PROJECTIVE HODGE LINES AND ORDINARY IITAKA SUBADDITIVITY
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ABSTRACT
We prove the ordinary Iitaka subadditivity conjecture for surjective
projective morphisms with connected fibers between smooth connected
projective varieties over algebraically closed fields of characteristic
zero. If F is the geometric generic fiber of f : X -> Z, then
kappa(X) >= kappa(F) + kappa(Z).
0. THE THEOREM, IN PLAIN WORDS
Fix once and for all an algebraically closed field k of characteristic
zero (for example the complex numbers). Let
f : X -> Z
be a surjective morphism of smooth connected projective varieties over
k with connected fibers. Think of X as a family of varieties
F_z = f^{-1}(z) parametrized by the base Z. The "geometric generic
fiber" F is the fiber over a point of Z that is as general as possible.
The theorem is:
kappa(X) >= kappa(F) + kappa(Z).
Here kappa is the Kodaira dimension. Roughly, kappa(V) measures how
fast the space of pluricanonical forms on V grows with the degree, and
it lies in {-infinity, 0, 1, ..., dim V}. The inequality says the
complexity of the total family is at least the complexity of a fibre
plus the complexity of the base. This is Iitaka's subadditivity
conjecture in the "ordinary" (no boundary divisor) case, and we prove
it in full below.
Three big inputs drive the proof:
(A) a positivity theorem for a single line of a variation of Hodge
structure (the "highest Hodge line");
(B) the canonical bundle formula, which converts the geometry of f
into a pair (Y, B) and a nef divisor M on a smaller base;
(C) log-general-type-fiber subadditivity (Kovacs-Patakfalvi,
Hashizume) together with Fujino's weak positivity.
The genuinely new content is (A): proving that the Hodge line M is
positive in the precise sense needed. Everything else assembles (A)
with (B) and (C). The appendices (Sections 10 and 11) give two other
proofs of (A); the main text proves (A) in Sections 3-7.
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1. BACKGROUND
This section states, in words, everything the proof uses. The two
facts marked (imported) are deep theorems from the literature; we
quote them as tools and prove nothing about them here. Everything
else is either elementary or is proved in this document.
1.1 KODAIRA DIMENSION
Every smooth variety V of dimension n has a canonical bundle K_V,
the line bundle of holomorphic n-forms. Its powers m K_V (m a positive
integer) are the pluricanonical bundles, and their global sections are
the pluricanonical forms, H^0(V, m K_V).
If every H^0(V, m K_V) is zero, define kappa(V) = -infinity (this
happens for example on the projective line). Otherwise, for each m
with a nonzero space, the forms define a rational map
V --> P( H^0(V, m K_V) )
to projective space, and its image has some dimension. Let kappa(V)
be the largest dimension of such an image, over all m. It is an
integer between 0 and dim V. Equivalently (and this equivalence is
the definition used in proofs), H^0(V, m K_V) grows like m^kappa(V)
once m is large.
A first concrete picture comes from smooth projective curves. There
kappa(C) = -infinity for the projective line, kappa(C) = 0 for an
elliptic curve, and kappa(C) = 1 for every curve of genus at least two.
Two features matter here:
- The conventions (-infinity) + a = -infinity and kappa(point) = 0.
- Kodaira dimension is a birational invariant and is unchanged by
finite surjective pullback. So we may freely replace a variety by
a birational model when computing kappa.
A useful fact we shall use: if V is a variety with kappa(V) = 0, then
the space of top-degree pluricanonical forms is at most one
dimensional in each degree. Two independent forms of the same degree
would give a nonconstant pluricanonical map, forcing kappa(V) >= 1.
1.2 VARIATIONS OF HODGE STRUCTURE AND PERIOD MAPS
Take a smooth family of smooth projective varieties over a base U.
The cohomology of the fibers, all in the same degree, forms a local
system: you can transport classes by walking along a path in the base,
and the transport depends only on the path up to a permutation of the
fiber (the monodromy). This local system carries much more, and the
extra structure is what "Hodge theory" is about.
A pure variation of Hodge structure of weight w on U consists of:
(1) a local system V of finite-dimensional real or complex vector
spaces, equivalently a flat connection;
(2) a decreasing filtration F^p of the associated holomorphic
vector bundle by subbundles, the Hodge filtration, such that
at every point the graded pieces recapture the Hodge
decomposition into types (a, b) with a + b = w;
(3) a flat bilinear form Q making each fiber a polarized Hodge
structure: Q is symmetric (or alternating) and the Hodge
components are all Q-orthogonal except for shifts by w.
"Integral" means the local system underneath comes from a local
system of free Z-modules (a lattice). "Rational" means the
polarization is rational. Griffiths transversality says the flat
connection sends F^p into F^{p-1} hat O. This is the horizontal
character of Hodge theory: the filtration may only move "downwards"
by one degree.
The filtration is bookkeeping for the finer decomposition. At every
point the fibre V breaks into a direct sum of Hodge pieces V^{a,b}
with a + b = w, and the subbundle F^p is the sum of those pieces with
a >= p. In particular F^w is H^{w,0}, the space of holomorphic
w-forms, and F^p = 0 for p > w.
The period domain D is the set of all Hodge filtrations of the type
fixed by (2); it is a complex manifold on which the orthogonal group
of Q acts. On the universal cover of U, a flat trivialization of V
identifies the Hodge filtration with an honest holomorphic map
U~ --> D
called the period map. Monodromy makes this equivariant for the
action of the image of pi_1(U); in practice we use the quotient of D
by an arithmetic group.
The two period maps we care about in this paper are:
- the "full" period map, remembering every step F^p;
- the "highest-line" period map, remembering only the top line.
The precise definitions are in 1.3. Everything else in the Hodge
part of the proof uses only these two maps and the extension theorems
of Deligne, Schmid, Cattani-Kaplan-Schmid, quoted in 1.4.
1.3 THE HIGHEST HODGE LINE AND ITS EXTENSION
Most of this paper studies one small piece of a variation. Let V be a
pure variation of weight w on a smooth variety U. Among the Hodge
subbundles F^p (they decrease as p grows), let m be the largest p for
which F^p is not zero. Suppose this top piece F^m is a line bundle
(rank one). We call it the highest Hodge line, or just the Hodge line,
and write J for its first Chern class once it has been extended.
A concrete case to keep in mind: a family of K3 surfaces. On H^2 of
each fibre there is a weight-2 variation, and its top piece F^2 =
H^{2,0} is one-dimensional, so this variation has a highest Hodge
line. A family of elliptic curves does the same one degree lower,
where F^1 = H^{1,0} is the line.
On a simply connected patch, choose a flat frame for V. Then the line
F^m becomes a holomorphic map
ell : patch -> P(V_ref)
into the projective space of a fixed reference fiber, recording "the
line spanned by a top form at this point". This is the line period
map. Its generic differential rank r (the dimension of its image) is
intrinsic: changing the flat frame conjugates ell by a constant linear
map.
Now compactify: let T be a smooth projective variety containing U whose
complement is a reduced divisor D with simple normal crossings (the
"boundary"). The variation does not extend across D, but its Hodge
line has a canonical extension, the Deligne-Schmid extended line. Two
facts we use constantly:
(i) the extended line bundle (with its natural rational structure)
is nef, and is big exactly when ell is generically immersive,
i.e. when the Kodaira-Spencer map is injective at a general
point;
(ii) this extension is functorial: pulling back along a map of
smooth log pairs (interior to interior) pulls back the extended
line, and it commutes with direct sums and tensor powers.
When the local monodromy is only quasi-unipotent (a finite part times
a unipotent part), one uses the rational parabolic extension: pass to a
finite cover making monodromy unipotent, take the Schmid extension
there, and descend a suitable tensor power. This keeps the finite
(root of unity) weights and is still nef and functorial. Throughout,
"the extension of the Hodge line" means this object in
Pic(T) tensor Q.
Numerical dimension. For a nef rational divisor N on a smooth
projective t-fold and an ample H, set
nu(N) = max{ a in {0,...,t} : c1(N)^a . H^{t-a} > 0 }.
This does not depend on H. For the highest Hodge line, the key
equality (proved in Section 5) is
nu(J) = r = generic rank of the line period map ell.
The whole difficulty of the paper is that r can be smaller than the
dimension of the full period image; boundary divisors force r to drop
strictly, and that strict drop is what removes the boundary.
1.4 THE CANONICAL BUNDLE FORMULA
The second tool converts a fibration into a divisor on the base. Let
g : X -> Y be a surjective morphism of smooth projective varieties with
connected fibers, whose geometric generic fiber has Kodaira dimension
zero. The canonical bundle formula (Fujino-Mori, with Ambro's moduli
theory) says that after birational changes of X and Y, keeping them
smooth and the boundary simple normal crossing, there are rational
divisors B, M, R with
K_X = g*(K_Y + B + M) + R (equality in Pic tensor Q),
where
- B >= 0 is effective with snc support and (Y,B) klt (Kawamata log
terminal: all discrepancies > -1);
- M is nef (the moduli part, a b-divisor descended to Y);
- R is the residual: g_* O_X(floor(k R^+)) = O_Y for all k>0, the
negative part R^- pushes to codimension >= 2 on Y, and R^- is
exceptional over the original smooth model.
The pair (Y,B) plus the nef M is the "log canonical model" of the
fibration: B records the singular fibers, M records how the fiber
moduli vary.
Why M is a Hodge line. The generic fiber has kappa = 0, so it has a
least positive plurigenus: there is a least b>0 with a nonzero section
of b K_fiber, unique up to scalar in every degree. Adjoin a b-th root
of that section to get the canonical root cover; it has geometric
genus one and a unique top form. As the fiber varies this top form is
the highest Hodge line of a rational polarizable variation (the b-th
root of the canonical eigensheaf). The formula identifies M, up to a
positive rational multiple, with the extension of that Hodge line on a
suitable alteration. This is the bridge: M inherits the "highest
Hodge line" positivity from Section 1.3.
1.5 WEAK POSITIVITY AND LOG GENERAL TYPE
Two imported positivity results finish the argument.
(a) Log-general-type fibers (Kovacs-Patakfalvi; Hashizume). Let
u : V -> T be a fibration with connected fibers, V normal projective,
T smooth projective, Delta >= 0 with (V,Delta) log canonical. If the
log canonical divisor of the geometric generic fiber G is big and
kappa(T) >= 0, then
kappa(V, K_V + Delta) >= dim G + kappa(T).
(b) Fujino's twisted weak positivity. For u : V -> T as above with
(V,Delta) an effective log canonical pair and k(K_V + Delta) Cartier,
the direct image sheaf
u_* O_V( n k (K_{V/T} + Delta) )
is weakly positive for every n>0. "Weakly positive" means: for every
ample Cartier A and every alpha>0 there is beta>0 such that
(Sym^{alpha beta} E)^{**} tensor O_T(beta A) is generated by global
sections at the generic point.
Part (b) turns a big divisor on the base into an effective divisor on
the total space; part (a) then supplies the growth of sections on a
general fiber. We use nothing else from the literature.
