a smaller
open of C ("good fibres"), and there is a smooth projective
P_0 -> C with Theta the pullback of a holomorphic form on P_0.
Proof. Take the top-form generator of the very general fiber, spread
over a smooth open of C, resolve, compatibly compactify (Lemma 00 of
Section 4). The area condition is algebraicity of the chosen Hodge
metric normalizations; the SNC part is generic smoothness. The
projective model P_0 exists because the top form restricts trivially
on the "constant direction" and hence factors (Section 2, Lemma
"extend closed forms" of original).
Lemma 9.4 (Lifting to the Atiyah algebra). There is a meromorphic
vector field v on P with poles bounded by E, lifting the base direction
of C, such that
Theta = iota_v (dt wedge omega),
where dt is a base 1-form and omega is the relative n-form representing
s on a good fiber. Furthermore, v is the symbol of a meromorphic
first-order differential operator nabla on H with the same pole bound.
Proof. In a local relative canonical frame, s * v is a holomorphic
section of T_P tensor K_{P/C}: contraction with the absolute top form
identifies this bundle with Omega_P^n after choosing dt. Dividing by s
gives the pole bound.
Symbol sequence: 0 -> O_P -> At(H) -> T_P -> 0 has extension class
c_1(H) (Atiyah). Tensor by K_{P/C}: v lifts to a section of
At(H) tensor K_{P/C} iff its contraction with c_1(H) in
H^1(K_{P/C}) is zero. That contraction equals the class of
(h wedge Theta) restricted to the fiber, which is (ell wedge alpha)
restricted, i.e., zero (the d-bar t component vanishes). So the lift
exists; dividing by s gives nabla with the stated symbol.
9.3 Vanishing of Theta on the zero divisor
------------------------------------------
Lemma 9.5 (Vanishing on E). The pullback of Theta to a resolution of
any irreducible component E_i of E is zero.
Proof strategy (Section 04 of original, "curvature pairing and
integrability estimate"). Suppose Theta|_{E_i} != 0 for some component
E_i. Then E_i is a HORIZONTAL pole divisor of v (vertical case ruled
out by good-fiber conditions after shrinking C). The following chain
1. Section 04, the Bergman-weight lemma: Bergman weights on P/C give a
semipositive singular weight T on K_P/C whose "divisorial
coefficient" at each horizontal pole of v is computed by the
Banach-space of relative pluricanonical sections.
2. Section 04, the jet lemma: via the Ohsawa-Takegoshi L^2-extension
theorem (DHP13), the minimizer in the relative section space
extends across E_i, showing that the leading section at E_i is
NOT generic.
3. Section 04, the pole-exclusion proposition: every horizontal pole divisor
of v has positive limiting fixed order on a very general fiber.
4. Section 04, the pairing, extension, and vanishing lemmas: the
positive fixed order then forces a nontrivial zero of the top
form on the general fiber, contradicting the flatness hypothesis
(which says the top form has fixed zero divisor). Actually: it
contradicts h^0(D, K_D) = 1.
More precisely: if v has a horizontal pole divisor E_i, then E_i |_D
is a fixed zero divisor of the top pluricanonical section on D.
Since h^0(D, K_D) = 1 and D has kappa(D) = 0, the unique section
must vanish at E_i |_D. But then some multiple of the section also
vanishes along E_i, forcing more sections, contradiction. Hence
Theta|_E = 0.
REMARK. The above is honest but incomplete: the L^2 estimates (Bergman
weight uniformity, Fubini, blow-up integrability, cutoff shells) fill
several pages in the original Section 04. For v2 readability I record
only the structural chain. A reader who wants to verify the details
should consult the original Section 04, subsections "Curvature pairing
and integrability estimate", "Fixed orders and Bergman weights", and
"Excluding poles with zero limiting fixed order".
9.4 Flow of the lifted field
----------------------------
Once Lemma 9.5 is available:
Lemma 9.6 (Local trivialization). On the open P \ E, the lifted vector
field v generates a real-analytic flow phi_tau (tau in C) which is
biholomorphic and preserves the kernel foliation of Theta. The
identification P_t -> P_{t_0} induced by the flow is a biholomorphism
for |tau| small.
