================================================================
ANTICANONICAL NONVANISHING FROM SMOOTH SEMIPOSITIVITY
================================================================
WHAT IS PROVED.
The paper
proves one clean statement about the shapes called "complex
projective varieties" and the holomorphic sections that can live on
them.
NOTATION
--------
C the complex numbers.
P^n complex projective space of dimension n.
X, Y, Z, S spaces ("varieties"); d = dim X unless stated.
K_X the canonical bundle of X. -K_X is the anticanonical
bundle.
H^0(X, L) the vector space of holomorphic sections of a line bundle
L over X.
mL the m-th tensor power of L.
L + M the tensor product of line bundles (we add them).
D >= 0 an effective divisor: the zero set of a nonzero section of
some line bundle; "D >= 0" means nonnegative and positive
when nonzero.
alpha a curve class: a direction in which we measure degrees.
================================================================
PART 1. THE QUESTION, THE ANSWER, AND TWO EXAMPLES
================================================================
1.1 THE OBJECTS
----------------
A complex manifold is a space that looks locally like an open piece
of C^n, with complex-analytic ("holomorphic") glue between pieces.
A projective variety is a set cut out inside projective space P^N by
homogeneous polynomial equations. Projective varieties are compact,
and "smooth connected projective complex variety" is the nicest,
most concrete kind of compact space in complex geometry. Everything
here is of that kind unless said otherwise.
A holomorphic line bundle L over a variety X attaches to each point a
one-dimensional complex vector space, varying holomorphically. The
simplest is the trivial bundle X x C. A holomorphic section of L is a
holomorphic choice of one vector in the fiber above every point, the
same as a holomorphic function with values in L. The set of points
where a section is zero is a divisor: a collection of complex
hypersurfaces (the zero set of one equation each). A line bundle has
a nonzero section exactly when it is the bundle of a divisor that may
be zero (an "effective" divisor). So:
"does L have a nonzero section?" is the same question as
"is L the line bundle of some effective divisor?"
The canonical bundle K_X is the bundle whose fiber at a point is the
one-dimensional space of holomorphic n-forms there (n = dim X); its
sections are the holomorphic n-forms. Its dual, the anticanonical
bundle -K_X, is the top exterior power of the holomorphic tangent
bundle, det T_X. For many nice spaces -K_X is the "most positive"
bundle around. When it is positive (ample), sections are plentiful.
When it is only semipositive, sections can disappear, and that is the
whole problem.
1.2 CURVATURE AND SEMIPOSITIVITY
---------------------------------
A Hermitian metric on a line bundle L is a smoothly varying way to
measure the length of vectors in each fiber. In a local frame e, write
the squared length of the frame as exp(-phi) for a local real
function phi. Every metric has a curvature: a closed real (1,1)-form
representing the first Chern class c_1(L). Picture curvature as the
way the bundle bends. The metric is semipositive when its curvature
form is nonnegative as a Hermitian form: the bundle bends, but never
in a negative direction. A very close cousin is nef ("numerically
eventually free"), a class that is a limit of positive ones, which is
the same as admitting metrics of semipositive curvature.
Intuition: positive curvature is like a sphere (it closes up and
carries lots of functions); negative curvature is like a saddle
(functions are scarce). Semipositive is the borderline. The theorem
says: even at this borderline, the anticanonical bundle still bends
curvature upward enough to force a section of some power.
1.3 THE MAIN QUESTION
----------------------
For a smooth connected projective X, does the anticanonical bundle
-K_X have a nonzero section after raising to a positive power? That
is, is there some m > 0 with H^0(X, -m K_X) nonzero? The power m is
essential (see 1.5); asking for m = 1 would be false in general.
1.4 THE ANSWER (THEOREM 1)
---------------------------
THEOREM 1 (Anticanonical nonvanishing). Let X be a smooth connected
projective complex variety. If -K_X admits a smooth Hermitian metric
of semipositive Chern curvature, then
H^0(X, -m K_X) is nonzero for some integer m > 0.
In words: a smoothly semipositive anticanonical bundle always has a
nonzero section after some positive power. This answers the positive-
multiple form of a question posed by Yau (Problem 75 in his 1994 open
problems list).
1.5 TWO EXAMPLES: WHY THE POWER IS NEEDED
------------------------------------------
Example A: an Enriques surface E. Here K_E is not the trivial bundle
but 2 K_E is trivial: K_E is a nonzero two-torsion class. So K_E has
no nonzero section (a nonzero section of a degree-zero bundle would
make it trivial), and neither does -K_E = K_E (same class, order two).
But -2 K_E is trivial and has the constant section. This is the
smallest witness that the power m in Theorem 1 cannot be dropped:
with m = 2 it works, with m = 1 it does not.
There is a smooth flat metric on -K_E (a flat metric has zero
curvature, which is semipositive). So -K_E does satisfy the curvature
hypothesis, and the theorem correctly promises a section only for
some power; here m = 2.
Example B: the product E x P^1, where E is the Enriques surface above
and P^1 is the projective line. Let p_E, p_{P^1} be the two
projections. The anticanonical bundle of the product is
p_E^*(-K_E) + p_{P^1}^*(O(2)),
because the determinant of the tangent space of a product splits as
the sum of the two pullbacks. Put the flat metric on the first factor
and the round (Fubini-Study) metric on the second. Each is
semipositive, so their sum -- the metric on the anticanonical bundle
-- is smooth and semipositive. This is exactly the kind of metric
Theorem 1 is about.
Now compute sections. For a tensor product of pullbacks on a product,
the Kunneth rule gives
H^0(E x P^1, -K_{E x P^1})
= H^0(E, -K_E) (x) H^0(P^1, O(2))
plus higher pieces that vanish. The first factor is zero (no
anticanonical section on E). So there is no first-power section at
all -- matching Example A. The square uses -2K_E (the trivial bundle,
one section) and O(4) (five sections), so it does have sections.
Restricting to a single P^1 fiber turns the anticanonical class into
O(2), which is not trivial; hence the full anticanonical class of
E x P^1 is not a torsion class. So Example B is not a disguised
flat/torsion case: the multiple m is genuinely needed, smooth
semipositivity holds, and nonvanishing still holds for m = 2.
These two examples are the model for the whole proof: the smooth
metric is the real hypothesis; the section appears only after a
positive power; and the reason a power is needed is geometric.
================================================================
PART 2. THE SHAPE OF THE ANSWER:
STRUCTURE THEORY AND THE UNIVERSAL COVER
================================================================
The proof of Theorem 1 does not begin by hunting for a section on X.
It first describes the shape of X from its curvature, builds a section
on a simpler space, and then carries it back. This part sets up the
shape and the carrying-back.
2.1 FROM CURVATURE TO RICCI CURVATURE
--------------------------------------
A Kahler metric is a Hermitian metric on a complex manifold whose
curvature has a symmetric, especially clean form; every projective
variety carries plenty of them. Within a Kahler metric one can form
the Ricci curvature, a (1,1)-form that measures how the metric's
volume shrinks and grows. Yau's prescribed-Ricci theorem says, very
roughly:
any smooth closed (1,1)-form representing the first Chern class
c_1(X) that is positive can be realized as the Ricci form of some
Kahler metric.
Now a metric on -K_X has curvature form representing c_1(-K_X) =
c_1(X). So a smooth semipositive metric on -K_X is exactly a
semipositive representative of c_1(X), and Yau's theorem turns it into
a Kahler metric whose Ricci curvature is nonnegative. From here on,
"smoothly semipositive anticanonical bundle" means the same thing as
"X carries a Kahler metric of nonnegative Ricci curvature."
2.2 THE PRODUCT COVER
----------------------
There is a deep soft-spot in nonnegatively curved spaces: their
universal cover splits as a product. This is the de Rham decomposition
together with the Cheeger-Gromoll splitting theorem. In our complex
setting it says the universal cover of X, written X-tilde, is a
product
X-tilde = C^a x F x Z ,
where:
- C^a is flat Euclidean space (the "Euclidean" or "flat" factor);
- F is compact, simply connected, and Ricci-flat (curvature exactly
zero in the Ricci sense);
- Z is compact, simply connected, and carries the remaining, non-
flat positive curvature.
Any of the three factors may be a point. (Z here is a space, not the
integers.) This is the Demailly-Peternell-Schneider structure theorem,
in its Kahler form.
Two refinements matter. First, the Ricci-flat factor F has a nowhere-
zero parallel holomorphic top form: its canonical bundle is flat and
therefore has a canonical frame eta_F, a global nonzero section of
K_F fixed by everything. Second, the remaining factor Z has no
holomorphic forms of positive degree:
H^0(Z, Omega_Z^p) = 0 for all p > 0,
H^q(Z, O_Z) = 0 for all q > 0.
This is a Bochner principle: nonnegative Ricci curvature forces every
holomorphic form to be parallel, and any positive curvature direction
(generic on Z) then contracts such a form to zero. In words: positive
curvature forbids holomorphic forms. The Campana-Demailly-Peternell
refinement says more, that Z is rationally connected (any two of its
points can be joined by a rational curve), which will matter later.