2. THE SHAPE OF THE PROOF
Here is the whole strategy, before any details. We want
kappa(X) >= kappa(F) + kappa(Z).
Step 0. If kappa(F) = -infinity or kappa(Z) = -infinity the left side
is -infinity by convention and there is nothing to prove. So assume
kappa(F) = d >= 0 and kappa(Z) >= 0.
Step 1 (Section 8). Run the relative Iitaka fibration of f. This
replaces the fibre F by its own Iitaka fibration, absorbing the
kappa(F)-dimensional part into a new base Y; the remaining fibres of
g: X~ -> Y are then Kodaira-dimension-zero, a far simpler case.
Concretely, it produces a birational model of X sitting over a new
variety Y:
X~ --g--> Y --q--> Z, dim(Y/Z) = d,
with g having Kodaira-dimension-zero geometric generic fiber, and with
kappa(X) >= kappa(Y, K_Y + B + M),
and K_Y + B + M big on the generic fiber of q.
Step 2 (Sections 3-7). Adjoint positivity for the Hodge line: show
that M is pulled back from a smaller base, M = p*L with L nef, and
that K_S + j L is big on S for all large j (S the target of p).
This is the technical heart and the only genuinely new theorem.
Step 3 (Section 9). With M = p*L in hand, combine the two fibrations
q : Y -> Z and p : Y -> S. The fiber of p has nonnegative log
Kodaira dimension. Weak positivity (1.5b) makes K_{Y/S} + B + p*P
effective for a big P; take P = K_S + jL - H. Interpolate this
effective divisor with the klt class K_Y + B + Gamma. Apply
log-general-type subadditivity (1.5a) to get the bound d + kappa(Z).
Step 4 (Section 9). Scalar extension moves the result from C to any
algebraically closed field of characteristic zero.
The appendices (Sections 10, 11) give independent proofs of Step 2,
one through an integral tensor replacement and transverse loops, one
through Chern intersection theory and Thom forms. The main text's
proof of Step 2 occupies Sections 3-7.
3. ADJOINT POSITIVITY FOR THE HIGHEST HODGE LINE
We now state precisely the theorem the main text proves, then sketch
why it is hard.
Theorem 3.1 (Adjoint positivity). Let Y be a smooth connected
projective complex variety, U a dense Zariski open subset, and V a
rationally polarized integral pure variation of Hodge structure on U
whose entire highest nonzero Hodge filtration step is a line. Let M
be a rational line bundle on Y. Assume that, after a smooth projective
birational modification of Y if necessary, there are a smooth
projective alteration tau : Yhat -> Y and a rational number c > 0
such that the pulled-back variation has unipotent monodromy along a
simple-normal-crossing boundary and
tau*M ~Q c J_Yhat,
where J_Yhat is its extended highest line. Then, after further smooth
projective birational modification, there are a surjective morphism
p : Y -> S with connected fibers onto a smooth projective variety S and
a nef rational line L on S such that
M ~Q p*L, and K_S + j L is big for all large j.
If S is a point, the conclusion is M ~Q 0.
The rest of the main text is its proof. Two features of the layout
matter.
Two period maps. The full period map of V records every step F^p; its
image, the full period image, is an algebraic variety B and gives us a
projective base on which an adjoint divisor can be big (Brunebarbe-
Cadorel: a generically immersive full period map forces log general
type). The line period map ell records only the top line F^m; its
numerical dimension nu(L) is what subadjunction later measures. These
two ranks can differ, and the proof must never identify them: it uses
the full period image to get bigness and the line rank to remove the
boundary.
Two kinds of boundary divisor. Descending to a projective model
introduces (1) boundary components with nonzero monodromy logarithm,
and (2) divisors of finite ramification of the finite quotient of the
full period image. Ordinary adjoint positivity needs BOTH removed from
an adjoint class. In both cases the engine is a STRICT drop of the
highest-line numerical dimension on the divisor, which Section 7 turns
into a section-growth computation.
Proof outline of Theorem 3.1.
(a) Replace V by its rationally minimal subvariation (Lemma 4.1),
which contains the whole top line and is unchanged on the
extension. This makes monodromy faithful on the moving top
line, so a symmetry acting trivially on that line is caught by
Lemma 4.2.
(b) Section 5: prove nu(J) = rank of ell, identify the boundary line
with a graded piece attached to the monodromy logarithm, and
prove the local rank principle (Lemma 5.4).
(c) Section 6: use (b) to force nu(J|_E) < nu(J) on every boundary
component E with nonzero monodromy, and on every divisor fixed
by a nontrivial finite inertia element. Both drops are proved
inside Theorem 6.1.
(d) Section 7: build the actual period quotient, apply the volume
lemma (Lemma 7.1) to subtract the boundary from an adjoint
class, and read off M ~ p*L with K_S + j L big.
Sections 10 and 11 give two other proofs of (b)-(c), independent of the
main text.
4. RATIONAL MINIMALITY AND FIXED PERIODS
Our first tool recognizes when a symmetry that fixes all highest lines
must fix the full periods. We first discard rational subvariations
that do not contribute to the top line. Rationality is essential: the
invariant spaces of rational normal monodromy subgroups will then be
Hodge substructures.
Let U be a smooth connected complex algebraic variety, and let
V = (V_Z, F^., Q) be a polarized integral pure variation of Hodge
structure of weight w. "Integral" means the rational local system
carries a monodromy-invariant lattice; Q is a rational, not necessarily
integral, flat polarization. Let m be the largest index with
F^m V_O nonzero, and assume throughout that rank F^m = 1. Call it the
Hodge line. On a simply connected flat chart the inclusion defines
ell : U_loc -> P(V_C,u),
and changing the flat chart conjugates ell by a constant linear map,
so its differential rank is intrinsic.
We call a lifted point Hodge-generic if it lies outside every proper
Hodge locus of a rational flat tensor built from V and its dual. These
are countably many proper closed analytic loci on the universal cover,
so their complement is dense (Baire).
Lemma 4.1 (Rationally minimal variation). There is a unique smallest
rational subvariation V^min of V whose Hodge filtration contains the
top line F^m. At a Hodge-generic point, its fibre is the smallest
rational Hodge substructure containing the top line, and that fibre is
simple. V^min carries an induced polarization and integral lattice,
and on a smooth compactification with unipotent boundary monodromy its
extended top line agrees with that of V.
Proof. Intersect all rational Hodge substructures containing the top
line. Finitely many already minimize the dimension, giving the unique
smallest one W. Semisimplicity shows W is simple: a decomposition into
simple summands has exactly one summand with nonzero top piece (that
piece has dimension one), and that summand already contains the line.
For the generic-fibre justification, trivialize the rational local
system on the universal cover. For each rational endomorphism e the
condition that e preserve every F^p is closed analytic; choose a lifted
point outside all proper such loci. A rational Hodge projector onto W
at that point preserves the filtration everywhere on the cover, acts as
the identity on the top line (an idempotent on a line bundle equal to
the identity at one point), and its constant image is a subvariation
containing the line. Uniqueness and the deck action imply monodromy
invariance, so it descends. Restriction of Q to a Hodge substructure is
nondegenerate and the lattice is V^min intersect V_Z; the polarization
gives a complement with zero top piece, and canonical extensions
preserve the direct sum. This is the pure case of Bakker-Filipazzi-
Mauri-Tsimerman's transcendental-part construction. QED
A variation is minimal for its top line if its generic fibre is the
smallest rational Hodge substructure containing that line. Lemma 4.1
supplies the replacement without changing the line. We use the
Mumford-Tate group MT(V): the smallest rational algebraic group through
which the Hodge homomorphism of a rational Hodge structure factors.
Lemma 4.2 (Normal subgroups and the top line). Let V be a rationally
polarized integral pure variation minimal for its top line. In a flat
trivialization on the universal cover let G be the connected algebraic
monodromy group and P = MT(V). If H is a connected normal algebraic
subgroup of G defined over Q, and V^H_C contains the top line at a
Hodge-generic lifted point, then H is trivial.
Proof. The monodromy and fixed-part theorems give that G is semisimple
and normal in the connected reductive P (Andre). Over an algebraic
closure a connected normal subgroup of a semisimple group is a product
of almost simple factors; conjugation by P preserves G and hence acts
trivially on this finite set of factors, so P preserves H. Thus V^H
is a rational P-subrepresentation, a rational Hodge substructure at a
Hodge-generic point, and it contains the top line. Minimality gives
V^H = V. Since G sits faithfully in GL(V), H = 1. QED
Lemma 4.3 (A flat automorphism over a fixed point). Let V be minimal
for its top line on a smooth connected U. Suppose an automorphism b of
U fixes a point e and is covered by a flat isomorphism of variations
Phi_y : V_y -> V_{b(y)}. Use the fibre V_e as reference and put
a = Phi_e. If, on a nonempty open subset of the universal cover, all
top lines lie in a single eigenspace of a, then a preserves the full
Hodge filtration at every lifted point, and locally at e the full
lifted period maps at y and b(y) agree.
Proof. Write rho for monodromy based at e. Flatness and b(e)=e give
a rho(gamma) a^{-1} = rho(b_* gamma), so a normalizes G. Let lambda be
the eigenvalue. The condition that the top line lies in ker(a - lambda)
is a holomorphic linear condition on the universal cover, so it holds
throughout. At a Hodge-generic lifted point let W be the span of the
monodromy orbit of its top line; W is G-invariant, and a acts on W as
lambda times the identity. Every commutator [a,g], g in G, acts
trivially on W, as do its G-conjugates; their normal algebraic closure
H in G is connected, defined over Q, and contains the top line in V^H.
Lemma 4.2 gives H = 1, so a centralizes G.
Pass to the connected finite cover U_1 -> U cut out by the inverse
image of G in monodromy. Since a centralizes monodromy on U_1, it is a
global flat section of End(V|U_1), a polarizable variation of weight
zero. At e_1 (over e) this section has Hodge type (0,0) because Phi_e
is a Hodge isomorphism. U_1 is Zariski open in a compact complex
variety, so the fixed-part theorem (Schmid) applies: the flat section
has type (0,0) everywhere, hence a preserves the full filtration.
Finally, lifting b to the cover fixing a lift of e, flatness gives
F^.(b y~) = a F^.(y~), and the previous sentence makes the right side
equal F^.(y~). QED
5. NUMERICAL RANK AND TANGENTIAL LIMITS
We compare a boundary restriction of the extended top line with the
limiting projective line map. Hodge theory identifies the limiting
graded line; the analytic rank principle then decides what happens if
its tangential rank fails to drop.
Setup. U and V as in Section 4. Take a smooth projective
compactification T of U with reduced simple-normal-crossing boundary D,
and suppose all local monodromies along D are unipotent. Write Vbar
for the Deligne extension and J = F^m Vbar for the Schmid extension of
the Hodge line. For a nef line J on a smooth projective t-fold and an
ample H,
nu(J) = max{ a in {0,...,t} : c1(J)^a H^{t-a} > 0 }.
A line on a point has nu = 0.
Theorem 5.1 (Extension and positivity inputs; imported).