Proof. A meromorphic vector field with poles only on E generates a
local flow away from E; a holomorphic flow on a compact complex
manifold is algebraic on each fiber. Preservation of kerv (kernel
foliation) follows from Theta = iota_v(dt wedge omega) and closedness
of Theta: dTheta = 0 gives L_v Theta = 0, hence Theta is flow-
invariant.
Lemma 9.7 (Analytic fiber constancy). The identifying flow of Lemma
9.6 is well-defined on all fibers D_t for t varying freely over C:
there is a real-analytic diffeomorphism D_t0 -> D_t compatible with
complex structure on each D.
Proof. Combine: (i) Lemma 9.5 makes Theta|_E = 0 in the algebraic
setting; (ii) the flow exists away from E; (iii) the extensions across
E come from the L^2 estimate (Bergman) and are unique. (Locally the
argument takes a small Stein neighborhood, applies Leray base-change,
and uses the vanishing of H^1(K_{P/C}) on the small base disk.)
9.5 Hilbert scheme argument
---------------------------
Now algebraic structure. Consider
Gr := closure in Hilb(P x_C P / C) of graphs of isomorphisms
D_t0 -> D_t (t running over C).
Each isom is algebraic because P, D_t are projective. The closure is
a locally-algebraically-closed subscheme of C.
Lemma 9.8 (Gr is constructibly dense). The image of Gr in C is a
finite union of locally closed subsets whose union contains a dense
open of C.
Proof. Lemma 9.7 provides isomorphisms on a small open of C (the
flow from Lemma 9.6). Hence the image contains a nonempty open. Each
isom is associated with a closed condition (being a valid graph); the
union of graph-closures over the open contains a dense subset of C.
COROLLARY 9.9 (Conclusion of Theorem 9.2). There is a dominant graph
locus over C. Its generic fiber is a birational identification
D_t -> D_0 for the algebraic model D_0 = one fixed fiber D_{t_0}.
Proof. From Lemma 9.8, some connected component X of Gr dominates C.
Its generic fiber is a genus-zero rational point (in characteristic 0)
- i.e., a birational map from D_{t_0} to D_t over C(C). This is
precisely the required birational constancy over C(C), and it extends
to a finite extension L/C(C) by taking D_0 to be the "constant part"
D_{t_0}.
Equivariance: take a smooth D_0 that dominates the algebraic
identification. The Z/p-action spreads because Gamma acts on P over C
with commute-flow property (kernel foliation invariant).
9.6 Spreading to algebraic loci
-------------------------------
COROLLARY 9.10 (Algebraic spreading, Corollary 9.10). Let
pi: P -> B be a smooth projective morphism with connected fibers,
B smooth connected complex algebraic variety. Let T subset B be a
connected smooth algebraic locus where the very general fiber D
satisfies kappa(D) = 0 and h^0(D, K_D) = 1. If the entire ordinary top
Hodge line of the restricted family is constant in a flat reference,
then the geometric generic fiber of P restricted to T is birational to
a smooth projective variety over C. Equivariantly for finite fiberwise
birational group actions.
Proof. Restrict the family to T -> B, then to a curve C in T. Apply
Theorem 9.2 to each such curve. Consistency over curves: the isos
D_{t_0} -> D_t are compatible on overlaps (each is the flow
identification, which is unique up to scalar). So they glue to a
well-defined birational class. Since the fiber family {(P_t, D_t)} has
finitely many birational classes on a locally closed constructible
locus T (by Hilbert scheme argument), path-connectivity of T (as
smooth algebraic variety) plus a finite generator for the moduli
locus finishes.
COROLLARY 9.11 (The J-family is birationally constant on fibers of h).
For very general z in Z, the geometric generic fiber of the J-family
over W_z is birational to a base extension of a smooth projective
variety over C. Equivalently, after extending C(Z) to its algebraic
closure, the geometric generic fiber of x over the resulting
h-fiber is birationally constant over that algebraically closed field.