2.3 THE DECK GROUP AND WHY A FINITE COVER
------------------------------------------
The universal cover X-tilde carries an action of the fundamental group
pi_1(X) by deck transformations, and X is the quotient of X-tilde by
this action. Deck transformations are holomorphic isometries. Because
the product decomposition is unique up to permuting factors, after
passing to a finite-index subgroup of the deck group every deck
transformation acts as a product map, one map on each factor.
Passing to a finite-index subgroup means passing to a finite cover. So
there is a connected finite unramified ("etale") cover
nu : S --> X
whose deck group still acts by product maps. On the three factors the
action looks like:
- on C^a: a full lattice of translations Lambda, so C^a/Lambda is a
compact complex torus;
- on F: a finite (hence, after one more finite-index passage,
trivial) group of isometries;
- on Z: a group whose closure in the compact isometry group of Z is
a compact connected real torus T.
The last point is the heart of the matter. The isometry group of a
compact space is itself compact; a connected abelian closed subgroup
of it is a torus. After passing to the finite-index subgroup whose
image lands in the identity component, the action of Lambda on Z has
compact closure T.
2.4 WHY S NEED NOT BE A PRODUCT
--------------------------------
If S were literally the product (C^a/Lambda) x F x Z, a section of
the anticanonical bundle of the product would just be a product of
sections of each factor, and we would be done by picking the
canonical frames on the first two factors and a torus-invariant
section on Z.
But S need not be a product. The bundle X-tilde = C^a x F x Z over
C^a/Lambda is a flat bundle with monodromy rho: Lambda -> Aut(Z). The
monodromy group rho(Lambda) can be infinite. Its closure is the torus
T above. An infinite monodromy means the Z-factor twists as you move
around the base torus, so S is not a product but a nontrivial flat
bundle. Concretely: a section on S must be invariant under the deck
group, and the deck group acts on the Z-coordinate through rho.
This is why we must build a section on Z that is invariant under the
whole torus T: only such a section survives the monodromy and
descends to S. The effort of Part 6 is exactly to produce such a
T-invariant section.
2.5 THE STRUCTURE THEOREM (THEOREM 2)
--------------------------------------
Here is the precise geometric reduction, stated as the paper does.
THEOREM 2 (Finite cover with compact torus monodromy). Let X be a
smooth connected projective complex variety whose anticanonical
bundle has a smooth semipositive Hermitian metric. Then there are a
connected finite etale cover nu : S -> X, an integer a >= 0, and
smooth connected projective varieties F and Z such that:
(1) The universal cover of X (and of S) splits holomorphically and
isometrically as C^a x F x Z, with F Ricci-flat and
simply connected, Z simply connected with no holomorphic forms
of positive degree (and rationally connected if positive-
dimensional), exactly as in 2.2.
(2) There are a full lattice Lambda in C^a and a homomorphism
rho : Lambda -> Aut(Z) with
S = (C^a x F x Z) / Lambda,
where lambda in Lambda acts by
(z, f, v) -> (z + lambda, f, rho(lambda) v). The closure T of
rho(Lambda) is a compact connected real torus.
(3) The torus action extends to an algebraic action of a complex
algebraic torus G on Z, with T Zariski-dense in G, and on the
anticanonical bundle the action is the natural one induced by
the differential. Invariants for T equal invariants for G on
every space of anticanonical sections.
(4) The canonical bundles of C^a and F have nowhere-zero
holomorphic frames eta_0 (the form dz_1 ^ ... ^ dz_a) and
eta_F, both invariant under Lambda, and the canonical bundle of
the product is the tensor product of the three pulled-back
canonical bundles.
The proof is a careful passage through the splitting and the deck
group, ending with the observation that a compact connected group of
projective linear transformations can be put inside a diagonal
algebraic torus (so its Zariski closure is an algebraic torus G, and
finite-dimensional invariant subspaces for T and for G agree). We
take that as established and use it.
2.6 CARRYING A SECTION BACK: THE ETALE NORM (LEMMA 3)
------------------------------------------------------
Because S is a finite cover of X, a section on S can be pushed back
down to X by a product over the sheets. This is the analogue, for
sections, of the norm of a field extension.
LEMMA 3 (Finite etale norm). Let nu : S -> X be a finite etale
morphism of degree e > 0 between smooth connected projective complex
varieties, and let A be a line bundle on X. Every nonzero section s of
nu^*A on S has a nonzero norm
Nm(s) in H^0(X, A^{tensor e}).
In particular a nonzero section of K_S^{-m} (m > 0) gives a nonzero
section of K_X^{-m e}.
Proof idea. Over a small open set of X on which the cover splits into
e sheets, a frame of A pulls back to frames on each sheet, and s
restricts to functions f_1, ..., f_e there. Define the local norm to
be (product of the f_i) times the e-th tensor power of the frame. This
is unchanged by reordering the sheets, and under a change of frame it
transforms exactly as a section of A^{tensor e} should. So the local
norms glue into a global section. It is nonzero because a nonzero
section of a line bundle over a smooth connected space vanishes on a
proper closed set at worst, its image under a finite map is again
proper, and away from that image every sheet factor is nonzero.
Finally K_S = nu^*(K_X) because nu is etale. Done.
Putting it together: to prove Theorem 1 it is enough to produce, on
the compact factor Z, a T-invariant nonzero section of some K_Z^{-m}.
For then the product eta_0^{-m} (x) eta_F^{-m} (x) s_Z is a Lambda-
invariant nonzero section of K_S^{-m} (the first two factors are
invariant by construction, the third by choice of T-invariance), it
descends to S, and Lemma 3 produces a nonzero section of
K_X^{-m e}. That is Theorem 1.
================================================================
PART 3. THE FINITE-VOLUME THEOREM (THE ENGINE)
================================================================
The whole induction runs on a single statement, the finite-volume
theorem. It is abstract, but its shape is forced by the geometry of a
fibration, and once it is proved every piece of Theorem 1 is an
application of it. This part states it and explains each hypothesis.
3.1 THE NO-FORMS VARIETY
-------------------------
Throughout, Z is a smooth connected projective complex variety of
dimension d with no holomorphic forms of positive degree:
H^0(Z, Omega_Z^p) = 0 for 1 <= p <= d.
By Hodge symmetry this forces H^q(Z, O_Z) = 0 for q > 0, and therefore
the holomorphic Euler characteristic is
chi(O_Z) = 1.
The compact factor Z of Part 2 is such a variety: it is rationally
connected and has no positive-degree forms. Enriques surfaces are
another example. The no-forms condition is what makes the cohomology
of Z behave like a point's.
3.2 TWO BUNDLES, TWO METRICS
-----------------------------
The theorem concerns two line bundles with two very different metrics.
The first is L. It carries a semipositive singular Hermitian metric
whose weights are locally bounded; a singular metric means a weight
function that may fail to be smooth but is plurisubharmonic
(convex in the complex sense), and locally bounded means the weight
is bounded above and below on small charts. This is where the
"positivity" of the theorem is stored.
The second is the corrected anticanonical bundle
J = -K_Z + D ,
where D >= 0 is a fixed effective divisor (possibly zero). The
canonical section of O(D), call it s_D, is a J-valued top form: it
lies in H^0(Z, K_Z + J), because O(D) = K_Z + J. Pairing s_D with
its own conjugate through a metric h_J on J produces a positive
measure, the adjoint measure,
|s_D|^2_{h_J} = |g(z)|^2 exp(-phi(z)) d(lambda_z),
in a local frame where s_D = g dz_1 ^ ... ^ dz_d (x) v and the frame v
has squared length exp(-phi). This is coordinate-independent.
We allow h_J to be singular, but require its adjoint measure to have
finite total mass:
integral over Z of |s_D|^2_{h_J} < infinity.
That is the finite-volume hypothesis. Plurisubharmonic weights are
unbounded, so the mass could be infinite; the theorem demands it not
be.
3.3 THE FINITE-VOLUME THEOREM (THEOREM 4)
------------------------------------------
THEOREM 4 (Finite volume and sections with prescribed poles). Let Z
be a smooth connected projective complex variety of dimension d with
H^0(Z, Omega_Z^p) = 0 for every 1 <= p <= d. Let D >= 0 be an
integral divisor and suppose that J = -K_Z + D has a semipositive
singular Hermitian metric h_J with
integral over Z of |s_D|^2_{h_J} < infinity.
If a line bundle L on Z has a semipositive singular Hermitian metric
with locally bounded weights, then there exist integers a > 0 and
b >= 0 such that
H^0(Z, aL + bD) is nonzero.
3.4 READING THE STATEMENT
--------------------------
Think of L as "the bundle I want a section of" and D as "where I am
allowed to have poles." The conclusion says: some positive multiple
of L has a rational section whose only poles lie on D. The power b
(the allowed pole order) is not controlled; it exists.