(i) The extended Hodge filtration is by subbundles of Vbar. Under a
morphism g : (T',D') -> (T,D) of smooth projective snc pairs with
g(T' minus D') subset U, the extended top line of g*V is g*J.
Extensions commute with direct sums and tensor products. If some
boundary monodromy logarithms vanish, the variation extends purely
across those components, including their crossings.
(ii) J is nef. It is big exactly when the top Kodaira-Spencer map
T_U -> Hom(F^m, F^{m-1}/F^m) is generically injective, equivalently
when ell is generically immersive.
The first assertions are Schmid's unipotent extension and nilpotent
orbit theorems (with Cattani-Kaplan-Schmid at crossings); the second is
Bakker-Brunebarbe-Tsimerman. Griffiths transversality identifies the
differential of ell with the indicated Kodaira-Spencer map.
Proposition 5.2 (Numerical rank). If r = generic rank of d ell, then
nu(J) = r.
Proof. Put t = dim T and fix an ample H. For 1 <= a <= t choose a
general smooth complete intersection C_a of dimension a from a large
multiple of H; it meets D transversely. At general points of C_a
intersect U, the restriction of d ell to T_{C_a} has rank min(a,r):
tangent conditions of first order are prescribed by a sufficiently
ample system, so Bertini gives this for general C_a. By Theorem 5.1,
J|_{C_a} is big exactly when a <= r; since it is nef this is equivalent
to (J|_{C_a})^a > 0, i.e. c1(J)^a H^{t-a} > 0. The case r = 0 and the
case dim 0 give the formula too. QED
The limiting line. Let E be a component of D, E^o = E minus the other
components of D, and N its monodromy logarithm. The weight filtration
W(N) is centred at w. The limiting filtration induces polarized pure
variations on the graded pieces over E^o, and if k is the unique weight
carrying the top line then J|_E is the Schmid extension of the top line
of Gr^{W(N)}_k.
Proposition 5.3 (Boundary line). On a transverse coordinate chart
(u,v) about a general point of E = (u=0), write a generator of the
lifted Hodge line as
e^{zN} a(u,v), z = log(u) / (2 pi i),
with a holomorphic across u=0 and a(0,v) != 0. After shrinking there
is an integer l >= 0 with
N^{l+1} a(0,v) = 0, N^l a(0,v) != 0,
the line lies in weight w + l, and
nu(J|_E) <= rank_{gen} d( v -> [N^l a(0,v)] ).
The map on the right is taken in the chosen flat coordinates. If N = 0
take l = 0; the variation then extends purely near the general point
of E.
Proof. Put a_0 = a(0,v). In the Deligne bigrading of the limiting
mixed Hodge structure F^m is a single one-dimensional summand I^{m,q}:
all first indices are at most m and dim F^m = 1. Set l = m + q - w.
The primitive decomposition on weight w+l writes each summand as a sum
of N^j P_{w+l+2j}, j >= max(0,-l), with P the primitive part. A term
with j>0 contributing to type (m,q) would come from type (m+j,q+j),
whose first index exceeds m; impossible. Hence j = 0, l >= 0, and the
top line is primitive. Hard Lefschetz for W(N) makes N^l nonzero and
N^{l+1} zero on the associated graded; because N has Deligne bidegree
(-1,-1), the same holds for a_0 itself. The integer l is constant on a
small stratum. Let J_E^gr be the extended top line of the pure
variation on Gr^{W(N)}_{w+l}; the boundary extension input identifies it
with J|_E, and projecting N^l a_0 to Gr^{W(N)}_{w-l} is (up to a fixed
linear map) this line. So the graded line has projective rank at most
v -> [N^l a_0(v)]. Apply Proposition 5.2 to the graded variation on
E^o and its compactification E. QED
Local rank principle. Lemma 5.4. Let B in C^n be a polydisc and
f : S_eps x B -> M a holomorphic map to a complex manifold, where
S_eps = { 0<|u|<eps, alpha < arg u < beta } is a sector of finite
opening. Suppose f(u,.) converges uniformly on compact subsets of B,
as u -> 0 through the sector, to a holomorphic f_0. Assume the full
generic differential rank of f is at most r and rank d f_0(v_0) = r at
some v_0 in B. Then, after shrinking the sector and choosing
v_0 in B' compact inside B_0 inside B,
f(S_{eps'} x B') is contained in f_0(B_0),
where the local image of f_0 is an embedded complex submanifold of
dimension r. In particular every fixed closed analytic subset that
contains this local image also contains the nearby image of f.
Proof. Work in a common chart of M, shrinking B and the sector. Cauchy
estimates give convergence of all derivatives in the B directions on
smaller polydiscs, so the (r+1)-minors of d f_0 vanish; thus f_0 has
constant rank r near v_0. The constant-rank theorem gives source
coordinates v = (x,y) in C^r x C^{n-r} and target coordinates in which
f_0(x,y) = (x,0). Write f = (F,G) with F(u,x,y) = x + E(u,x,y). On
fixed smaller polydiscs E and d_x E tend uniformly to zero, so for
small eps' the maps x -> w - E(u,x,y) send a fixed closed x-ball into
itself with Lipschitz constant < 1/2, uniformly in (u,w,y). The
contraction theorem gives a unique holomorphic solution x = chi(u,w,y)
of F(u,chi,y) = w, with chi -> w uniformly. In the new coordinates
f(u,chi,y) = (w, H(u,w,y)). The w-derivatives already have rank r, so
the rank bound forces d_u H = 0 and d_y H = 0 (any nonzero such
derivative would add an image vector with zero first r coordinates).
Since the sector and Y are connected, H depends only on w; its limit as
u -> 0 is 0, so H = 0. Shrinking proves the assertion. The case r = 0
makes f constant on the connected source and gives the same
conclusion. QED
Lemma 5.5 (Nilpotent tangential convergence). Let N be nilpotent on a
finite-dimensional complex space and a(u,v) holomorphic on
Delta x B. Suppose for some l >= 0 that N^{l+1} a(0,v) = 0 and
N^l a(0,v) != 0 throughout B. On a sector with z = log(u)/(2 pi i),
[e^{zN} a(u,v)] -> [N^l a(0,v)]
normally on small v-charts, with all tangential derivatives. Hence
Lemma 5.4 applies whenever the rank of this limit equals an upper bound
for the full rank of the line map; its nearby image then lies in P(ker N).
Proof. Choose k with N^{k+1} = 0 and write a(u,v) = a_0(v) + u q(u,v).
Set b(v) = N^l a_0(v)/l!. Direct expansion gives
z^{-l} e^{zN} a(u,v) = b(v) + sum_{j<l} (z^{j-l}/j!) N^j a_0(v)
+ u sum_{j=0}^{k} (z^{j-l}/j!) N^j q(u,v),
and for every fixed tangential derivative the remainder is
O(|z|^{-1}) + O(|u| |z|^{k-l}). On the sector |z| is comparable to
|log|u|| so both terms go to zero. Choose a functional nonzero on
b(v_0), shrink the v-chart, and divide; this gives convergence with all
tangential derivatives in a fixed affine chart. Finally N b = 0, so the
limit lies in P(ker N) and Lemma 5.4 finishes. QED
Remark 5.6. All numerical-rank statements are unchanged under
replacing J by a positive rational multiple. The local statements hold
with N = 0, when ell extends holomorphically across the divisor.
6. RANK LOSS ON THE BOUNDARY AND THE RAMIFICATION DIVISOR
The previous lemmas apply to a smooth model of the full period image.
At a neat level the variation is unipotent at the boundary; returning
to the original level also introduces ramification divisors. We prove
the strict numerical rank drop for both kinds.
Data. T smooth connected projective complex variety of positive
dimension, D_T a reduced snc divisor, U = T minus D_T. V a rationally
polarized integral pure variation on U, minimal for its top line, with
unipotent boundary monodromy. J the extended top line. Let D_N be the
union of components of D_T with nonzero monodromy logarithm, and
U_N = T minus D_N; V extends purely over U_N. Suppose a finite group H
acts on T preserving D_T and acts on V by flat Hodge isomorphisms
preserving the lattice and polarization and covering the action. Put
G_f = H / ker( H -> Aut(T) ),
the effective group on T; keep H for the action on V (its kernel on T
may act nontrivially on fibres). Fix a connected period domain D and a
neat arithmetic group Gamma' containing the monodromy, and let
phi : U_N -> Gamma' minus D be the full period map. Assume the
normalization B of the closure of its actual image is algebraic, that
T dash-> B is birational, and that phi is generically immersive; in
particular C(T) = C(B).
Theorem 6.1 (Boundary and ramification rank loss). With these data let
r = nu(J). Then r > 0. If E is either a component of D_N, or a
component of D_T fixed pointwise by a nonidentity element of G_f, then
nu(J|_E) < r.
In particular, after placing divisorial fixed loci in D_T on an
equivariant smooth model, the conclusion holds for every divisorial
ramification component of T -> T/G_f.
Proof. Proposition 5.2 identifies r with the generic rank of ell. We
first show r > 0, then do the nilpotent and finite cases.
If r = 0, the top line is constant on the connected universal cover of
U_N. Monodromy preserves it, and its connected algebraic monodromy
group G (semisimple) acts trivially on it. Lemma 4.2 with H = G gives
G = 1; after a finite etale cover the monodromy is trivial, and the
fixed-part theorem makes the whole variation constant, so its full
period map has rank zero, contradicting generic immersivity of the
positive-dimensional T. Hence r > 0.
Component with nonzero monodromy logarithm. Let E be in D_N, N != 0.
In coordinates (u,v) near a general point of E = (u=0), the lifted line
map is [e^{zN} a(u,v)]. Proposition 5.3 gives l with
N^{l+1}a(0,v) = 0, N^l a(0,v) != 0 and
nu(J|_E) <= rank (v -> [N^l a(0,v)]). Lemma 5.5 gives normal tangential
convergence, and the interior (r+1)-minors vanish, so by Cauchy the
limit has rank at most r. Suppose nu(J|_E) >= r. The limit has rank
exactly r at a general point, its image lies in P(ker N), and Lemma 5.4
forces all nearby interior lines into that same P(ker N). The
statement "N annihilates the top line" is holomorphic, so it extends
over the universal cover of U_N. The one-parameter group exp(tN) lies
in G, and every monodromy conjugate of it fixes the top line pointwise;
let G_N be the algebraic subgroup they generate. It is connected,
defined over Q, and normal in G, and V^{G_N} contains the top line at a
Hodge-generic point. Lemma 4.2 gives G_N = 1, contradicting N != 0.
Divisor fixed by finite inertia. Let b in G_f be nonidentity and fix E
pointwise. If E lies in D_N the previous case applies. Otherwise a
general e in E lies in U_N. Choose a lift h in H of b; the H-action
gives a flat isomorphism Phi_y : V_y -> V_{b(y)} on U_N. Use V_e as
reference; put a = Phi_e. If x is the full lifted period map, flatness
gives x(by) = a x(y). The operator a is a rational Hodge isometry at
e, and a^{ord(h)} = 1. Since b fixes E, every top line near E is an
eigenline of a with a constant eigenvalue lambda on a connected small
patch, so those lines lie in ker(a - lambda). Proposition 5.3 with
N = 0 bounds nu(J|_E) by the rank of this extending line map.