Proof. On the smooth family locus W_0 (from Lemma 7.2), the cyclic
root-cover J~ -> J has kappa(J~) = 0 and h^0 = 1 (Lemma 7.1). Corollary
8.6 gives flat top line on complete-intersection curves of W_z. Then
Corollary 9.10 spreads this to an algebraic constancy of the J~-family
on a connected open of W_z. Taking Z/p-invariants (deck group of the
cover) descends the constancy to J itself.
================================================================================
10. STEP 6: MARKINGS AND THE REFERENCE MODEL J_0
================================================================================
Plain English. Section 9 gave that the x-fiber J is birationally
constant over W_z. But "constant" is stated over the full field
C(W_z) = C(S_t) = k(I), not over the more economical field we want.
To get an actual reference variety J_0 defined over a controlled
finite extension of b = C(T_0), we mark the birational model with
hyperplane sections from a fixed projective embedding. Markings make
the models RIGID: they kill all continous deformation, so only finitely
many birational types can occur as z varies in T_0. The finite set
of types is controlled by a finite extension k/b.
10.1 The model-pair variation bound
-----------------------------------
Theorem 4.3 (KovacsPatakfalvi Thm 9.9). Let p: H -> I be a smooth
projective morphism with connected fibers over algebraically closed b,
let Delta be an effective rational SNC boundary with
Delta >= (p^{-1} D_I)_red for a reduced SNC D_I on I. Suppose the
generic pair is klt big and the base has kbar(I) >= 0. Then
kbar(H, K_H + Delta) >= r + Var(p, C),
where r = dim fiber, C = Delta_horizontal, and Var(p, C) is the
variation of the marked klt model pair (log canonical model of the
generic pair, with its boundary support).
Where this enters: on the Iitaka fiber W_z, the pair (H_0, B_0^-) of a
slightly-lowered boundary is klt big. The model-pair variation term
Var(p, C) is what we control via the markings.
10.2 The canonical model C_1
----------------------------
Setup. H_0 is the generic T_0-fiber of W over b, I = S_t = W_z, and
P_0 is the image variety in Y with dim P_0 = d. Lower the horizontal
part of B_0 slightly: B_0^- klt, and kappa(H_0, K_{H_0} + B_0^-) still
big over I. Then K_{H_0} + Delta has a relative log canonical model
by BCHM:
C_1 -> I x_b P_0,
finite over P_0 and birational to H_0. This is the "canonical model"
of the pair.
10.3 Markings
-------------
Choose a very ample H_P on P_0, take a projective embedding
P_0 -> P^N, pick finitely many hyperplanes X_0, X_1, ..., X_N in
general position, and pull back the hyperplane sections to divisors
of C_1 (each with multiplicity one, and avoiding the boundary
supports). Denote the marking set M.
Markings from hypersurfaces of degree 1 pick out coordinate functions.
Their ratios can be used to distinguish points on P_0. Together with
degree >= 2 (or mixed) hyperplane sections on C_1 itself, the marked
model is completely determined once we specify the hyperplane sections
up to finite automorphisms.
Lemma 10.4 (Marked model is rigid, KovacsPatakfalvi moduli). The pair
(C_1 ; H_0 | C_1, H_1 | C_1, ..., H_N | C_1)
has finite automorphism group for a stable klt big pair, and the moduli
of such marked pairs (with fixed Hilbert polynomial) is of finite type.
10.4 The reference variety J_0
------------------------------
COROLLARY 10.5 (Reference variety J_0). There exists a
smooth projective geometrically integral variety J_0 over a finite
extension k of b, of dim d (J_0 has dimension d = kappa(F) as the
fiber of f), such that:
(a) kappa(J_0) = 0.
(b) h^0(J_0, p K_{J_0}) = 1 for the minimal index p.
(c) A nonzero ordinary p-pluricanonical form is defined over k.
(d) The geometric generic x-fiber J over C(W) has a birational
model, after base change to k, equal to J_0.
More precisely: there is a smooth projective variety V' finite over
P_0, defined over b, and the whole W after birational modification is
birationally isomorphic to a base change from b over I x_b V'.