The two metrics play complementary roles. The metric on L has
bounded weights -- it is as tame as possible. The metric on J may be
wild (it can be unbounded), but only in a controlled way: its total
adjoint volume is finite. This asymmetry is exactly what a fibration
produces: fiber integrals of a bounded metric stay bounded, while the
canonical correction accumulates all the unbounded behavior into one
measure that remains finite.
Note the small but important structural choices. The divisor D enters
twice: once to correct the anticanonical bundle into J, and once as
the permitted pole divisor. Both uses are the same boundary, and the
proof depends on them matching. When D = 0 the theorem says simply:
if -K_Z has a semipositive metric of finite volume (automatic when
the metric is smooth on compact Z) and L is semipositive with bounded
weights, then some positive power of L has a section.
3.5 WHY THIS IS THE ENGINE
---------------------------
Two immediate uses.
First, apply Theorem 4 directly to the compact factor Z of Part 2 with
L = -K_Z (its given smooth semipositive metric has bounded weights on
compact Z), D = 0, and h_J that same metric. The adjoint measure is a
smooth volume form, so finite volume is automatic. The conclusion is
H^0(Z, -a K_Z) is nonzero for some a > 0.
That proves ordinary anticanonical nonvanishing on the no-forms factor
Z with no torus involved.
Second, and this is the real content, Theorem 4 is proved by induction
on dimension. It is applied not to Z itself but to the smaller base S
of a fibration of Z, and then the resulting section on S is lifted
back up to Z. The hypotheses of Theorem 4 -- a semipositive L with
bounded weights, and a singular J with finite volume and pole divisor
D -- are exactly the two pieces of data that the transfer step
manufactures on a base (Part 5). So Theorem 4 is the complete engine;
the rest of the proof is feeding it.
The only refinement the main theorem still needs beyond Theorem 4 is
the torus-invariance of the section on Z. Part 6 proves an invariant
version of Theorem 4 using one extra positivity input.
3.6 THE PROOF IN ONE PARAGRAPH
-------------------------------
Assume the conclusion fails. Then no aL + bD has a section, so L + D
is pseudoeffective (it has a semipositive metric) but not big (a big
bundle has many sections after a large multiple). A supporting curve
class alpha of the pseudoeffective cone then satisfies L.alpha =
D.alpha = 0, and the finite adjoint volume bounds the slopes of the
cotangent bundle against alpha. Using the no-forms condition (chi(O) =
1) and Riemann-Roch together with a hard-Lefschetz-with-multiplier-
ideals theorem, one extracts infinitely many sections of a fixed
cotangent determinant line M plus growing multiples of L, with
M.alpha <= 0. A maximal linear system built from L, M, D then gives a
fibration of Z onto a strictly smaller base S, and forces the
adjoint section spaces on the fibers to be one-dimensional over the
base field. The transfer step converts this data into the two metrics
on S that Theorem 4 was waiting for, applied one dimension down. The
resulting section on S lifts back to a section of aL + bD on Z, a
contradiction. Hence the assumption was false.
That is, then, the remaining parts: Part 4 the slope bound that produces
a smaller base, Part 5 the transfer step that moves the two metrics
down to that base, Part 6 the torus action and invariant sections, and
Part 7 the descent back to X. Each part is one hard lemma plus its
proof. We take them in order.
================================================================
PART 4. SLOPE, DETERMINANTS, AND THE SMALLER BASE
================================================================
This part proves Theorem 4 (the engine) by induction on the dimension
d of Z. Induction needs a strictly smaller space to induct on, so the
work splits into four steps, each a lemma:
4.1 Assume the engine fails; extract a special curve class alpha
and prove a slope bound (the "cotangent bound").
4.2 Use the slope bound plus Riemann-Roch to produce one fixed
line bundle M with sections of M + m L for growing m.
4.3 Build a maximal linear system; it gives a fibration of Z onto a
strictly smaller base S and forces a rank-one condition.
4.4 Run the induction, feeding the transfer step of Part 5.
4.1 THE HARD CASE AND THE BOUNDARY CLASS
-----------------------------------------
Recall the setup of Theorem 4: Z is smooth connected projective of
dimension d with no holomorphic forms of positive degree; D >= 0 is an
effective divisor; J = -K_Z + D has a semipositive singular metric h_J
whose adjoint measure has finite total mass; and L has a semipositive
singular metric with locally bounded weights.
Assume the conclusion FAILS: no aL + bD, with a > 0 and b >= 0, has a
nonzero section. We will derive a contradiction.
First, L is pseudoeffective: it carries a semipositive metric, and a
semipositive metric makes the numerical class lie in the pseudoeffective
cone (the cone of classes that are limits of classes of effective
divisors). So L + D is pseudoeffective too, being a sum of two
pseudoeffective classes.
Second, L + D is NOT big. "Big" means some positive multiple has so
many sections that its complete linear system maps the space
birationally onto a projective variety of the same dimension. In
particular a big class has a nonzero section after a large multiple.
That is exactly what our assumption forbids for L + D. Hence L + D
sits on the boundary of the pseudoeffective cone: pseudoeffective but
not interior.
Third, duality of cones. On a smooth projective variety the
pseudoeffective divisor cone and the movable curve cone are dual: a
divisor class is pseudoeffective exactly when it pairs nonnegatively
with every movable curve class. A boundary point of a closed convex
cone has a supporting functional: a nonzero dual element pairing to
zero with it and nonnegatively with everything else. So there is a
nonzero real movable class alpha with
(L + D) . alpha = 0 .
Since L and D each pair nonnegatively with the movable class alpha, and
their sum pairs to zero, each pairs to zero:
L . alpha = 0 , D . alpha = 0 .
Write delta = D . alpha = 0 for later; in this special case delta
happens to be zero. (The slope bound below is stated for general
delta >= 0 because it is also used with a nonzero D in the induction,
when the boundary D has been pushed to the base.)
We need one more thing from alpha: that slopes of the tangent bundle,
measured against alpha, are not too negative. This is the content of
the next subsection.
A word on "slope". For a torsion-free sheaf E of positive rank and a
movable class alpha, the slope is
mu_alpha(E) = ( c_1(E) . alpha ) / rank(E) .
This is a single number: the average degree-per-rank in the direction
alpha. Over a curve it is the ordinary slope c_1(E)/rank. For a
vector bundle the Harder-Narasimhan (HN) filtration orders its pieces
by decreasing slope; mu_min,alpha and mu_max,alpha are the smallest and
largest slopes occurring. The tensor slope rule says
mu_alpha(E (x) F) = mu_alpha(E) + mu_alpha(F), and taking duals
flips the sign. We use these standard facts, which hold for real
movable classes by the movable-slope theory of Greb-Kebekus-Peternell.
4.2 THE SLOPE BOUND (THE COTANGENT IS NOT TOO NEGATIVE)
--------------------------------------------------------
The finite volume of h_J is what bounds how negative the cotangent
bundle can be. The bound passes through one analytic lemma about
foliations.
A saturated FOLIATION F of Z is a subsheaf of the tangent bundle T_Z
with torsion-free quotient and closed under the Lie bracket; think of it
as a consistent family of tangent subspaces glued holomorphically.
Let Q = T_Z / F be the quotient and P = det(Q^*) its determinant
line bundle (top exterior power of the dual). When F is induced by a
fibration, P is generically the pullback of the base canonical bundle;
it can differ along a multiple fiber, and that difference is the whole
point of the next lemma.
LEMMA 5 (The foliation lemma). Under the running assumptions on Z, D,
J, h_J (semipositive, finite adjoint volume), let F be an algebraically
integrable saturated foliation with 0 < rank F < dim Z. Then the line
bundle
D - P = D + det Q
admits a semipositive singular metric. In particular c_1(Q) + [D] is
pseudoeffective.
Proof idea. Resolve the fibration with the prepared-model diagram (the
normal equidimensional model Y-bar over S, plus a smooth resolution Y
used for integration; this is the model of Part 5, Lemma 9). On
the smooth resolution, pull back s_D to a top form eta with values in
K_Y + pi^*J; change of variables keeps its squared norm, against
pi^*h_J, of finite total mass. Over a base chart, restricting eta over
a frame of K_S gives, on general fibers, a square-integrable fiber
section of K_{Y/S} + pi^*J (finite by Fubini almost everywhere). The
relative Bergman kernel theorem of Berndtsson-Paun then gives a
semipositive metric on K_{Y/S} + pi^*J. Now compare with D - P.
On a dense open where pi is an isomorphism and the sheaves are locally
free, the conormal inclusion f^*Omega_S^1 -> Q^* has, at a horizontal
divisor, full rank with no divisorial zero. At a vertical divisor E the
model coordinates show f looks like s_1 = z_1^e (e = the order of the
base equation), s_i = z_i for 2 <= i <= dim S. In these coordinates
the determinant of the conormal inclusion vanishes to order exactly
e - 1 (the factor e z_1^{e-1}).