Suppose nu(J|_E) >= r. The rank on E is at most r because all
(r+1)-minors of the extending line map vanish by continuation, so it
equals r. Lemma 5.4, with N = 0, puts a nonempty open set of top lines
in ker(a - lambda); Lemma 4.3 then shows a preserves every full period,
so x(by) = x(y). Projecting to Gamma' minus D gives phi(by) = phi(y) on a
nonempty open set. Since B is the normalization of the actual image,
its map is birational, hence injective over a dense open subset of that
image, and b acts trivially on C(B). The chosen model has
C(T) = C(B), so b acts trivially on T, contradicting its nonidentity in
G_f.
Finally, in a finite quotient in characteristic zero, ramification at a
divisorial valuation is detected by a nontrivial inertia group acting
identically on that divisor's function field; its elements fix the
divisor pointwise, and the previous case applies. QED
Remark 6.2. The conclusion concerns the numerical dimension of the top
line, while generic immersivity is a hypothesis on the full period map.
The proof and conclusion never identify these ranks. The function-field
equality with the actual period image is used only at the last
contradiction in the finite case; it cannot be replaced by an arbitrary
finite extension of that function field.
7. THE PERIOD QUOTIENT AND ADJOINT BIGNESS
We prove Theorem 3.1. The full period image supplies an algebraic
base, and Section 6 controls the divisors introduced by compactification
and finite descent. Two elementary lemmas first show that numerical
rank drops suffice to remove divisors from an adjoint class.
Lemma 7.1 (Boundary subtraction). Let T be a smooth projective
t-fold, A a rational divisor, J a nef rational divisor with r = nu(J)
> 0, and D = sum_i d_i E_i an effective integral divisor with smooth
prime components. Suppose nu(J|_{E_i}) < r for every i. If A + D is
big, then A + j J is big for every sufficiently large integer j.
Proof. Choose an ample H_1 with A + D - H_1 effective. Let m tend to
infinity through sufficiently divisible integers. For fixed j >= 0,
asymptotic Riemann-Roch for the ample divisor H_1 + j J gives
liminf (t!/m^t) h^0( T, m(A+D+jJ) ) >= (H_1 + jJ)^t >= c j^r
for a constant c>0, using J^r H_1^{t-r} > 0. Now remove mD one prime
divisor at a time by the exact sequences
0 -> O(m(...) - E_i) -> O(m(...)) -> O_{E_i}(...) -> 0. Each quotient
is a line bundle on some E_i with at most h^0(E_i, m(H_2+jJ)|_{E_i})
sections for a single ample H_2 on T built from globally generated
divisors (choose B_0 with a globally generated multiple clearing the
denominator of B_0 - (A+D), and B_h with B_h and B_h + E_h globally
generated; take H_2 = B_0 + sum_h d_h B_h). There are m sum_i d_i
quotient terms; asymptotic Riemann-Roch on each E_i bounds their total
contribution at scale m^t/t! by t sum_i d_i (H_2+jJ)^{t-1}.E_i, which is
O(j^{r-1}) because any intersection containing at least r factors of
J|_{E_i} vanishes. Hence
liminf (t!/m^t) h^0( T, m(A+jJ) ) >= c j^r - O(j^{r-1}) > 0
for all large j, proving bigness. QED
Lemma 7.2 (Exceptional pullback). Let pi : T -> S be a surjective
generically finite morphism of smooth projective varieties, and let
E >= 0 be a rational divisor on T whose components have image of
codimension at least two in S. For a rational divisor D on S, the
divisor pi*D + E is big if and only if D is big.
Proof. Factor pi as T -> Sbar -> S, with Sbar the normalization of S
in C(T). The first map is birational and the second finite. Every
component of E is exceptional for the birational map, since a finite
map preserves dimensions. Normality gives rho_* O_T(mE) = O_Sbar in
divisible degrees (a rational function with poles only on exceptional
divisors has no codimension-one pole below), and the projection
formula identifies the sections of m(pi*D+E) with those of m tau*D.
Bigness is invariant under surjective finite pullback. QED
Proof of Theorem 3.1 (adjoint positivity). We proceed in steps.
(1) Minimal variation and the actual period image. Make the birational
modification allowed by the hypothesis. Replace V by the minimal
rational subvariation of Lemma 4.1 with its intersection lattice and
restricted polarization; the top line and tau*M are unchanged.
Choose a rational reference space with lattice and polarization, let
D be the connected period-domain component containing a lift of the
periods, and let Gamma be the arithmetic group of integral polarizing
isometries of D; it contains the monodromy. Let Gamma' be a normal
neat subgroup of finite index. The kernel of pi_1(U) -> Gamma/Gamma'
defines a connected finite Galois cover U' -> U with deck group H.
The period map of V|U' factors through B = normalization of the closure
of its actual image in Gamma' minus D; B is normal quasiprojective and its
period realization is generically immersive (Bakker-Brunebarbe-
Tsimerman). We do NOT replace B by its relative algebraic closure.
(2) The descended variation. Neatness makes the arithmetic action
free; the universal local system (D x V_Z)/Gamma' with tautological
filtration is horizontal on the smooth open of B (dominance + Griffiths
transversality), hence a polarizable integral variation there.
Gamma/Gamma' acts honestly because Gamma' is normal, so H acts on this
variation covering its algebraic action on B. Let G_f be the effective
image in Aut(B); keep H for the action on V.
(3) The quotient line. Take a G_f-equivariant smooth projective
compactification S' of a smooth dense invariant open of B with snc
boundary. All boundary monodromies are unipotent: they are
quasi-unipotent and lie in the neat group Gamma', so their root-of-unity
parts are 1. Let J' be the extended top line and L' = cJ'; it is nef
and carries the H-action. Let q : S' -> Q_0 = S'/G_f be the finite
quotient. An element of ker(H -> G_f) may act on J' fibres through a
finite-order character; a tensor power divisible by |H| kills the
fibre action of every stabilizer, so some rational Lbar on Q_0 satisfies
q*Lbar ~Q L'. It is nef. Choose a resolution s : S_0 -> Q_0 and set
L_0 = s*Lbar (nef).
(4) M is pulled back. The equivariant period map descends to a
dominant rational map Y dash-> Q_0, hence to S_0; resolving its graph by
a smooth projective birational modification gives p_0 : Y -> S_0. We
claim
M ~Q p_0* L_0. (*)
To verify, take a common smooth projective alteration W dominating the
alteration in the hypothesis and the level cover, resolve its graph to
S', and arrange snc boundaries containing the relevant inverse images.
The pulled-back variations agree on a dense open subset; unipotent
Deligne-Schmid extensions commute with such pullbacks between snc pairs
(BFMT Lemma 5.6), so the extended top line J_W is the pullback of the
top line on either model. Then, writing mu : W -> Y and t : W -> S',
mu*M ~Q c J_W ~Q t*L' ~Q mu* p_0* L_0.
Pullback by a proper generically finite map is injective on rational
line-bundle classes (pushforward, or the norm of a principal rational
function, multiplies the class by the degree). This proves (*); in
particular no boundary twist is inferred from an equality merely on the
open locus.
If dim S_0 = 0, (*) gives M ~Q 0 and we are done. Assume dim S_0 > 0.
(5) A common equivariant model. Normalize S_0 in C(S'), take a
G_f-equivariant smooth projective resolution of the resulting model and
its birational map to S', and denote the maps eta : T -> S' (birational)
and pi : T -> S_0 (generically finite, Galois function-field group
G_f), with G_f acting on T over S_0. Choose the resolution so that a
G_f-invariant snc boundary D_T contains the complement of the open
period locus and the support of the Jacobian divisor of pi. The
pulled-back variation has unipotent boundary monodromy; if J_T is its
extended top line, then
L_T := c J_T ~Q eta* L' ~Q pi* L_0.
Let D_N be the reduced sum of components of D_T with nonzero monodromy
logarithm. Over U_N = T minus D_N the variation extends as a pure
polarized variation (nilpotent-orbit extension), and the extension
preserves H-equivariance.
(6) Rank drops and boundary bigness. The full period map on U_N is
generically immersive because T is birational to B. The
log-general-type theorem for polarized variations with generically
immersive period map (Brunebarbe-Cadorel) gives
K_T + D_N is big.
We now have exactly the data of Theorem 6.1: T is birational to the
actual normalized period image, the full period map is generically
immersive, and the retained finite group H acts on its integral
polarized variation, with effective group G_f. Minimality is preserved
(the variation on T and the original have the same Hodge structures at
Hodge-generic points). So
r := nu(L_T) > 0, nu(L_T|_E) < r
for every component E of D_N and every divisorial ramification
component of pi mapping onto a divisor of S_0 (the inertia group of
C(T)/C(S_0) fixes that divisor pointwise). Positive rescaling from J_T
to L_T leaves these numerical dimensions unchanged. Using the actual
image before Stein factorization is exactly the function-field
hypothesis needed in the finite-inertia part of Theorem 6.1.
(7) Removing boundary and ramification. Write the effective Jacobian
divisor R_pi = K_T - pi* K_{S_0} = R_div + R_exc, where R_div is
supported on components mapping onto divisors of S_0 and each component
of R_exc maps to codimension >= 2. Apply Lemma 7.1 with
J = L_T, D = D_N + R_div,
A = K_T - R_div = pi* K_{S_0} + R_exc.
Every component of D satisfies the rank inequality, and
A + D = K_T + D_N is big. So
A + j L_T ~Q pi* (K_{S_0} + j L_0) + R_exc
is big for j >> 0. Lemma 7.2 removes the effective exceptional divisor
R_exc and descends bigness through pi:
K_{S_0} + j L_0 is big for every sufficiently large integer j.
(8) Connected fibers. Take the Stein factorization of p_0 and resolve
its finite normal intermediate variety to smooth projective S; resolving
the induced rational map from Y gives, by further smooth projective
birational modification, a morphism p : Y -> S with connected fibers
(its Stein factor is finite birational over the normal S, because C(S)
is algebraically closed in C(Y), hence is S itself). If sigma : S ->
S_0 is the resulting generically finite morphism, set L = sigma* L_0; it
is nef and (*) pulls back to M ~Q p*L. Since S, S_0 are smooth, the
Jacobian divisor of sigma is effective, so
K_S + j L = sigma* (K_{S_0} + j L_0) + R_sigma
is big for j >> 0. This proves Theorem 3.1. QED
8. CANONICAL BUNDLE DATA AND THE LEAST-INDEX ROOT COVER
We return to the original algebraic fibre space f : X -> Z. This
section builds the rational line to which Theorem 3.1 applies and keeps
the exact section comparison with X. All varieties here are complex
and projective (the move to characteristic zero is Section 9).