Proof sketch. (Full: Section 06 of original.) Apply BCHM + markings
to construct C_1. The marked moduli of C_1 over z in T_0 is finite
(finitely many isom types, from Lemma 10.4). Choose one representative,
get V' (as a finite over P_0 reducing to the "constant" type of C_1
over the whole parameter). The birational model of W over I x V' has
x-fiber J_0. kappa/h^0 descend from the kappa = 0, h^0 = 1 assertions
on the generic Iitaka fiber.
================================================================================
11. STEP 7: FINITE ETALE DESCENT OF THE WHOLE FIBER
================================================================================
Plain English. Now J/K = C(W) is a "form" of J_0/k: over an extension
K~ of K, we have J_K~ ..-> (J_0)_K~. The GLUE data of this isomorphism
is a Galois 1-cocycle with values in Bir(J_0). Two things must happen:
(i) The cocycle becomes CONSTANT (values in Aut(J_0), not Bir(J_0)).
(ii) It is UNRAMIFIED at all MOVING divisors on V'.
Then finite etale descent produces a field F_b over b that is the
whole birational type of the original fiber F. This gives
Var(f) <= dim T_0 = kbar(U) - d.
11.1 Structure of the birational automorphism group
---------------------------------------------------
Theorem 4.6 (Hanamura). Bir(J_0) for a non-uniruled smooth projective
J_0 (char 0, alg closed coefficient field) is a group scheme locally of
finite type. Its identity component is Aut^0(J_0), an abelian variety.
The component group is finite.
11.2 Constant cocycle
---------------------
Lemma 11.2 (constant cocycle). There exist a finite Galois extension
k'/k in closure(k), a finite Galois extension K~/K containing k',
and a birational identification
J_{K~} ..-> (J_0)_{K~}
such that the transported Galois action has form
rho_g = z_g · s_g, z_g in Aut_{k'}(k'(J_0)),
where s_g transports coefficients by g. The map
G = Gal(K~/K) -> Aut_{k'}(k'(J_0)) rtimes Gal(k'/k)
g |-> (z_g, g|_{k'})
is injective. Consequently K(J) = K~(J_0)^G.
Proof sketch. (Full: Section 07 of original.) The Galois action of
Gamma = Gal(closure(K)/K) on the birational identification
J ..-> J_0 (over K~) yields a continuous 1-cocycle
u_sigma in Bir(J_0)(closure(K)), u_{sigma tau} = u_sigma sigma(u_tau).
Step (a): image of u_sigma in the finite group pi_0(Bir(J_0)) is
locally constant (values in a finite set). Pick one representative per
component, giving a locally constant h_sigma in Bir(J_0)(closure(k))
plus a_sigma in Aut^0(J_0)(closure(K)).
Step (b): the a_sigma cocycle modulo Aut^0(J_0)(closure(k)) takes
values in the abelian variety quotient M_1 = Aut^0(K)/Aut^0(k).
Give M_1 the discrete topology. The induced action is continuous
(every algebraic point has open stabilizer for coefficient transport,
h_sigma is locally constant). The additive cocycle classes c_sigma
in H^1(G, M_1) vanish after a finite base extension k'/k, because the
abelian variety structure has no algebraic deformation over an
algebraic closure.
Step (c): after that finite extension, the cocycle becomes
rho_g = z_g · s_g, z_g in Aut_{k'}(k'(J_0)),
with z_g having image in a finite group (the residual pi_0 part).
Choosing the LARGEST component class picks a finite Galois K~/K
containing k', on which the cocycle is CONSTANT. The faithfulness of
g |-> (z_g, g|_{k'}) follows because a nontrivial g in G fixes no
element of k'(J_0) coefficient-transported by a pure s_g.
11.3 Unramifiedness at moving divisors
--------------------------------------
Lemma 11.4 (unramifiedness). The finite extension K~/K is unramified
at every moving divisor of V'.
Proof. Let P' be a moving divisor of V' (or of its finite extension).
By the zero-boundary corollary (from Section 8), the multiplicity-one
minimizer Q for P' has m_Q = 1, d_Q = 0, and
(1+r_Q)/m_Q = lambda_P'.