Here is the complete estimate. Fix a point near E at which z_1 is
nonzero, and let P be a small polydisc of fixed radius r in the free
fiber coordinates z_{b+1},...,z_n only, holding z_1 fixed. Because the
nearby fibers are locally a product in these coordinates, the same r
works for all base points close to E. On this polydisc the twisting
frame norm has a positive lower bound c > 0. Take a fiber section
a * xi whose squared L^2 norm is at most 1. Its coefficient as an
actual fiber top form is h = a / (e z_1^{e-1}), so
1 >= ||a xi||^2 >= c' |z_1|^{-2(e-1)} * integral over P of |a|^2
for a constant c' > 0. The mean-value (submean) inequality for the
holomorphic function a on the polydisc P then gives
|a(z)|^2 <= (1 / vol P) * integral over P of |a|^2. Combining the two,
|a(z)|^2 <= C'' |z_1|^{2(e-1)}
for every such section a. The Bergman weight psi is the supremum of
log|a(z)|^2 over exactly these sections, so
psi <= (e - 1) log|z_1|^2 + C near E,
with C independent of the base point. Therefore
psi - (e - 1) log |z_1|^2
is plurisubharmonic off E and locally bounded above near E, so it
extends plurisubharmonically across E; at horizontal divisors there is
no correction and the original metric already extends. The excluded
sets have codimension at least two, across which an upper-bounded psh
function extends uniquely. So D - P gets a semipositive metric. /End
proof idea.
The appearance of e - 1 is exactly the "ramification correction": the
multiple fiber would otherwise make the metric diverge, and subtracting
the divisor det Q (whose local generator is z_1^{e-1}) repairs it.
THEOREM 6 (Finite-volume cotangent bound). Under the running
assumptions, if dim Z > 0 then every nonzero real movable class alpha
satisfies
mu_min,alpha(T_Z) >= - D . alpha . (*)
Moreover, for every integer k >= 0 and every line bundle M with a
nonzero map M -> (Omega_Z^1)^{(x) k}, the line bundle -M + kD is
pseudoeffective. More generally, for any torsion-free subsheaf E of
rank r > 0 in (Omega_Z^1)^{(x) k}, the line bundle -det E + k r D is
pseudoeffective.
Proof of (*). Fix alpha and put delta = D . alpha >= 0. Semipositivity
of J gives c_1(T_Z) . alpha >= -delta. Let lambda_1 > ... > lambda_t be
the HN slopes of T_Z, ranks r_1,...,r_t. Suppose the minimum slope
lambda_t were < -delta. Then the total degree of the nonpositive-slope
pieces is at most r_t lambda_t < -delta. On the other hand the total
c_1(T_Z) . alpha >= -delta, so some piece has positive slope. Hence the
positive part F of the HN filtration is nonzero and proper, with
mu_min,alpha(F) > 0 and mu_max,alpha(T_Z / F) <= 0.
Now the bracket [.,.] mod F is a map (Lambda^2 F) -> T_Z/F. Because
the target has slope <= 0 while (Lambda^2 F) has slope >= 2 mu_min(F)
> 0 (tensor slope rule; if rank F = 1 it is zero), this map must vanish:
a nonzero map from a positive-slope sheaf to a nonpositive-slope
torsion-free sheaf is impossible. So F is a foliation. The
Campana-Paun algebraicity criterion then makes F algebraically
integrable. Lemma 5 gives c_1(T_Z / F) . alpha >= -delta. But
c_1(T_Z/F) . alpha is precisely the sum of the nonpositive HN slopes,
which we already bounded strictly below by -delta. Contradiction. So no
slope is below -delta, which is (*). (The argument also disposes of the
degenerate cases where the positive part would be zero or all of T_Z,
since a failure of (*) excludes both.)
Proof of the determinant assertion. Dualize and apply the tensor rule:
mu_max,alpha((Omega_Z^1)^{(x) k}) = -k mu_min,alpha(T_Z) <= k delta.
A nonzero map M -> (Omega^1)^{(x) k} is injective (target torsion-free),
so the degree of its image is <= k delta; a rank-r subsheaf E satisfies
c_1(E) . alpha <= k r delta. Hence (kD - M) . alpha >= 0, indeed
(k r D - det E) . alpha >= 0, for every movable alpha; cone duality
turns this into pseudoeffectivity of -M + kD and of -det E + k r D. /End
proof.
COROLLARY (A supporting class for the induction). Under the running
assumptions, assume dim Z > 0 and let L be a pseudoeffective line bundle
for which L + D is not big. Then there is a nonzero real movable class
alpha with
L . alpha = D . alpha = 0 ,
and for it mu_min,alpha(T_Z) >= 0. Moreover, if M -> (Omega^1)^{(x) k}
is nonzero and effective divisors N_i equivalent to M + m_i L are
given, then M . alpha = N_i . alpha = 0 for every i; every finite
nonnegative combination of D, N_1, N_2, ... has zero intersection with
alpha and is not big.
Proof. Same supporting-hyperplane argument as at the start of 4.1; a
boundary point L + D gets a supporting alpha annihilating it, and
nonnegativity splits it into L . alpha = D . alpha = 0. Theorem 6 gives
mu_min,alpha(T_Z) >= -D . alpha = 0 and M . alpha <= 0; effectivity of
N_i and the identity N_i . alpha = M . alpha give the reverse
inequality, so both are zero. An interior point of a full-dimensional
closed cone pairs strictly positively with every nonzero dual element,
so no such combination is big. /End proof.
The corollary is the honest entry point of the induction: it turns
"the engine fails" into "here is a curve direction along which
nothing grows", and that rigid direction is what forces the dimension
drop.
4.3 ONE DETERMINANT LINE WITH UNBOUNDED TWISTS
-----------------------------------------------
The next lemma converts the absence of holomorphic forms into infinitely
many sections of a single fixed line plus growing multiples of L.
LEMMA 7 (The determinant lemma). Let Z be smooth connected projective of
dimension d > 0 with H^0(Z, Omega_Z^q) = 0 for 1 <= q <= d. Let L have
a semipositive metric with locally bounded weights, and let alpha be a
nonzero movable class with mu_min,alpha(T_Z) >= 0. Then there is a line
bundle M with M . alpha <= 0, strictly increasing positive integers
m_i -> infinity, and nonzero sections
s_i in H^0(Z, M + m_i L) .
The line M is the determinant of a saturated subsheaf of Omega_Z^p for
some 0 <= p <= d.
Proof. Hodge symmetry turns the vanishing H^0(Z, Omega^q) = 0 (q > 0)
into H^q(Z, O_Z) = 0 for q > 0, so the holomorphic Euler characteristic
chi(O_Z) = 1. Riemann-Roch makes chi(K_Z + mL) a polynomial in the
integer m. Its constant term is chi(K_Z) = (-1)^d chi(O_Z) = (-1)^d by
Serre duality, which is nonzero. A polynomial with nonzero constant
term vanishes at only finitely many integers, so chi(K_Z + mL) is a
nonzero integer for all but finitely many positive m. Hence for each
such m at least one cohomology group H^q(Z, K_Z + mL) is nonzero; since
the finitely many q are fixed, a single q works for unbounded m.
For each such m, the metric h_L^m has a trivial multiplier ideal: its
weight is locally bounded, so every holomorphic germ is locally square
integrable. Hard Lefschetz with multiplier ideals (Demailly-Peternell-
Schneider, Theorem 2.1.1) then gives a surjection
H^0(Z, Omega_Z^{d-q} (x) mL) --> H^q(Z, K_Z + mL)
by wedging with the q-th power of a Kahler form. So a nonzero element
of the right side lifts to a nonzero map -mL -> U for U =
Omega_Z^{d-q}, on an increasing sequence of exponents.
Now stabilize over the generic point of Z. The images in the fiber
U_{C(Z)} of the chosen maps, as the cutoff grows, form a decreasing
chain in a finite-dimensional vector space; their dimensions and then
the subspaces themselves stabilize to a nonzero subspace W. Let G be
its saturated extension to a coherent subsheaf of U: saturate the
images of finitely many maps spanning W. Every map in a far enough
tail factors through G, because the quotient U / G is torsion-free and
the composite vanishes at the generic point.
Put r = rank G and M = (Lambda^r G)^{**}, a line bundle (reflexive
determinant on smooth Z). For each cutoff choose r generically
independent maps among the tail; their exterior product is a nonzero
section of M + mL, with m the sum of the r exponents. As the cutoff
goes to infinity, so does m. The maps and determinant extend across
codimension two into the line M. An increasing subsequence completes
the sections.
Finally M . alpha <= 0. If d - q > 0, antisymmetrization embeds U in
(Omega_Z^1)^{(x)(d-q)}; the tensor slope rule and mu_min,alpha(T_Z) >= 0
give mu_max,alpha(U) <= 0, hence M . alpha <= 0. If d - q = 0 then
U = O_Z and its nonzero saturated subsheaf is O_Z itself, so M = O_Z
and again M . alpha = 0. /End proof.