Conventions. A rational pair (Y,B) is klt if, on a log resolution, the
boundary defined by crepant pullback of K_Y + B has all coefficients
< 1, and log canonical (lc) if all coefficients <= 1. We allow
negative coefficients for a sub-pair. A b-divisor records compatible
traces on all birational models; "the moduli part descends to Y" means
every higher trace is the pullback of its trace M on Y. We keep
"exceptional over a model of the total space" apart from "image on the
base of codimension >= 2".
Theorem 8.1 (Parabolic canonical bundle formula; imported). Let
g_0 : X_0 -> Y_0 be an algebraic fibre space of smooth projective
varieties with geometric generic fibre of Kodaira dimension zero.
After birational modifications of source and base there is a
commutative square
X' --alpha--> X_0
| g | g_0
Y --beta--> Y_0
with X', Y smooth and rational divisors B, M, R such that
K_{X'} ~Q g*(K_Y + B + M) + R,
with B >= 0 and snc support, (Y,B) klt, M nef, and
g_* O_{X'}(floor(k R^+)) = O_Y (k > 0),
codim_Y g(Supp R^-) >= 2, R^- is alpha-exceptional.
M is the trace of the moduli b-divisor and can be assumed to descend to
Y. On a sufficiently high generically finite base change admitting
semistable reduction, its pullback is Q-linearly equivalent to a
positive rational multiple of the Schmid extension of the distinguished
canonical Hodge eigensheaf of the canonical root cover.
The residual and discriminant assertions, normalized by the least
nonzero pluricanonical index, are Fujino-Mori; the moduli and Hodge
identifications are Ambro. The auxiliary sub-boundary may have
negative coefficients, so no abundance of M is assumed.
Normalization of the sub-pair. On the geometric generic fibre C let b
be the least positive integer with H^0(C, bK_C) nonzero, and D_C the
divisor of its unique section up to scalar. Set R_C = D_C/b and
Delta_C = -R_C. The least positive integer making K_C + Delta_C
principal is exactly b: bK_C - D_C is principal, and conversely if
m(K_C - D_C/b) is principal then mD_C/b is an integral effective
divisor linearly equivalent to mK_C, forcing H^0(C, mK_C) nonzero and
m >= b. This identifies the least-index normalization with the
pluricanonical index used for the root cover.
The direct-image condition holds in every positive degree. A section
of O_{X'}(floor(kR^+)) over g^{-1}(V), restricted to C, has poles
bounded by floor(k D_C/b), an integral multiple of D_C; since D_C ~ bK_C
and kappa(C) = 0, every such function is constant on C. Because
g_* O_{X'} = O_Y, C(Y) is algebraically closed in C(X'), so the section
is g* psi for a rational function psi on V. On a component E over a
base prime P with residual coefficient zero, regularity gives
0 <= ord_E(g*psi) = mult_E(g*P) ord_P(psi), so psi has no
codimension-one pole; normality makes it regular. Hence
g_* O_{X'}(floor(kR^+)) = O_Y for all k > 0.
The rank-one hypothesis of the sub-pair holds even though Delta_C is
negative: kappa(C, R_C) = 0, and a section of the rounded discrepancy
sheaf pushes down to a rational function with poles bounded by a large
multiple of D_C, a one-dimensional space containing the constant, so
rank g_* O_{X'}(ceil A(X', -R)) = 1.
Lemma 8.2 (Canonical root cover). Let C be a smooth connected
projective variety with kappa(C) = 0, b > 0 least with H^0(C, bK_C)
nonzero. Choose 0 != s in bK_C and a nonzero rational canonical form
omega. Let T be the normalization of C in the field obtained by
adjoining a root xi with xi^b = s / omega^b. Then T -> C has degree b,
and every smooth projective resolution W of T satisfies kappa(W) = 0
and h^0(W, K_W) = 1 with the unique canonical line spanned by
xi h* omega, where h : W -> C is the induced map.
Proof. Put n = dim C and D_C = div(s); n = 0 is immediate. First,
every nonzero pluricanonical space of C is one-dimensional: two
independent sections in one degree give a nonconstant pluricanonical
map, hence positive Kodaira dimension. If s/omega^b = u^p for a prime
p | b, the rational (b/p)-form u omega^{b/p} has p-th power s, hence no
divisorial poles (its orders are (1/p) times the nonnegative orders of
s), so it is regular, contradicting minimality of b. The Kummer
irreducibility criterion gives a degree-b field extension; let pi : T ->
C be normalization and rho : W -> T a resolution, h = pi rho.
The rational form eta = xi h* omega satisfies eta^{tensor b} = h*s, a
regular section of bK_W; so b ord_E(eta) >= 0 for every prime E and
normality makes eta regular, giving h^0(W,K_W) >= 1.
For the upper bound, take K_C = div(omega). Outside Supp D_C the order
of s/omega^b is divisible by b, and locally the defining equation is a
b-th root of a unit, unramified (characteristic zero); hence all
ramification lies over Supp D_C. For Q over P in Supp D_C, with e =
e(Q/P) and d = ord_P(D_C) >= 1, tameness gives ramification e-1, and
e-1 <= ed = ord_Q(pi*D_C). The finite-map canonical formula at
codimension one gives K_T <= pi*(K_C + D_C). On W the same holds off
the rho-exceptional divisors, so K_W <= h*(K_C + D_C) + E for an
effective integral rho-exceptional E. Normality gives
rho_* O_W(mE) = O_T, and the projection formula gives
kappa(W, K_W) <= kappa(T, pi*(K_C + D_C)) = kappa(C, K_C + D_C).
The last equality is invariance of Iitaka dimension under finite
surjective pullback (monic characteristic polynomial of a section).
Since D_C ~ bK_C, kappa(C, K_C + D_C) = kappa(C, (b+1)K_C) = 0.
Together with the nonzero form eta this gives kappa(W) = 0 and
h^0(W,K_W) = 1. QED
Family form. Over a dense open U of Y, a rational relative
pluricanonical section builds the root cover in families; resolving and
shrinking U gives a smooth projective h_U : W_U -> U whose geometric
generic fibre has the invariants of Lemma 8.2. Constancy of Hodge
numbers gives geometric genus one on every fibre, and if n is the
dimension,
V = R^n (h_U)_* Q, rank F^n (V tensor O_U) = 1.
V has an integral lattice modulo torsion and is polarizable (relative
ample class and Lefschetz; primitive middle cohomology contains
H^{n,0}). The cyclic deck xi -> zeta xi acts on the top form by zeta,
so the distinguished canonical eigensheaf is the entire top Hodge line
of a rational polarizable variation, even when the character is not
defined over Q. We never replace the rational variation by that
character eigenspace. After a smooth projective alteration,
resolution of the boundary, and semistable reduction,
tau* M ~Q a J, a in Q_{>0},
J the Schmid extension of the top Hodge line of tau*V with unipotent
monodromy. The extension comparison is Ambro's, not an identification
only on U.
Proposition 8.3 (Canonical bundle reduction). Let f : X -> Z be an
algebraic fibre space of smooth connected projective varieties with
geometric generic fibre F of Kodaira dimension d >= 0. There exist
smooth projective varieties X~, Y, a birational morphism mu : X~ -> X,
and algebraic fibre spaces
X~ --g--> Y --q--> Z, q g = f mu,
with dim Y - dim Z = d, the geometric generic fibre of g of Kodaira
dimension zero, and data B, M, R of Theorem 8.1 with R^- exceptional
over X. Moreover M has the realization above for a polarizable
rational variation whose top Hodge piece has rank one, and with
D = K_Y + B + M,
kappa(X) >= kappa(Y, D), D|_{Y_etabar} is big,
where etabar is a geometric generic point of Z; the same bigness holds
on very general smooth fibres of q.
Proof. Over C(Z) take a divisible pluricanonical map of the generic
fibre and its Iitaka fibration, including the relative algebraic
closure of the image field; this gives image dimension d and Kodaira
dimension zero for the geometric generic fibre. Extend the rational
map to projective models over Z, resolve source and target, and take
the connected-fibre factorization. This gives X_0 -> Y_0 -> Z with the
required dimensions, and the second map has connected fibres (push O
twice). Apply Theorem 8.1 to X_0 -> Y_0; all modifications can be done
over Z, and exceptionality over X_0 implies over X. Lemma 8.2 and the
family form give the rational variation and its top line. The two
comparisons are Lemma 8.4. QED
Lemma 8.4 (Comparison of section growth). In the situation of
Proposition 8.3, kappa(X) >= kappa(Y,D) and D|_{Y_etabar} is big.
Proof. The formula gives K_{X~} + R^- ~Q g*D + R^+. Using compatible
canonical divisors, K_{X~} = mu*K_X + E_mu with E_mu effective and
mu-exceptional; E_mu + R^- is also mu-exceptional. For divisible m,
normality and the projection formula give
H^0(X~, m(K_{X~}+R^-)) = H^0(X, mK_X). Multiplying by the canonical
section of mR^+ injects H^0(Y, mD) into the left (using
g_* O_{X~} = O_Y). Hence kappa(Y,D) <= kappa(X).
Now base change to etabar. X~_etabar and Y_etabar are smooth, the
former birational to F. Restrictions of canonical divisors agree up to
linear equivalence, so
K_{X~_etabar} <=Q g_etabar*(D|_{Y_etabar}) + (R^+)|_X~_etabar,
where <=Q means Q-linearly equivalent up to an effective divisor.
Proper direct image commutes with this flat base change, so
(g_etabar)_* O(m (R^+)|_X~_etabar) = O_{Y_etabar} for divisible m, and the
projection formula gives
h^0(X~_etabar, mK) <= h^0(Y_etabar, mD|). Hence
d = kappa(F) <= kappa(Y_etabar, D|) <= dim Y_etabar = d; so D on the
geometric generic fibre is big (including dimension zero).
For the general-fibre statement fix a relatively ample H. Bigness on
the geometric generic fibre gives a nonzero section of mD - H there;
flat scalar extension gives one over C(Z); shrinking the base, it
extends and stays nonzero on fibres, so mD|_{Y_z} is ample plus
effective for general z. QED
Lemma 8.5 (Further birational modifications). Suppose Y, B, M, D
satisfy Proposition 8.3, and let nu : Y' -> Y be a smooth projective
log resolution, with K_{Y'} + B^cr = nu*(K_Y + B). Set B' = (B^cr)^+,
M' = nu*M, D' = K_{Y'} + B' + M'. Then B' is effective, (Y',B') klt,
M' is nef with the same Hodge realization after pullback,
D' = nu*D + (B^cr)^-,
with the extra divisor effective and nu-exceptional. Hence
kappa(Y',D') = kappa(Y,D) and D' is big on the geometric generic fibre
over Z.
Proof. Nonexceptional coefficients of B^cr are those of the effective
B, so its negative part is exceptional; klt coefficients are < 1, and
the positive part has snc support in [0,1), so (Y',B') is klt. The
identity is direct. Exceptional effective twists do not change section
spaces of a pullback to a normal base, so Iitaka dimension is preserved.