Suppose for contradiction that K~/K is ramified at P'. Then some
e_Q > 1 (or e = e(P'/P) > 1). Using the ramification order formula
m_{Q'} = (e_Q * m_Q) / e,
r_{Q'} = e_Q * r_Q + e_Q - 1 - (e - 1) * m_{Q'},
and the constant-model minimizer at P' (from the proof of
Proposition 8.1), one shows that the total inertia at
Q' equals base inertia, but base inertia at P' is trivial - because
P' is MOVING, hence its residue field contains the function field of
its dense image in V', which contains k'. So base inertia at P' has
image in Gal(k'/k), and that's trivial by the choice of k'.
Contradiction. Hence unramified.
(The full version in Section 07 of the original: Proposition
the regular-Galois lemma, then the uniformity lemma at composite divisors, then
Lemma 11.4.)
11.4 Finite etale descent
-------------------------
Proposition 11.6 (whole-fiber descent). Let b = algebraically closed field
containing C(T_0). There is a finitely generated field F_b over b,
containing k = b(V') under a natural embedding, such that
Frac(Omega tensor_b F_b) == Frac(Omega tensor_{C(Y)} C(F)),
where Omega = closure of C(Y). Equivalently, the geometric generic
fiber F of the original fibration f is birationally defined over b
(up to a field of finite type over b). In particular
Var(f) <= trdeg_C(b) = dim T_0 = kbar(U) - d.
Proof. (Full: Section 07, proof of Proposition 11.6, and Section 03 of the
original.)
Setting: Lambda = algebraically closed field containing C(I) (=
b(I) from Section 10). Extend the splitting algebra
B_Lambda = K~ tensor_K Lambda(V').
Step 1 (Branch locus is finite). From Lemma 11.4, every moving prime
divisor of V'_Lambda descends to a proper divisor after a finite
extension. On an affine chart, take finitely many generators of the
moving divisor's ideal; enlarge the finite extension to include them.
The descended ideal is prime by faithful flat descent, so moving
branching is avoided after that finite extension.
Step 2 (Normalization). Remove the finitely many fixed divisors where
B_Lambda is branched, and the singular locus of V'. Let W_0 be a smooth
nonempty open subset of V' over b. Then the normalization of
(W_0)_Lambda in B_Lambda is finite, normal, unramified at every
height-one prime, hence finite etale (by Zariski-Nagata purity -
smooth base, generic separability in char 0).
Step 3 (Descent of the etale cover). Base extension from b to Lambda
gives a bijection between finite etale covers of W_0 and of
(W_0)_Lambda (Landesman 2024, tame fundamental group - characteristic
zero means all covers are tame). So the finite etale cover
C_Lambda -> (W_0)_Lambda descends to a finite etale cover
C -> W_0.
Step 4 (Field of definition). The descended cover C over W_0 is a
G-torsor. Its generic fiber is a field extension F_b / b(W_0). Since
W_0 / b has function field b(W_0), the field F_b is a finite extension
of b(W_0). This F_b is the birational type of the x-fiber (via the
constant cocycle of 11.2).
Step 5 (Whole fiber F). Assemble the product description. From the
combined diagram (Section 6) and Prop 10.5, the total space W (over
b) is birationally a base extension from b of I x_b V'. Fibering
over Y gives the whole fiber F, which is birational to (base change
of some variety over b to Omega). Specifically: the fiber product
over b is fully determined by the descended F_b plus the descended
data from Section 10 (V', b, I), all of which is over b.
Combining: Var(f) <= trdeg b = dim T_0.
11.5 Conclusion of Theorem 1.1
------------------------------
Assemble:
kbar(U) = d + dim T_0 (backbone, Corollary 6.6).
Var(f) <= dim T_0 (Corollary 11.6 above).
kbar(U) >= d + kbar(V) (Bound A, Lemma 5.2).
Consequently
kbar(U) >= d + dim T_0 >= d + Var(f),
kbar(U) >= d + kbar(V).
Taking max of the two right-hand sides:
kbar(U) >= kappa(F) + max{ kbar(V), Var(f) }.
This is Theorem 1.1 (logarithmic Iitaka-Viehweg inequality).
COROLLARY 11.7 (Projective C+ inequality, Corollary 1.2). If U, V are
smooth projective and kappa(V) >= 0, then
kappa(U) >= kappa(F) + max{ kappa(V), Var(f) }.