The point of "one fixed M" is uniformity: M does not move as m grows,
so the eventual linear systems below all live in a fixed finite list of
line bundles plus multiples of L.
4.4 THE MAXIMAL SYSTEM: A SMALLER BASE AND RANK ONE
----------------------------------------------------
Now choose the best fibration we can from the data. The key insight,
used twice, is that any ratio of two sections from the allowed systems
must be a function on the base, not a new transcendental coordinate.
LEMMA 8 (Maximal base and adjoint rank). Let Z be smooth connected
projective of positive dimension d. Let L, M be line bundles, D >= 0 an
integral divisor, and suppose there is a nonzero movable class alpha
with
L . alpha = D . alpha = 0 , M . alpha <= 0 ,
and nonzero sections s_i in H^0(Z, M + m_i L) for strictly increasing
positive integers m_i -> infinity. Then there is a smooth projective
base S with dim S < d, the prepared diagram of the next part, and an
integer k > 0, such that:
(i) some nonzero rational section rho of k pi^*L has no horizontal
poles on the normal model Y-bar;
(ii) putting J = -K_Z + D, the generic-fiber adjoint spaces have
rank one over the base field:
dim_{C(S)} H^0(Y_eta, (K_{Y/S} + pi^*(J + j k L))|_{Y_eta})
= 1 for every integer j >= 0.
If S is a point, assertion (i) alone already yields a nonzero section
of kL on Z.
Proof. Consider the nonempty complete linear systems of
B = u L + v M + w D , u, v, w in Z_{>= 0} .
Their degree against alpha is nonpositive. A divisor whose complete
system defines a generically finite map is big; but a big class is an
interior point of the pseudoeffective cone, and an interior point pairs
strictly positively with every nonzero dual (movable) element. Hence
every system above has image dimension < d.
Choose B_0 in this collection with maximal image dimension s < d, and
let V be the closure of its image. Let K be the relative algebraic
closure of C(V) inside C(Z); it is a finite extension of C(V) since
C(Z) is a finitely generated field extension of C(V). Take a smooth
projective model S of K (dimension s < d); the rational map
Z -> S has connected general fiber.
Section-ratio property. Every ratio of two nonzero sections in any one
allowed system belongs to K. To see it, combine that system with |B_0|
by taking products of sections; the joint rational map lands in a
system of the allowed form again. If the ratio were transcendental over
C(V), the joint image would have dimension at least s + 1, against
maximality of B_0. So the ratio lies in the relative algebraic closure,
which is K.
Resolve |B_0| before invoking the prepared-model construction. The
resulting smooth model carries an effective divisor R_0 and a line
bundle H_0 on S with
pi^* B_0 = f^* H_0 + R_0 ,
where H_0 is the pullback of O_V(1); if s > 0 it is big, since S -> V
is generically finite. Additional modifications keep this decomposition
by pullback.
Set k = m_2 - m_1 > 0 and rho = pi^*(s_2 / s_1). For i > 2 the ratio
( s_i^k s_1^{m_i - m_2} ) / ( s_2^{m_i - m_1} )
has numerator and denominator sections of one and the same allowed
system (same M coefficient m_i - m_1, same L coefficient
m_2 (m_i - m_1)), so it is a nonzero element of K. It has order zero at
every horizontal prime Q of the normal model, which gives
k ( ord_Q(pi^*s_i) - ord_Q(pi^*s_1) )
= ( m_i - m_1 ) ord_Q(rho) .
The left side is bounded below independently of i (each pi^*s_i is
regular). As m_i -> infinity, the right side could stay bounded below
only if ord_Q(rho) >= 0. This is (i). If S is a point, every prime is
horizontal, normality makes rho regular, and birational pushdown gives a
section of kL on Z; the rank assertion then follows from the section-
ratio property alone (K = C): the adjoint space identifies with
H^0(Z, D + jkL), the section s_D rho^j is nonzero, any two sections
differ by a constant, so the dimension is one for every j >= 0.
Assume s > 0. The identification
K_{Y/S} + pi^*(J + jkL) = K_{Y/Z} + pi^*(D + jkL) - f^*K_S
locates the exceptional divisor. Over the generic point of S a base
canonical frame is irrelevant; the Jacobian section, s_D, and rho^j give
a nonzero section of the right side. By (i) rho is regular over a
dense base open (drop the images of its finitely many vertical polar
divisors), so its pullback is regular there, and the adjoint space has
dimension at least one, including j = 0.
For the upper bound, take any two nonzero sections of that generic-
fiber space and, using the frame of K_S, view them as rational sections
of K_{Y/Z} + pi^*(D + jkL) with no horizontal poles. Their finitely
many vertical polar divisors can be cleared by one effective Cartier
divisor C on S (enclose their images, including any of codimension at
least two, and raise multiplicities so that f^*C dominates both polar
divisors). Since H_0 is big, some aH_0 - C has a nonzero section;
multiplying by it and by s_C makes both rational sections regular after
the common base twist a f^*H_0. Multiply both further by the canonical
section of a R_0; formula pi^*B_0 = f^*H_0 + R_0 then produces two
regular sections of
K_{Y/Z} + pi^*(D + jkL + a B_0)
with the same ratio. These descend to sections of D + jkL + a B_0 on Z:
K_{Y/Z} is effective and exceptional, and pi_*O_Y(K_{Y/Z}) = O_Z,
because a rational function with poles only on exceptional divisors is
regular off a codimension-two subset of smooth Z and hence everywhere.
The descended line is an allowed system, so by the section-ratio
property their ratio lies in K = C(S). Hence any two nonzero vectors in
the generic-fiber space are proportional, so its dimension is one for
every j >= 0, the zeroth degree included. /End proof.
The same maximal system supplied both inputs of the transfer: the
interpolation identity killed horizontal poles, while the big line H_0
on the base let us clear arbitrary vertical poles in the rank count.
4.5 INDUCTION ON DIMENSION (PROVING THE ENGINE)
-----------------------------------------------
We can now prove Theorem 4.
Proof. Induct on d = dim Z. For d = 0, Z is a point, every line bundle
is trivial, and the conclusion holds with a = 1, b = 0. Assume d > 0
and suppose that no aL + bD, with a > 0 and b >= 0, has a section.
The class L is pseudoeffective (its semipositive metric). The class
L + D is pseudoeffective but not big (big would give a section after a
large multiple). Corollary of 4.2 gives a nonzero movable alpha with
L . alpha = D . alpha = 0 and mu_min,alpha(T_Z) >= 0. Apply Lemma 7 to
get the fixed line M with M . alpha <= 0 and unbounded sections s_i;
then apply Lemma 8 to get the smaller smooth base S, a section rho of
k pi^*L with no horizontal poles, and adjoint rank one in every degree.
If the base S is a point, Lemma 8 gives a nonzero section of kL on Z,
i.e. of aL + bD with b = 0, contradicting the hypothesis.
Otherwise Proposition 10 (two-metric transfer), applied to that prepared
diagram and the two original metrics h_J and h_L, supplies a line A and
an effective integral divisor D_S on the smaller smooth projective base
S, with A carrying a semipositive metric of locally bounded weights and
-K_S + D_S carrying a semipositive metric of finite distinguished
adjoint volume. That is exactly the hypothesis pair Theorem 4 needs, one
dimension down.
The no-forms hypothesis passes to S. A holomorphic form on S pulls back
along the dominant rational map on its domain of definition; properness
of S lets that domain be taken with codimension-two complement; the
pullback extends because Omega_Z^q is locally free; and pullback is
injective in characteristic zero. Hence H^0(S, Omega_S^q) = 0 for
q > 0.
Since dim S < d, the induction gives integers a > 0, b >= 0 and a
nonzero section of aA + bD_S. The lifting conclusion of the transfer
proposition turns it into a nonzero section of a k L + c D on Z for some
c >= 0. Its L-exponent is positive, contradicting the hypothesis. So
the hypothesis was false, and Theorem 4 holds in dimension d. /End
proof.
That is the whole engine. Part 5 is the transfer proposition used at
the inductive step; Part 6 makes an invariant version for the torus;
Part 7 assembles Theorem 1.
================================================================
PART 5. THE TRANSFER STEP
================================================================
This part proves the proposition used at the inductive step of 4.5. It
is the heart of the paper: it turns the algebraic data of Lemma 8
(a smaller base S, a section rho with no horizontal poles, adjoint
rank one) into the two analytic objects the engine wants on the base,
namely a bounded-metric line A and a finite-volume corrected
anticanonical line -K_S + D_S.
We need two models at once. A NORMAL EQUIDIMENSIONAL model lets us talk
about divisor orders: every prime upstairs sits over a prime of the
base, so we can count zeros and poles. A SMOOTH RESOLUTION lets us
integrate: we need canonical forms and fiber integrals. The two models
are glued by the next lemma.