On the geometric generic fibre over Z the map is birational and the
identity is big-plus-effective. Nefness is preserved by pullback; the
Hodge statement follows on a common smooth alteration (Deligne-Schmid
extensions commute). QED
Remark 8.6. After Lemma 8.5, B' need not be the discriminant of the
original sub-pair. What is used later is exactly: effectiveness and
klt, the nef Hodge realization of M', and the two comparisons. Since
the big cone of Y_etabar is open, there is a rational 0 < v < 1 with
(K_Y + B + vM)|_{Y_etabar} big, and this persists on very general
fibres by the spreading argument.
9. ORDINARY SUBADDITIVITY
We assemble the main theorem. Section 8 reduced it to an effective klt
pair (Y,B) with a nef Hodge line M on the base of q : Y -> Z. Theorem
3.1 writes M = p*L for a second fibration p : Y -> S. The two bases Z
and S need not map to one another. Their combination first gives
nonvanishing on a p-fibre, then weak positivity and one interpolation
identity give the total-space sections.
Two imported positivity results, stated in the forms we use.
Theorem 9.1 (Subadditivity with log-general-type fibre). Let
u : V -> T be a surjective morphism with connected fibers, V normal
projective, T smooth projective. Let Delta >= 0 be a Q-divisor with
(V,Delta) log canonical. If the log canonical divisor of the geometric
generic fibre G is big and kappa(T) >= 0, then
kappa(V, K_V + Delta) >= dim G + kappa(T).
This is the log-general-type-fibre theorem of Kovacs-Patakfalvi (normal
total-space form due to Hashizume). It does not need abundance of K_T.
Theorem 9.2 (Twisted weak positivity). Let u : V -> T be as above
with (V,Delta) effective log canonical and k(K_V + Delta) Cartier. Then
u_* O_V( n k (K_{V/T} + Delta) ) is weakly positive for every n > 0:
for every ample Cartier A and every alpha > 0 there is beta > 0 such
that (Sym^{alpha beta} E)^{**} tensor O_T(beta A) is generated by global
sections at the generic point.
Lemma 9.3 (Period-fibre nonvanishing). Let p : Y -> S and q : Y -> Z be
algebraic fibre spaces of smooth projective varieties, (Y,B) effective
klt, M ~Q p*L for a Q-divisor L on S. Suppose kappa(Z) >= 0 and
D = K_Y + B + M is big on the geometric generic fibre of q. Then the
geometric generic fibre of p has nonnegative log Kodaira dimension.
Proof. Take the Stein factorization of Y -> (image in S x Z), resolve
the intermediate variety, and resolve the induced map from Y, choosing
the model to be also a log resolution of (Y,B). Get Y -r-> W with
connected fibres, t : W -> S, h : W -> Z, p = t r, q = h r, W -> S x Z
generically finite onto its image. The replacement is harmless (positive
part of crepant boundary, exceptional correction). Fibres of t are
connected. For general s the fibre W_s is smooth connected and h_s :
W_s -> Z is generically finite onto its image; write a = dim W_s, m =
dim Z, so a <= m. Claim kappa(W_s) >= 0 (clear if a=0). Along a smooth
fibre of t the cotangent sequence
0 -> Omega^1_{S,s} tensor O_{W_s} -> Omega^1_W|_{W_s} -> Omega^1_{W_s}
-> 0 splits off: since Omega^1_{W_s} has rank a, the m-th exterior power
of Omega^1_W|_{W_s} lies in the filtration term with at least m-a base
factors, giving a nonzero morphism
h_s^* omega_Z -> Lambda^{m-a} Omega^1_{S,s} tensor omega_{W_s}. A linear
functional on the constant factor yields a nonzero h_s^* omega_Z ->
omega_{W_s}. Choose 0 != tau in H^0(Z, nK_Z): for general s the image
h_s(W_s) is not in div(tau) (else dominance fails), so h_s^* tau maps to
0 != section of nK_{W_s}. Claim proved.
Let G be a general fibre of r; for general z in Z these cover Y_z, so a
big divisor restricts big to a general G. Since G maps to a point of S,
M|_G ~Q 0 and adjunction gives D|_G ~Q K_G + B|_G. For general s,
r_s : Y_s -> W_s has connected fibres, (Y_s, B|) is klt, and the general
fibre of r_s has big log canonical divisor; Theorem 9.1 gives
kappa(Y_s, K + B|) >= dim G + kappa(W_s) >= 0. Take s outside the
countable jumping loci of all divisible log pluricanonical bundles;
generic base change then gives the same nonvanishing on the geometric
generic fibre. QED
Lemma 9.4 (Effectivity). Let p : Y -> S be an algebraic fibre space of
smooth projective varieties, (Y,B) effective klt, and suppose the
geometric generic fibre of p has nonnegative log Kodaira dimension.
Then for every big Q-divisor P on S the divisor K_{Y/S} + B + p*P is
Q-linearly equivalent to an effective divisor.
Proof. Flatten p over a dense open by a projective birational base
change, then resolve the base keeping flatness of the main transform;
resolve the transform Y_1 -> S_1 with composite mu : Y_1 -> Y a log
resolution of (Y,B). A prime divisor of Y_1 over a base subset of
codimension >= 2 is mu-exceptional (flatness). Let B_1 be the positive
part of the crepant boundary, so (Y_1, B_1) is klt and
K_{Y_1} + B_1 = mu*(K_Y + B) + E, E effective mu-exceptional.
Generic-fibre nonvanishing and base change give divisible k > 0 with
E_sheaf = p_{1*} O_{Y_1}(k(K_{Y_1/S_1}+B_1)) of positive rank; Theorem
9.2 applies. Fix ample A and alpha large with sigma*P - A/(k alpha) big.
Weak positivity gives beta with (Sym^{alpha beta} E_sheaf)^{**} tensor
O(beta A) generically generated. On the locally free locus U, symmetric
powers of sections map into p_{1*}O(kl(K_{Y_1/S_1}+B_1)); the product map
is nonzero at the generic point, so a global section survives over
p_1^{-1}(U) of kl(K_{Y_1/S_1}+B_1) + beta p_1^*A. The big divisor
kl sigma*P - beta A = kl(sigma*P - A/(k alpha)) has an effective
multiple; multiply to get a nonzero section over p_1^{-1}(U) of a
multiple of D_1 = K_{Y_1/S_1}+B_1+p_1^*sigma^*P, with poles only above
S_1 minus U. So there are n>0, rational phi != 0, and effective
mu-exceptional T with div(phi) + n D_1 + T >= 0. Put D_0 = K_{Y/S}+B+
p*P, R = K_{S_1} - sigma*K_S >= 0; then D_1 = mu*D_0 + E - p_1^*R. Push
forward to the smooth Y using mu_*E = mu_*T = 0:
div(phi) + n D_0 >= n mu_* p_1^*R >= 0, so D_0 is Q-linearly effective.
QED
Proof of the theorem over C. If kappa(F) or kappa(Z) is -infinity the
claim is by convention; assume d = kappa(F) >= 0 and kappa(Z) >= 0.
Proposition 8.3 and Lemma 8.4 give X~ -> Y -q-> Z and an effective klt
pair (Y,B) with a nef Q-divisor M of the required Hodge realization;
writing D = K_Y + B + M,
dim(Y/Z) = d, kappa(X) >= kappa(Y,D), D|_{Y_etabar} big.
Apply Theorem 3.1 (and Lemma 8.5 for the compatible modifications) to
get p : Y -> S with S smooth projective, M ~Q p*L, L nef. If dim S = 0
then M ~Q 0 and Theorem 9.1 applied directly to q plus the above gives
the result. So dim S > 0 and K_S + j L is big for all large j.
Openness of the big cone on Y_etabar gives a rational 0 < v < 1 with
(K_Y + B + vM)|_{Y_etabar} big. By Lemma 9.3 the geometric generic
p-fibre has nonnegative log Kodaira dimension. Choose large integer
j > 1 and an ample Q-divisor H on S with P_0 = K_S + j L - H big; Lemma
9.4 gives
E_1 = K_Y + B + jM - p*H
= K_{Y/S} + B + p*P_0 ~Q an effective divisor.
Set lambda = (1 - v)/(j - v), so lambda in (0,1) and
lambda j + (1-lambda) v = 1. Put
A_S = v L + (lambda/(1-lambda)) H, E_2 = K_Y + B + p*A_S.
Then A_S is ample, and the coefficient balance gives
D ~Q lambda E_1 + (1 - lambda) E_2.
Choose divisible N with |N p*A_S| basepoint-free and a general member
T_N, so Gamma = T_N/N ~Q p*A_S and (Y, B + Gamma) is klt (Bertini). Then
E_2 ~Q K_Y + B + Gamma is the log canonical class of an effective klt
pair, and its restriction to Y_etabar is (K_Y + B + vM)| plus a nef
divisor, hence big. Theorem 9.1 gives kappa(Y, E_2) >= d + kappa(Z).
Multiplying by powers of the nonzero section in the E_1 line embeds the
linear systems of (1 - lambda) E_2 into those of D, and rational scaling
leaves Iitaka dimension unchanged, so
kappa(X) >= kappa(Y,D) >= kappa(Y,E_2) >= kappa(F) + kappa(Z).
QED
Lemma 9.5 (Field extension). Let K subset K' be a field extension and V
a smooth projective geometrically integral variety over K. Then
kappa(V) is unchanged after extending scalars to K'. The same holds for
the geometric generic fibre of an algebraic fibre space of smooth
projective geometrically integral varieties.
Proof. Flat base change for proper coherent cohomology gives
H^0(V, nK_V) tensor_K K' = H^0(V_{K'}, nK_{V_{K'}}), and the canonical
bundle commutes with smooth scalar extension. Nonvanishing is unchanged;
when nonzero, a K-basis base-changes to the complete pluricanonical map
and dimensions of finite-type images are unchanged. Taking the supremum
over n proves the statement including -infinity. For the fibre, with
E = K(Z) and E' = K'(Z_{K'}), the generic fibre over E' is
F_K x_E E'; choosing algebraic closures overline E in Omega of E gives
F_K x_E overline E and F_K x_E Omega, the latter a scalar extension of
the former, so the Kodaira dimensions agree by the first part. QED
Completion for any algebraically closed field k of characteristic zero.
The varieties, projective embeddings, and morphism descend to a finitely
generated subfield K subset k over Q. Smoothness, geometric integrality,
and surjectivity descend by faithful flatness; proper flat base change
and faithful flatness preserve f_* O_X = O_Z, hence connected fibers.
Embed K into C. Lemma 9.5 identifies all three Kodaira dimensions with
their complex counterparts, using the geometric generic fibre for F.
The complex case proves the theorem over k. This completes Theorem 1
(ordinary Iitaka subadditivity).
10. APPENDIX: AN INTEGRAL PROJECTIVE PROOF OF ADJOINT POSITIVITY
We give a second proof of adjoint positivity for the whole top Hodge
line of a rationally polarized integral pure variation, with the same
conclusion as Theorem 3.1. An integral tensor replacement removes every
monodromy element acting trivially on the moving line. A transverse
loop then detects strict loss of the line's numerical dimension at each
remaining boundary divisor. The full period map gives an algebraic
base; the line map controls the boundary subtraction. The two ranks
need not agree.