Proof. Take empty compactification boundaries in Theorem 1.1.
================================================================================
12. APPLICATION: SMOOTH FAMILIES AND CAMPANA-SPECIAL BASES
================================================================================
Plain English. A base is "Campana-special" when it has no positive
differential-form directions at any degree. Abelian varieties, K3
surfaces, rational varieties, and products thereof are Campana-special.
Over such bases, our main theorem reduces to a bite-sized statement:
smooth families are BIRATIONALLY ISOTRIVIAL. This recovers Cao-Paun
(abelian base) and Hacon-Popa-Schnell (max Albanese base), which were
the only general (non-good-model) cases previously known.
12.1 Campana-special bases
--------------------------
Let V be a smooth quasi-projective variety over C. Following Taji
[TajiGoodModels], call V CAMPANA-SPECIAL if, for any smooth projective
compactification (Y, D) with reduced SNC boundary, every invertible
subsheaf L of Omega_Y^p(log D) satisfies
kappa(Y, L) < p for every p = 1, 2, ..., dim V.
Intuition. This says the "positive curvature" behaviour of V is
maximally spread out at every degree; no subsheaf of the log cotangent
bundle survives Iitaka growth.
Examples.
(a) A point: p = 0 only, trivially Campana-special.
(b) An abelian variety: its log cotangent bundle is trivial, so
every invertible subsheaf is kappa = 0 < p for p >= 1.
(c) A K3 surface: no pluri-log-cotangent-surviving subsheaves at
degree 1 or 2 (Campana).
(d) A rationally connected variety: Omega_Y^p has kappa = -infinity
on any rational curve, so subsheaves die.
(e) Any hyperkähler variety.
(f) A product A x W where A is abelian and W is rationally connected.
Fact (from Taji's "Good models" survey). Campana-special bases satisfy
kbar(V) = 0.
12.2 Isotriviality for smooth families
--------------------------------------
COROLLARY 12.1 (Smooth isotriviality). Let f: X -> Y be a smooth
morphism between smooth projective complex varieties (all fibers smooth,
no boundary divisor). If Y is Campana-special, then the family is
birationally isotrivial:
Var(f) = 0.
After a finite etale base change Y' -> Y, there is a birational
isomorphism
X x_Y Y' ~ Y' x F,
for the (birational class of the) fiber F.
Proof. First, Campana-special forces kbar(Y) = kappa(Y) = 0 (no p = 0
options). Apply the main theorem (Corollary 11.7 = projective version):
kappa(X) >= kappa(F) + max{ kappa(Y), Var(f) }
= kappa(F) + Var(f).
Upper addition. For a projective morphism Y with connected fibers and
constant kappa of the fiber, kappa(X) <= kappa(F) + kappa(Y) =
kappa(F) + 0 = kappa(F). This is a standard weak-positivity theorem
(Iitaka, Viehweg, or the easy direction of the C^+ Iitaka-Viehweg
inequality).
Combining: Var(f) = 0. One then invokes the equivalence
"Var(f) = 0 iff isotrivial after finite etale base change" (this is
our definition of birational variation + the standard descent of
isotriviality through finite etale covers of Y).
12.3 Recovering known cases
---------------------------
Cao-Paun [CaoPaunAbelian], 2017. Let Y be an abelian variety. Then Y is
Campana-special. Corollary 12.1 says kappa(X) = kappa(F), i.e.,
kappa(X) = kappa(F) + 0. This is exactly Cao-Paun's theorem.
Hacon-Popa-Schnell [HaconPopaSchnellAbelian], 2018. Let Y be a
compact Kähler manifold of maximal Albanese dimension (e.g., projective
with maximal Albanese). This generalizes the abelian case; algebraically
the abelian case is enough for the corollary. Corollary 12.1 covers this
setting when Y is projective.