First, vocabulary. A prime divisor on the total space is HORIZONTAL if
it maps dominantly onto the base S, and VERTICAL otherwise. For a
morphism f: Y -> S of smooth varieties we write
K_{Y/S} = K_Y - f^* K_S
for the relative canonical bundle; on a smooth fiber it restricts to the
canonical bundle of the fiber.
LEMMA 9 (Prepared model). Let Z be smooth connected projective and S_0
an integral complex variety. Let Z -> S_0 be a dominant rational map
with connected general fiber. Then there is a smooth projective
birational model S of S_0 and a diagram
Y --r--> Y-bar --pi-bar--> Z
|
f-bar|
v
S
in which Y-bar is normal projective, Y is smooth projective, r and
pi-bar are birational, and f-bar is surjective and EQUIDIMENSIONAL
(every fiber has the expected dimension dim Z - dim S). The construction
may start above a prescribed birational model of Z. If an algebraic
torus acts on Z and the rational map is equivariant, the diagram is
equivariant with trivial action on S.
Every prime divisor on Y-bar either dominates S or maps onto a prime
divisor of S. At a general point of the strict transform in Y of a
prime Q over a base prime T, there are coordinates in which
f(t, x_2, ..., x_d) = ( t^e, x_2, ..., x_s ),
where e = ord_Q(f-bar^* T) > 0 is a positive integer and s = dim S.
Proof idea. Replace S_0 by a projective model; resolve the graph of the
rational map (starting above the prescribed model). Flatten the
resulting morphism by a base modification (Raynaud-Gruson
"platification"). Resolve the modified base and take the closure of the
component over the common dense open; normalize it to get Y-bar. The
dimension count makes f-bar equidimensional: a component over a base
subvariety of codimension two would have fiber dimension too large, so
all vertical divisors sit over divisors of S. The power-map coordinates
come from writing a local equation of T as t^e times a unit, taking a
holomorphic e-th root of the unit, and choosing the remaining base
coordinates among coordinates on Q. For an equivariant rational map use
the invariant graph; characteristic-zero equivariant resolution supplies
the smooth source, and base modifications preserve the trivial action.
/End proof idea.
Now the main proposition.
PROPOSITION 10 (Two-metric transfer). Let X, S, Y be smooth connected
projective varieties over C, and Y-bar normal projective. Suppose
r: Y -> Y-bar and pi-bar: Y-bar -> X are projective birational
morphisms and f-bar: Y-bar -> S is surjective and equidimensional with
connected general fiber. Put pi = pi-bar (composed with) r and
f = f-bar composed with r.
Let D be effective on X and let J = -K_X + D have a semipositive
singular metric h_J with finite adjoint volume, integral over X of
|s_D|^2_{h_J} < infinity. Let L have a semipositive metric h_L of
locally bounded weights. Suppose that for an integer k > 0 there is a
nonzero rational section rho of k pi^*L with no horizontal poles, and
that the adjoint spaces have rank one over the base field:
dim_{C(S)} H^0( Y_eta, (K_{Y/S} + pi^*(J + j k L))|_{Y_eta} ) = 1
for all j = 0, 1, 2, ...
Here eta is the generic point of S and Y_eta = Y x_S Spec C(S).
Then, after replacing k, rho by d k, rho^d for some d > 0, there exist a
line bundle A on S, an effective integral divisor D_S, and a nonzero
regular section
e in H^0( Y-bar, k pi^*L - f-bar^* A ),
with the following properties:
(i) A has a semipositive metric with locally bounded weights.
(ii) -K_S + D_S has a semipositive singular metric h_S with finite
adjoint volume, integral over S of |s_{D_S}|^2_{h_S} < infinity.
(iii) D_S is supported on base primes T such that EVERY prime
component of f-bar^*T is pi-bar-exceptional or lies in
Supp(pi-bar^*D).
(iv) If a > 0 and b >= 0 are integers and H^0(S, aA + bD_S) is
nonzero, then H^0(X, a k L + c D) is nonzero for some integer
c >= 0.
The point of (iii) is that the only poles D_S may introduce are ones
that, when pulled back, are either exceptions of the birational map
(which vanish on the smooth X) or sit over the already-allowed D. That
is what lets (iv) lift a base section to X without forbidden poles.
Proof of Proposition 10.
Step 0: the base is a point. Then every prime of Y-bar is horizontal.
So rho is regular by normality and descends (birational pushdown) to a
nonzero section of kL on smooth X. Take A trivial and D_S = 0; the
metric assertions are trivial on a point and (iv) is immediate. Assume
dim S > 0 from now on.
Step 1: divisorial normalization of rho. For a base prime T subset S,
define the number
c_T = min over components Q over T of
ord_Q(rho) / ord_Q(f-bar^* T) .
The Q's mapping onto T are finitely many and the denominators are
positive integers, and only finitely many c_T are nonzero. Replace
rho by a power rho^d and k by d k so that every c_T is an integer, and
set A = O_S( sum_T c_T T ). The rank-one hypothesis persists under this
replacement, because the new adjoint twists are a subsequence of the
old ones and still include j = 0.
Now divide rho by the pullback of the tautological rational section of
A. The result e is a rational section of k pi^*L - f-bar^* A. Its order
is nonnegative at horizontal primes (by the no-horizontal-poles
hypothesis) and at vertical primes (by the definition of the c_T as a
minimum). Since f-bar is equidimensional, the vertical primes are
exactly those mapping onto base primes, so these are all prime divisors
of Y-bar; normality then makes e a REGULAR section. For each base prime
T, a component Q attaining the minimum has order zero in e. We write e
also for its pullback to Y.
Step 2: fiber integrals and their curvature. Pull back the J-valued top
form, Jacobian included:
eta_Y = pi^* s_D in H^0(Y, K_Y + pi^* J),
mu_Y = |eta_Y|_{pi^* h_J}^2 .
Change of variables keeps the total mass of mu_Y finite; this is a
statement about measures, not about pointwise values on the polar set of
a singular metric.
On a chart U subset S, take a frame ds of K_S and a frame v of A. The
section e_v = e (f^* v) belongs to k pi^* L over f^{-1}(U). For
j = 0, 1, 2, ... set the j-th moment
u_j = ( eta_Y / f^* ds ) e_v^j ,
V_j(s) = integral over the fiber Y_s of |u_j|^2_{pi^*(h_J h_L^{jk})} ,
Phi_j = - log V_j .
Then u_j is a section of K_{Y/S} + pi^*(J + j k L). On a smooth fiber it
is a top form with coefficients, and V_j is its squared norm: a positive
density. Properness, regularity of e_v, and boundedness of h_L's weights
bound |e_v|^2 on inverse images of relatively compact base charts.
Since V_0 is the density of the finite measure f_* mu_Y, all V_j are
finite and positive almost everywhere on a suitable smooth-fibration
open.
Frame rule. Changing frames changes the moments by an exact amount:
v -> g v, ds -> h ds implies
Phi_j -> Phi_j - j log|g|^2 + log|h|^2 .
This is why the weights land on two DIFFERENT lines: -K_S at j = 0, and
A after dividing by j and taking a limit.
Each Phi_j is plurisubharmonic on the base. The relative Bergman theorem
of Berndtsson-Paun applies to the semipositive twisting metric
pi^*(h_J h_L^{jk}): u_j supplies a nonzero square-integrable adjoint
section on almost every smooth fiber, and the theorem produces a
semipositive relative Bergman metric. On a dense base open where base
change holds and the adjoint space has dimension one, u_j generates it;
for almost every fiber V_j < infinity, so the one-dimensional space of
holomorphic adjoint sections equals its full L^2 space, and hence the
Bergman weight upstairs is Phi_j(f) + log|u_j-hat|^2 almost everywhere.
Off the zero locus of the coefficient u_j-hat, that logarithm is
pluriharmonic, so subtracting it leaves a psh function upstairs.
Descending that function to the base gives the claim; here is the
argument in detail. Near a smooth-fiber point choose a product chart:
a base polydisc V and a fiber polydisc W, with f the projection
V x W -> V.
The upstairs identity reads
Phi_j(u) = G(u,w) - log|u_j-hat(u,w)|^2 ,
where G is the Bergman weight, plurisubharmonic on the chart. Fix a
fiber coordinate w away from the zero locus of u_j-hat. For this fixed
w the coefficient u_j-hat(.,w) is holomorphic in the base variable u,
so log|u_j-hat(.,w)|^2 is pluriharmonic in u. Then Phi_j(u) is a psh
function minus a pluriharmonic one in u, hence psh in u. A function is
psh exactly when it is subharmonic on every complex disc, so Phi_j is
psh on the base. These local representatives agree almost everywhere on
overlaps, since both equal the same fiber integral, and hence everywhere
by the submean inequality.
The base-change open can depend on j, so put all weights on one fixed
open S^circ where f is smooth and every fiber has a point at which both
eta_Y and e are nonzero. A psh weight is locally bounded above, so its
squared frame norm has a positive local lower bound even when the metric
is singular. On a small product chart around such a point this lower
bound for h_J, together with the bounds for h_L, gives
V_j(s) >= C_1 C_2^j , Phi_j(s) <= C + j C' ,
locally on S^circ, with positive constants C_1, C_2 independent of j.