Setup. P smooth connected projective complex, D a reduced snc divisor,
P^o = P minus D. A top Hodge line of a rationally polarized integral
pure variation V on P^o means F^p V_O with F^{p+1} = 0 and rank F^p = 1.
"Integral" means the local system is free Z-modules of finite rank; the
polarization is rational and flat. The line is a complex Hodge
subbundle with no rationality assumption. The line period map is
ell : universal cover of P^o -> P(V), x |-> F^p_x,
and its generic differential rank is intrinsic.
Theorem 10.1 (Adjoint positivity). P, D as above, V a rationally
polarized integral pure variation on P minus D whose top Hodge piece is
a line, L its rational parabolic extension. Then there are a projective
birational morphism pi : P~ -> P with P~ smooth, a smooth projective Z,
a surjective h : P~ -> Z with connected fibers, and a nef rational A on
Z such that
pi* L ~Q h* A, K_Z + aA big for all large rational a.
A divisor on a point is big by convention.
Extension facts. Local monodromies are quasi-unipotent. For unipotent
monodromy L is the top filtration subbundle of the Deligne-Schmid
extension; for quasi-unipotent monodromy the rational extension is
defined so that after an adapted finite cover making monodromy
unipotent its pullback is the unipotent extension, as a rational line
bundle. This includes the parabolic weights.
Lemma 10.2 (Extension facts).
(i) L is nef, and compatible with positive tensor powers and pullback
along morphisms of smooth projective log pairs mapping interior to
interior.
(ii) In the unipotent case, on each connected boundary stratum the
limiting mixed Hodge structures determine a unique weight-graded
piece containing the top line; these glue to a polarizable pure
variation with rank-one top filtration, whose extended top line on
the closure of the stratum is L restricted.
(iii) The same after an adapted cover in the quasi-unipotent case; so
numerical dimensions of restrictions are computed on a smooth
adapted cover.
Proof. (i)-(ii) are nilpotent-orbit theory (Schmid, Cattani-Kaplan-
Schmid), graded pieces polarized by the primitive decomposition. On a
stratum take the weight filtration at an interior point of the monodromy
cone; strictness and rank F^p = 1 select one highest graded piece. For
gluing, replacing a normal coordinate t by u(s) t changes the limiting
filtration by exp(-(log u(s))/(2 pi i) N), which is trivial on each
weight graded, so the graded data glue. Rational convention: take
divisible roots of boundary coordinates; the unipotent extension is
equivariant for the finite deck group; a common positive power kills
the characters of stabilizers on line fibres and descends. Nefness
descends under proper surjective maps, and nef numerical dimension is
invariant under generically finite pullback, giving (iii). QED
Lemma 10.3 (Curvature and intersections). nu(L) equals the generic
rank of the line period map. In the unipotent case with alpha the
Chern form of the Hodge metric on L|P^o and omega a Kahler form, for
0 <= k <= d = dim P,
integral over P^o of alpha^k wedge omega^{d-k}
= c1(L)^k . [omega]^{d-k},
absolutely convergent.
Proof. The closed-test-form version is Lemma 11.1; take the test form
omega^{d-k}. Griffiths' curvature formula gives ker alpha = ker d ell.
If r is the generic rank, alpha^{r+1} = 0 on the open set while
alpha^r wedge omega^{d-r} > 0 on a nonempty open subset; the
intersection identity plus nefness give nu(L) = r. QED
Lemma 10.4 (Integral tensor replacement). U smooth connected complex
algebraic, V rationally polarized integral pure with top line L^o. There
are b > 0 and a rationally polarized integral pure variation V_1 on U,
satisfying
gamma ell(x) = ell(x) for all x ==> gamma = id
(gamma in the monodromy image),
whose top line is (L^o)^{tensor b}. On any smooth projective snc
compactification the rational parabolic top line of V_1 is the b-th
power of that of V.
Proof. By semisimplicity (Deligne) the local system is a sum of
isotypic summands V_j, which are complex Hodge subvariations. The top
line is in one summand V_j0; retain V_j0 and all rational conjugates
(defined over a finite splitting field, Galois invariant). Their sum V_0
is a rational subvariation carrying the rational polarization and the
intersected lattice.
Suppose gamma in the monodromy image fixes the moving line projectively
everywhere. Its projective fixed locus is a finite union of
projectivized eigenspaces, so gamma is scalar on the linear span S of
the line image; equivariance makes S a subrepresentation of V_j0. With
V_j0 = E tensor M (E irreducible, monodromy trivial on M), S contains a
copy of E, so gamma is scalar on E and on all of V_j0; conjugation makes
it scalar on every retained V_j. Hence gamma|_{V_0} preserves the Hodge
filtration, polarization, and lattice. The stabilizer of a polarized
Hodge structure in its polarization group is compact, so gamma|_{V_0}
has finite order. Take
b = lcm{ q >= 1 : phi(q) <= R }, R = rank V_0,
where phi is Euler's totient; an integral finite-order eigenvalue has
order q with phi(q) <= R. Set V_1 = direct sum over retained j of
V_j^{tensor b}: direct, Galois invariant, rationally polarized,
integral, with unique top line (L^o)^{tensor b}. Every gamma above acts
trivially on V_1. Conversely a monodromy element fixing the new moving
line maps back to one fixing the original (injectivity of the Veronese
map P(V_j0) -> P(V_j0^{tensor b})) and is killed. QED
Lemma 10.5 (Full-period descent). After Lemma 10.4 and smooth
birational modifications, the variation descends over a dense open of a
smooth projective Z; its full period map is generically immersive and
satisfies the condition of Lemma 10.4. If A_1 is its top Hodge
extension then b pi* L ~Q h* A_1 for h : P~ -> Z with connected fibres.
Proof. The full period map factors through an algebraic dominant
P^o -> Y_0 with Y_0 -> Gamma_ar minus D a closed analytic immersion (Bakker-
Brunebarbe-Tsimerman). Normalize Y_0 in the relative algebraic closure
of C(Y_0) in C(P); this is finite and leaves C(Y_0) relatively
algebraically closed, so the generic fibre is geometrically connected.
Shrink to a smooth topologically trivial fibration with connected
fibres. A local lift of the full period map along a fibre is locally
constant in fibre directions (values in one discrete Gamma_ar-orbit);
continuing around a fibre loop fixes the lifted full period, so
transporting it to nearby fibres gives monodromy fixing the top line on
an open set. This continues to the universal cover and Lemma 10.4 kills
it; the homotopy sequence then factors the representation through
pi_1(Z). The filtration descends likewise (constant on connected fibres
on a simply connected chart), with transversality and polarization by
pullback. The descended period map has the dimension of the original
full period image and is generically immersive; the faithful-line
condition is preserved by analytic continuation. Choose a smooth
projective compactification of the base, resolve, and resolve the
rational map P -> base to get h : P~ -> Z with connected fibers.
Lemma 10.2 identifies the b-th power. QED
Lemma 10.6 (Transverse-loop criterion). Let U = Delta x Delta^{d-1}
with coordinates (z,s) and U* = Delta* x Delta^{d-1}. Suppose
ell : universal cover of U* -> P(V) is holomorphic, equivariant for
T in GL(V) under one positive turn around z=0, and rank d ell <= r.
Suppose q : U -> C^r is holomorphic with tangential differential of rank
r at a point of z=0, and on U* its lift satisfies ker d ell subset
ker d q. Then T fixes every line in the image of ell on a nonempty
open set; if ell is a line period map on a connected universal cover, T
fixes the whole image. This includes r = 0.
Proof. The submersion theorem gives coordinates (z,q_1,...,q_r,
u_1,...,u_{d-r-1}); on the punctured neighborhood the rank bound and
kernel inclusion give rank d ell = r and ker d q = ker d ell. Fix q, u
and turn once in z: the derivative of ell vanishes along every lift of
the loop, so ell is constant; its endpoints differ by T, which fixes
the initial line. Varying q, u gives an open set, and the projective
equality extends by the identity theorem. The case r = 0 makes ell
locally constant. QED
Lemma 10.7 (Boundary rank drop). Z smooth projective, D reduced snc,
a polarizable integral pure variation on Z minus D with top line
satisfying the Lemma 10.4 condition. Write A_1 for its rational
extension and r = nu(A_1). If D_i has nonidentity transverse monodromy
then nu(A_1|D_i) < r.
Proof. Otherwise single-valued functions of the moving line would have
the same differential kernel as the line, and a transverse loop would
force the monodromy to fix the moving line on an open set, contradicting
Lemma 10.4.
At a general point of D_i with coordinates (z,s), D_i = (z=0). Write
T = T_s T_u for the Jordan decomposition of the transverse monodromy,
N = log T_u, choose e with T_s^e = 1, and set z = t^e. The nilpotent
orbit extension gives a holomorphic line map Psi on the root polydisk
with
ell(t,s) = exp((e log t)/(2 pi i) N) Psi(t,s),
Psi(zeta t, s) = T_s Psi(t,s), zeta = exp(2 pi i / e).
Let W = W(N) be centered at the weight, with unique index k containing
the top line. Since F^{p+1} = 0, F^p intersect NV = N(F^{p+1}) = 0, so
the limiting line survives in C = V/NV. In the primitive decomposition
Gr^W_k V = P_k + N Gr^W_{k+2}V with P_k ~-> Gr^W_k C and the second
summand of zero F^p, the graded moving top line lies in P_k with
unchanged projective derivative rank; projecting to the graded piece
recovers the pure line period map, whose rank is r_i = nu(A_1|D_i) by
Lemmas 10.2 and 10.3.
Now Psi(0,s) is an eigenline of T_s with constant eigenvalue lambda. Let
E = C_lambda and q_0 : V -> E the quotient followed by the eigenspace
projection. Then Q(t,s) = P(q_0)(Psi(t,s)) has boundary rank >= r_i,
equals P(q_0)(ell(t,s)) because N vanishes on C, and satisfies
Q(zeta t, s) = Q(t,s); so Q descends holomorphically to the original
(z,s)-polydisk including z=0. Suppose r_i >= r: choose r affine
coordinates q_j of Q independent along D_i; q = (q_j) factors through
ell on the punctured chart, so ker d ell subset ker d q, and rank d ell
<= r. Lemma 10.6 makes T fix the moving top line on the universal cover,
contradicting Lemma 10.4. The cases r = 0 and r > d-1 are immediate.
QED
Lemma 10.8 (Boundary subtraction). Z smooth projective of dimension
d > 0, A_1 nef rational, J = sum_i D_i reduced with smooth components.
Suppose K_Z + J big and nu(A_1|D_i) < r := nu(A_1) for every i. Then
K_Z + aA_1 is big for all sufficiently large rational a.
Proof. If r = 0 then J = 0 and K_Z is big. Otherwise, by Kodaira's
lemma write K_Z + J ~Q H + E with H ample, E >= 0, and set B_a = H + aA_1.