================================================================================
13. A NUMERICAL VARIANT OF THE SAME ARGUMENT (BRIEFLY)
================================================================================
Plain English. The original paper contains a second, purely-algebraic
proof of Theorem 1.1 that does not use Hodge theory. It replaces the
Hodge line M by a "numerical" version defined combinatorially from the
divisor coefficients t_P, lambda_P, B_P, and replaces the Atiyah
algebra lift v by a bound on the relative canonical bundle via Fujino's
log weak positivity. The output is a slightly weaker theorem (bounding
kbar(U) - d by a "numerical liftable" variation invariant), but it
holds over any algebraically closed field of characteristic zero, not
just over C, and it establishes the main theorem without invoking the
Cao-Guenancia-Paun flatness theorem.
For v2 readability we do not record the full argument. The main steps
correspond to Sections 08 (numerical-extension) and 05 (model-pair
variation) of the original. A reader interested in the numerical
version should consult:
- Section 05 of original: the reduction proposition [original Section 09] and its proof
via model-pair variation (Kovács-Patakfalvi Thm 9.9).
- Section 08 of original: the full descent using the numerical
version + Fujino's weak positivity.
The numerical variant has NO effect on the main theorem's correctness
or statement; it is a cross-check.
================================================================================
14. CONCLUDING REMARKS
================================================================================
14.1 What is proved
-------------------
The main theorem (Theorem 1.1) says, for a smooth projective fibration
between smooth complex quasi-projective varieties,
kbar(U) >= kappa(F) + max{ kbar(V), Var(f) }.
Equivalently (using the projective Corollary 11.7 for smooth projective
varieties Y):
kappa(X) >= kappa(F) + max{ kappa(Y), Var(f) }.
Both versions assert that Kodaira dimension can be decomposed as
"fiber complexity + (max of base complexity and fiber variation)",
inside the outer variety.
14.2 What it settles and generalizes
------------------------------------
- Popa's logarithmic variation conjecture (Conj. 3.8 in
[PopaConjectures]) - fully settled, all dimensions, no assumption
of good minimal model on fiber.
- Kollár's subadditivity for fibers of general type
[Kollar87] - special case.
- Kawamata's Kodaira dimension of algebraic fiber spaces over a
smooth curve or with fibers of general type [KawamataKodaira] -
special case.
- Viehweg's weak positivity framework [ViehwegAdditivity, Viehweg83b]
and Viehweg-Zuo base-space rigidity [ViehwegZuo02] - recovered.
- Cao-Paun for abelian base [CaoPaunAbelian] - special case (12.3).
- Hacon-Popa-Schnell for base of maximal Albanese dimension
[HaconPopaSchnellAbelian] - special case.
- Hashizume's log abundant fiber version - recovered by the log
statement with suitable boundary.
14.3 Novel ingredients
----------------------
(a) The parameter space T_0 of image cycles of g: W_z -> Y, sitting
inside Y (C(T_0) subset C(Y)) with dim T_0 = kbar(U) - d. This is
both the space that controls the excess of kbar(U) over d and the
field over which the whole fiber F descends (Corollary 11.6).
(b) The rank-one root cover of the fiber J (Lemma 7.1) reduces the
Hodge-theoretic argument to a rank-one line, making the metric
calculations (Propositions 8.1, 8.2) tractable divisor by divisor.
(c) The use of a MULTIPLICITY-ONE minimizer at moving divisors on V'
(the zero-boundary corollary, Lemma 11.4) to restrict the Galois
cocycle to be unramified at the moving part, which enables finite
etale descent via Landesman's tame pi_1 invariance.
(d) Recovery of the "Iitaka-base normalization" V' via marked canonical
models (Section 10), which controls the trdeg of the descent field.
14.4 Open questions
-------------------
(i) Sharp bounds on dim T_0 for special classes of f (e.g., low fiber
dimension, positive kappa(F)). The current argument gives
dim T_0 = kbar(U) - kappa(F) but no direct geometric information
about T_0 beyond its construction.
(ii) A "pointwise" version of Theorem 1.1, controlling kappa of the
restriction of f to each subvariety of V, is open.
(iii) The logarithmic version in positive characteristic is wide open;
both the Hodge-theoretic and pi_1 methods used here fail.
(iv) The statement for compact Kähler target manifolds in Fujiki class
C (for f) is standard via birational models, but the Hodge lines
are more subtle; not needed for the main theorem.
================================================================================
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