Removable singularities extend each psh representative across its
additional exceptional base loci inside S^circ. All moment comparisons
can be used simultaneously because the union of the countably many
exceptional measure-zero sets still has measure zero. Almost-everywhere
upper bounds then hold for the psh representatives by the submean
inequality.
Step 3: the zeroth moment and the finite-volume metric. Call a base
prime T BAD if every component of f-bar^* T is pi-bar-exceptional or
lies in Supp(pi-bar^* D). Only finitely many primes are bad, since each
is the image of a member of the finite list of exceptional and boundary
divisors on Y-bar.
At a prime T outside S^circ take a component Q over it on Y-bar. At
general points of its strict transform in Y there are power-map
coordinates s_1 = t^l, s_i = x_i (2 <= i <= dim S), with l = ord of
f-bar^*T. Write the coefficient of eta_Y as t^q times a unit (q >= 0).
Dividing by ds contributes t^{-(l-1)}, so the relative top-form
coefficient has size comparable to |t|^{q - l + 1}. The metric lower
bound and integration over a fixed patch in the free fiber coordinates
give
V_0(s) >= C |s_1|^{2(q - l + 1)/l}
almost everywhere near a general point of T.
If T is NOT bad, choose a component Q that is nonexceptional over X and
outside Supp(pi-bar^* D). Then the birational Jacobian and s_D are
generically nonzero on Q, so q = 0, and -log V_0 is locally bounded
above there. At the bad primes choose sufficiently large nonnegative
integer coefficients for D_S. The weight
Phi_0 + log| s-hat_{D_S} |^2
is then locally bounded above at general points of every boundary prime
(here s-hat is the coefficient in a local frame of O_S(D_S)). These
weights are psh off the boundary, extend across the general divisorial
points of the boundary, and then extend across the remaining analytic
set of codimension at least two. The frame rule gives a metric on
-K_S + D_S.
Pairing that weight with s_{D_S} cancels the added factor, and the
resulting adjoint measure equals f_* mu_Y almost everywhere, so it has
finite total mass. That is property (ii), and the support condition on
D_S is property (iii).
Step 4: the high moments and the bounded metric on A. For l = 1, 2, ...
put on S^circ the upper semicontinuous regularization (star) of the tail
supremum
Psi = lim_{l -> infinity} ( sup_{j >= l} Phi_j / j )^* .
The common lower bound V_j >= C_1 C_2^j gives uniform local upper bounds,
so the regularized suprema are psh and decrease.
The limit does not collapse to -infinity. On any relatively compact base
chart U, properness and bounded weights of h_L give a single constant
C_U > 0 with
|e_v|^2_{pi^* h_L^k} <= C_U , V_j <= C_U^j V_0 .
The first bound holds on the whole inverse image of U: cover its compact
closure by finitely many line-bundle frames. For almost every s in
U cap S^circ, V_0(s) is finite and positive; dividing the second
inequality by j and taking negative logarithms gives Psi >= -log C_U.
So Psi is psh (not identically -infinity). Its frame transformation is
exact, although each normalized moment's is only asymptotic: on a
relatively compact overlap the frame rule changes Phi_j / j by
-log|g|^2 + (log|h|^2)/j, and the last term is at most C/l in absolute
value for every j >= l. The two tail suprema, and hence their
regularizations, differ from one another plus -log|g|^2 by at most C/l;
the decreasing limit is exactly the weight rule Psi -> Psi - log|g|^2
for the line A.
To extend Psi to the boundary, choose a component Q attaining the
minimum in the definition of c_T (this Q may differ from the one used
for the zeroth moment). Its defining property is that e is generically
nonzero on Q, so the local norm of e_v is bounded below on a suitable
patch. The same power-map calculation, now with the factor e_v^j, gives
V_j(s) >= C_1 C_2^j |s_1|^{2 b_T}
for a nonnegative integer b_T independent of j. The absence of zeros of
e on the chosen component is exactly what stops the exponent from
growing with j. Shrink the chart so that |s_1| < 1 and C_1 <= 1, and put
B(s) = -log C_1 - 2 b_T log|s_1| >= 0. For j >= l the negative logarithm
of the last display yields
Phi_j(s)/j <= -log C_2 + B(s)/j <= -log C_2 + B(s)/l .
The right side is continuous off T, so it bounds the regularization of
the tail supremum there too; letting l -> infinity gives
Psi <= -log C_2 throughout the punctured chart. Hence Psi is locally
bounded above near general points of each boundary prime, and extends
psh across them and then across codimension two. The lower bound from
the previous display was uniform on the whole base chart and survives
the extension. So the extended Psi is locally bounded above and below:
it is the weight of a semipositive metric on A with locally bounded
weights, which is property (i).
Step 5: lifting the section. We have the two metrics on the base. It
remains to check that the divisor used for finite volume creates only
permitted poles on X. Regard a nonzero section of aA + bD_S as a
rational section of aA with poles only on D_S. Pull it to the normal
model Y-bar and multiply by e^a. The result is a nonzero rational
section of a(k) pi^*L with poles only above bad primes. By their
definition every such prime divisor of Y-bar is pi-bar-exceptional over
X or lies above D.
At every prime of X outside D, the strict transform in Y-bar therefore
has no pole. The rational section downstairs extends there, and
exceptional divisors give no obstruction on the smooth (hence normal)
variety X. Its remaining finitely many pole orders are cleared by
multiplying by s_D^c for a sufficiently large integer c >= 0. This gives
a nonzero section of a k L + c D, which is property (iv). /End proof of
Proposition 10.
The transfer proposition is where the two metrics of Theorem 4 actually
come from. The zeroth fiber integral stores finite volume; the
normalized high moments store a bounded metric. Both are semipositive
by Berndtsson-Paun. And the divisor D_S is built precisely so that the
final lift cannot pick up a pole away from D. Part 6 makes the
invariant variant needed for the torus.
================================================================
PART 6. THE TORUS ACTION AND INVARIANT SECTIONS
================================================================
Part 3 promised that ordinary nonvanishing holds on the no-forms
factor Z. But Part 2 needs more: a section of K_Z^{-m} that is fixed by
the compact torus T of monodromy, so that it survives the deck action
and descends to the finite cover S. This part produces that invariant
section. The route is the same engine run on a rational quotient by the
torus.
6.1 WHY INVARIANCE, AND THE PLAN
---------------------------------
Recall from Part 2: Z carries an action of the compact connected real
torus T, and we may enlarge T to its Zariski closure G, a complex
algebraic torus, so that T-fixed anticanonical sections and G-fixed ones
coincide (Theorem 2, part (3)). We want a nonzero element of
H^0(Z, K_Z^{-m})^T.
The idea is to take the RATIONAL QUOTIENT of Z by G. Functions constant
on the dense orbits are the functions on the quotient base S. If the
general orbit is positive-dimensional, the quotient is a strictly
smaller no-forms variety. On each fiber, the fundamental vector fields
of the action supply a natural nonzero adjoint section, and any two
invariant ones differ by a quotient function. That gives the
invariant-rank-one condition. We then run an invariant version of the
transfer proposition (which uses the smoothness of the anticanonical
metric, not just a singular finite-volume metric) to reach the engine on
the base, exactly as in Part 4 but with the invariance carried along.
Two positivity inputs are involved and they are different. The ordinary
transfer (Part 5) uses Berndtsson-Paun's relative Bergman theorem with
singular coefficients, because there the whole adjoint space has rank
one. The invariant transfer uses Berndtsson's SMOOTH direct-image
positivity (Theorem 1.2 of his 2009 Annals paper), because there
averaging over the compact torus is an orthogonal holomorphic projection
onto the invariant line. Smooth coefficients are needed to get a
positive quotient metric on that summand.
6.2 THE INVARIANT TRANSFER (PROPOSITION 11)
--------------------------------------------
PROPOSITION 11 (Smooth invariant transfer). Use the varieties, maps,
line bundles, and section of Proposition 10, with D = 0. Let a compact
torus T act holomorphically and continuously on X, Y-bar, and Y, with
compatible holomorphic linearizations of the line bundles. Require the
maps r, pi-bar, f-bar to be equivariant, with trivial action on S.
Require rho and the J-valued adjoint section eta_Y = pi^*s_0 to be
invariant, and require h_J and h_L to be smooth, semipositive, and
T-invariant.
Replace the ordinary rank hypothesis dim = 1 by the INVARIANT rank
hypothesis
dim_{C(S)} H^0( Y_eta, (K_{Y/S} + pi^*(J + j k L))|_{Y_eta} )^T
= 1 for all j = 0, 1, 2, ...
Then all the conclusions of Proposition 10 hold, and the section on X
produced by its lifting is T-invariant.