Choose ample integral C_0 with both C_0 and C_0 + J globally generated;
put H' = H + C_0. For divisible m, filter by kJ and use the restriction
sequences; since J is reduced, O_J injects into sum_i O_{D_i}, and
mC_0 + kJ = (m-k)C_0 + k(C_0+J) is globally generated, so multiplying by
a section nonzero on D_i bounds the restricted terms:
h^0(Z, m(B_a - J)) >= h^0(Z, mB_a)
- m sum_i h^0(D_i, m(H' + aA_1)|_{D_i}).
Fix a and divide by m^d/d!; asymptotic Riemann-Roch gives
vol(B_a - J) >= B_a^d - d sum_i (H' + aA_1)^{d-1}.D_i.
The first polynomial has degree r with positive leading coefficient
binom(d,r) A_1^r H^{d-r} > 0, and each restriction polynomial has degree
at most nu(A_1|D_i) <= r-1. The right side is positive for large a, and
K_Z + aA_1 ~Q (B_a - J) + E is big. QED
Proof of Theorem 10.1. Lemmas 10.4 and 10.5 give a variation on a
dense open of a smooth projective Z, satisfying the Lemma 10.4
condition, with generically immersive full period map, and A_1 with
b pi* L ~Q h* A_1. Resolve the boundary to snc and delete boundary
components of identity transverse monodromy; call the remaining reduced
divisor J. The variation extends to Z minus J (nilpotent orbit with all
logarithms zero is a pure extension), and the full period map is still
generically immersive, so Brunebarbe-Cadorel gives K_Z + J big. Lemma
10.7 applies to every component of J, and Lemma 10.8 gives K_Z + a A_1
big for large rational a. Set A = A_1/b; it is nef, pi* L ~Q h* A, and
the adjoint positivity survives rescaling. QED
11. APPENDIX: CHERN INTERSECTIONS AND THOM FORMS
We prove the Chern-intersection formula used in Lemma 10.3 and give a
third proof of the boundary restriction estimate, using smooth Thom
forms concentrating on a boundary component. A product Poincare bound
controls crossings. Throughout R is a smooth connected projective
complex variety, D_R a reduced snc divisor, V a polarizable integral
pure variation on R minus D_R. A top line is F^p with F^{p+1}=0,
rank F^p = 1; A is its rational parabolic extension; for n = dim R,
nu(A) = max{ j in {0,...,n} : c1(A)^j . H^{n-j} > 0 },
H any ample class, A nef. Write ddc = (i/2pi) d dbar.
Lemma 11.1 (Chern intersections with closed test forms). nu(A) equals
the generic differential rank of the line period map, provided A is nef.
In the unipotent case, with alpha the Chern form of the Hodge metric on
A over the open part, for every 0 <= j <= n and every smooth closed form
gamma of type (n-j,n-j) the integral of alpha^j wedge gamma over the
open part is absolutely convergent and equals c1(A)^j . [gamma].
Proof. Compute after a generically finite smooth alteration making all
boundary monodromies unipotent: pass to a finite level cover with neat
monodromy, compactify, resolve; quasi-unipotence then gives unipotence
of the commuting logarithms. Pullback preserves nef numerical dimension
and generic differential rank, so we may assume unipotent.
alpha is the semipositive Chern form of the top-line Hodge metric.
Griffiths' curvature formula makes alpha(v,v) a positive fixed multiple
of the squared norm of the projective differential in direction v, so
its rank equals the line-map rank.
Take a crossing chart with coordinates z_1,...,z_n and boundary
z_1...z_c = 0, put rho_i = -log|z_i|^2 > e. The singular
Ahlfors-Schwarz estimate gives
0 <= alpha <= C Omega,
Omega = sum_{i<=c} i dz_i dzbar_i /(|z_i|^2 rho_i^2)
+ sum_{i>c} i dz_i dzbar_i,
a product cusp metric (Hermitian Cauchy-Schwarz controls off-diagonal
terms even where the differential degenerates). A nonvanishing top
extension frame in flat coordinates is
v(z) = exp( sum_i (log z_i)/(2 pi i) N_i ) u(z),
with commuting nilpotent N_i and u holomorphic nonvanishing; the Hodge
metric and its inverse are bounded between products of powers of the
rho_i, and the frame factor is a polynomial in the logarithms, so the
ratio phi of the metric to a fixed smooth metric satisfies
|phi(z)| <= C ( 1 + sum_i log rho_i ).
The canonical Hodge extension represents c1(A) by the current alpha
(Cattani-Kaplan). Cut off with chi_M = product of chi((log rho_i)/M),
chi smooth 1 on (-inf,1], 0 on [2,inf). Since every curvature is
closed, alpha^j - beta^j = ddc( phi sum_k alpha^k beta^{j-1-k} ),
beta the curvature of the fixed smooth metric. Integrating by parts
against chi_M gamma, the error is at most a constant times
(1 + sum_i log rho_i) Omega^n, which is integrable because
integral (1 + log rho) rho^{-2} drho converges; so it tends to 0. This
proves the intersection formula (rational bundles by an integral tensor
power).
Now let r be the generic line-map rank. Then alpha^{r+1} = 0 on the
open part, while alpha^r is positive on a nonempty open subset. The
intersection formula with complementary powers of a Kahler form gives
vanishing of the (r+1)-st and strict positivity of the r-th
intersection; nefness gives nu(A) = r. QED
Lemma 11.2 (Boundary rank drop by Thom forms). In the setting of Lemma
11.1, suppose monodromy is faithful on the moving top line. If D_i is a
boundary component with nonidentity transverse monodromy, then
nu(A|D_i) < nu(A).
Proof. Put r = nu(A). Near a general point of a unipotent boundary
component use coordinates (t,s), divisor t=0, monodromy logarithm N,
weight w, largest index p. The limiting mixed Hodge structure is
polarized by N and its graded pieces vary as polarized variations along
the stratum. Limiting F^p is one-dimensional with F^{p+1} = 0, hence a
single Deligne summand, and the line is primitive: N^l u_0 != 0,
N^{l+1} u_0 = 0 for u_0 = u(0,s). Substituting the exponential frame
into the Hodge norm gives, with y = -log|t|,
log ||v(t,s)||^2 = l log y + log h_l(s) + c
+ O(y^{-1}) + O(|t| y^M)
for some M, so the tangential curvature converges, locally uniformly
with tangential derivatives, to that of the primitive graded top line;
its rank is at most q_i, the generic rank of that line's projective
differential. We claim nu(A|D_i) <= q_i.
The following estimate avoids uniform convergence at crossings. Fix an
ample Kahler form omega and j with q_i < j <= n-1. Choose a smooth
metric on O(D_i), curvature beta_i, put tau = log||s_{D_i}||^2, let g be
a fixed smooth convex function with g' = 0 on (-inf,0], g' = 1 on
[log 4, inf), and set f_eps(x) = g(x - log eps^2). Then
eta_eps = beta_i + ddc f_eps(tau)
= (1 - f_eps'(tau)) beta_i
+ (i/2pi) f_eps''(tau) dtau dbar tau
is a smooth closed representative of c1(O(D_i)); f_eps(tau) is constant
near D_i and extends smoothly, and eta_eps vanishes outside
||s_{D_i}|| <= 2 eps.
On a crossing chart put lambda = i dt dbar t /|t|^2, rho = -log|t|^2,
Omega = lambda/rho^2 + Omega_s (finite-volume product cusp metric), and
T = alpha^j omega^{n-1-j}, so 0 <= T <= C Omega^{n-1}. Write tau =
log|t|^2 + psi, L_eps = |log eps|. The smooth first term contributes at
most C integral_{|t|<=C eps} Omega^n = O(L_eps^{-1}). For the second,
dtau = a_0 + b_0 with a_0 = dt/t and b_0 = d psi; its support lies in a
shell c eps <= |t| <= C eps. Positivity gives integral i a_0 dbar a_0
wedge T = O(1) (wedging by lambda selects only the tangential part) and
integral i b_0 dbar b_0 wedge T <= C integral_{shell} Omega^n =
O(L_eps^{-2}); Cauchy-Schwarz bounds the mixed terms by O(L_eps^{-1}).
For the pure normal-shell term set t = eps u; the coefficient becomes
g''(log|u|^2 + psi(eps u, s)), supported in a fixed annulus, and the
integrand is bounded by C (i du dbar u)/|u|^2 wedge Omega_s^{n-1},
integrable including all tangential crossings. For almost every s the
tangential curvature limit has rank < j, so the tangential j-th power
tends to 0; dominated convergence makes the normal-shell integral tend
to 0, uniformly under truncating tangential cusp variables. Apply the
intersection formula with gamma = omega^{n-1-j} wedge eta_eps. Hence
c1(A|D_i)^j . [omega|_{D_i}]^{n-1-j} = 0, and nefness gives
nu(A|D_i) <= q_i.
Return to the original boundary component with transverse monodromy
T_s exp N, T_s of finite order, and work over a transverse root
coordinate killing T_s. Set C_N = coker N with the induced weight
filtration; the fixed quotient W_k C_N -> Gr^W_k C_N recovers the
primitive graded line, and projective differential rank cannot increase.
The limiting line in C_N lies in one T_s-eigenspace; composing the
quotient by N with the eigenspace projection gives a nonzero holomorphic
projective map near the boundary point. Deck rotation acts by the
scalar T_s on that eigenspace, so the projective map descends to the
original chart; on the punctured chart it factors through ell, and its
boundary differential rank is at least q_i.
Suppose nu(A|D_i) >= r. Then the projected map has at least r
independent tangential differentials at a general boundary point.
Choose r affine components g_1,...,g_r of it with that property; their
differentials are independent along the boundary, and g = (g_j) extends
holomorphically over the original chart, factoring through ell on the
punctured chart. Lemma 11.1 gives rank ell <= r, so the hypotheses of
the transverse-loop criterion (Lemma 10.6) hold, and the original
transverse monodromy fixes the moving line on the universal cover,
contradicting faithfulness. The cases r = 0 (empty list of functions)
and r > n-1 (impossible) are immediate. QED
Connection. The hypotheses here arise from the tensor replacement and
full-period descent of Lemmas 10.4 and 10.5. Lemma 11.2 gives the same
strict inequality as Lemma 10.7, with the boundary intersection
justified directly by Thom forms, and the subtraction Lemma 10.8 applies
to the union of divisors with nonidentity transverse monodromy. The
full-period map supplies bigness of K_R + J; the top-line rank supplies
the strict degree drop. The two different period ranks keep their
separate roles.
Remark 11.3. On a smooth complete curve, degree zero gives zero
integral of the semipositive Hodge curvature by Lemma 11.1, after an
adapted cover, so the curvature vanishes and the top line is
projectively constant. On a connected smooth algebraic locus where the
top line is projectively constant, applying the tensor replacement to
the restricted variation itself shows that the original top-line
character on that locus has finite image; the tensor exponent may depend
on the restricted variation.
Remark 11.4. The adjoint conclusion is preserved under further smooth
projective birational modification pi : R' -> R with a compatible
modification of the source: parabolic pullback preserves the line
identity and nefness, and K_{R'} + b pi* A = pi*(K_R + bA) + E_pi with
E_pi >= 0 preserves bigness. No equality is asserted between
highest-line rank, full-period rank, and whole-fiber birational
variation.
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