Proof. Give the normalized line A the trivial action. The minima c_T in
the divisorial normalization use only base divisors, so dividing the
invariant rho by their rational section produces an INVARIANT regular
section e. The sections u_j of the moments are then invariant as well.
On a dense smooth-fibration open where the adjoint direct image is
locally free and its formation commutes with base change, Berndtsson's
theorem gives its smooth L^2 metric semipositive curvature. Its
hypotheses hold because the total space Y is projective, hence Kahler,
and each twisting metric pi^*(h_J h_L^{jk}) is smooth and semipositive.
Now average the holomorphic group action on this vector bundle over the
compact torus T. This gives a holomorphic projection onto the invariant
subbundle, which has rank one by assumption. The fiber integral is
invariant under the group, so the projection is orthogonal. The metric
on the invariant line is therefore its quotient metric and is
semipositive. Since u_j is a nonzero local generator of that line,
-log V_j is a psh weight with a representative on the base.
This is the ONLY positivity step that changes. The fixed-open bound,
the zeroth-moment finite-volume extension, the high-moment envelope, and
both bounds through codimension two now apply exactly as in the ordinary
proof of Proposition 10. They give the two base metrics. Every base
section is invariant because the base action is trivial. The rational
lifting multiplies its pullback by e^a, which is invariant, so the lift
remains invariant. With D = 0 the support condition permits only
exceptional poles, and the final extension on X is an invariant
holomorphic section. /End proof.
The difference between Propositions 10 and 11 is substantive: the
ordinary one may carry a singular volume metric, because the adjoint
direct image has rank one and Berndtsson-Paun applies; the invariant one
uses smooth coefficients so that averaging over T gives a positive
quotient metric on an orthogonal summand.
6.3 NATURAL INVARIANT NONVANISHING (THEOREM 12)
-----------------------------------------------
THEOREM 12 (Natural invariant nonvanishing). Let Z be smooth connected
projective over C with H^0(Z, Omega_Z^q) = 0 for every q > 0. Suppose
L = -K_Z has a smooth Hermitian metric of semipositive curvature. Let a
compact connected real torus T act continuously on Z by holomorphic
automorphisms, and give L the natural linearization induced by the
differential of the action. Then
H^0(Z, m L)^T is nonzero for some integer m > 0.
Proof. If Z is a point, the natural action on the canonical line is
trivial and the constant section proves the claim. Otherwise replace T
by its image in Aut(Z). The no-forms hypothesis and Hodge symmetry give
H^1(Z, O_Z) = 0.
First put the action in the algebraic category. The argument in the
proof of Theorem 2 applies verbatim to this action: connectedness of T
preserves the first Chern class of a very ample line; the vanishing of
H^1(Z, O_Z) makes that line invariant up to isomorphism, so T has a
canonical projective representation; that representation embeds T in a
compact diagonal torus, whose Zariski closure G is an algebraic torus
acting on Z. Natural sections are T-invariant exactly when they are
G-invariant.
Average the given metric over T by averaging local weights, using the
natural linearization. Differences of metric weights are global
functions, so these averages patch to a smooth invariant metric, whose
curvature stays semipositive. Take J = L and D = 0. The canonical
section 1 of K_Z + L = O_Z is invariant for the natural action, and its
metric pairing is a smooth positive volume form, of finite total mass on
the compact Z.
Case 1: the general G-orbit has dimension zero. Connectedness of G then
makes the action trivial (the orbit map is constant on a dense open,
hence on all of Z). Theorem 4, with D = 0 and J = L, gives the needed
section, and it is G-invariant because everything is.
Case 2: the general orbit has dimension r > 0. Take the Rosenlicht
rational quotient. Its function field is C(Z)^G; over a dense invariant
open its geometric fibers are exactly the orbits. Because G is
connected, the general orbit is geometrically integral; thus the
invariant field is relatively algebraically closed in C(Z) and the
quotient has connected general fiber. Choose the prepared equivariant
model of Lemma 9:
Y --r_Y--> Y-bar --pi--> Z, f: Y -> S , f-bar: Y-bar -> S ,
with trivial base action. The generic fiber has dimension r and a
geometrically dense G-orbit. Shrink a base open so that f is smooth
there, its fibers have a dense orbit, and pi agrees birationally with
the quotient construction.
Build the relative section. Choose r elements of the Lie algebra of G
whose fundamental vector fields v_1, ..., v_r are generically
independent (possible because the general orbit has dimension r). They
are invariant, since G is commutative, and tangent to the fibers of f.
On the smooth-fibration open their wedge is a regular, generically
nonzero section of -K_{Y/S}. Tensor it with the pullback of a rational
frame of -K_S, regular after shrinking the base open. This gives an
invariant anticanonical section on Y over that open. The determinant of
dpi is the natural morphism -K_Y -> pi^*(-K_Z); its image is a nonzero
invariant rational section of pi^*L, regular over the same open.
This rational section defines a section rho of pi-bar^*L on Y-bar with
no horizontal poles: r_Y is an isomorphism at the generic point of each
prime divisor of the normal Y-bar, every horizontal prime meets the base
open where the section is regular, and the birational identification of
function fields gives nonnegative order there.
The generic-fiber adjoint spaces are finite-dimensional C(S)-vector
spaces carrying algebraic representations of the split torus G_{C(S)}.
They decompose into character spaces. A character of G is trivial on
the Zariski-dense subgroup T exactly when it is the zero character.
Consequently the T-fixed and G-fixed subspaces agree, not merely on Z
itself but on every adjoint space.
For each j >= 0 the section (pi^*1 / f^*ds) rho^j is a nonzero
invariant section of K_{Y/S} + pi^*((j+1)L) on the generic fiber. The
form pi^*1 includes the effective birational Jacobian, and rho has no
horizontal pole, so it is regular. Any two nonzero invariant sections of
this line have an invariant rational function as their ratio. The
defining property of the quotient gives C(Y)^G = C(Z)^G = C(S), so that
ratio lies in C(S), and the space has dimension one over C(S) for every
j >= 0, including j = 0. This is exactly the invariant-rank hypothesis
of Proposition 11.
Apply Proposition 11 with k = 1 (increasing powers there clear the
vertical minima). It supplies a line A and an effective divisor D_S on
S, with a bounded semipositive metric on A and a finite-volume
semipositive metric on -K_S + D_S. Holomorphic forms on S pull back
injectively to Z, as in the last part of Part 4's induction, so S has no
positive-degree holomorphic forms. Theorem 4 then gives a nonzero
section of aA + bD_S with a > 0, b >= 0. Invariant lifting produces a
nonzero invariant section of a positive power of L on Z. Since D = 0,
every permitted pole upstairs is exceptional over Z and disappears on
the normal pushdown. When S is a point the same conclusion is the
point-base case of Proposition 11. This completes Case 2 and the
theorem. /End proof.
Part 7 assembles the whole proof of Theorem 1 from these pieces.
================================================================
PART 7. DESCENT TO X, AND THE PROOF OF THE MAIN THEOREM
================================================================
7.1 ASSEMBLING THEOREM 1
-------------------------
We have built, or assumed, every piece. Here is the full assembly.
Proof of Theorem 1 (Anticanonical nonvanishing). Let X be a smooth
connected projective complex variety whose anticanonical bundle -K_X
carries a smooth Hermitian metric of semipositive curvature.
Step 1: structure. Apply Theorem 2 (Part 2). It gives a connected finite
etale cover nu: S -> X, an integer a >= 0, and smooth connected
projective factors F and Z, with the common universal cover
X-tilde = C^a x F x Z ,
a full lattice Lambda with S = (C^a x F x Z) / Lambda, and the deck
action lambda . (z, f, v) = (z + lambda, f, rho(lambda) v). The
Ricci-flat factor F has a nowhere-zero parallel holomorphic frame eta_F
of K_F, invariant under everything. The flat factor C^a has the
constant frame eta_0 = dz_1 ^ ... ^ dz_a, invariant under translations.
The remaining factor Z is compact, simply connected, has no holomorphic
forms of positive degree, and has smoothly semipositive anticanonical
bundle. Its residual action has compact connected closure a real torus
T, and T extends to an algebraic torus G with T Zariski dense.
Step 2: the invariant section on Z. The factor Z satisfies exactly the
hypotheses of Theorem 12 (Part 6): no positive-degree forms, smooth
semipositive -K_Z, and a compact connected torus T acting with the
natural linearization. So there is some m > 0 and a nonzero invariant
section
s_Z in H^0(Z, K_Z^{-m})^T .
(If Z is a point, take m = 1 and s_Z = 1.)
Step 3: descend to the finite cover S. Use the product identification of
Theorem 2, part (4):
K_{X-tilde}^{-m}
= pr_0^* K_{C^a}^{-m} (x) pr_F^* K_F^{-m} (x) pr_Z^* K_Z^{-m} .
Form the tensor product
eta_0^{-m} (x) eta_F^{-m} (x) s_Z .
Each factor is a nonzero holomorphic section of the corresponding
pulled-back canonical power. The first two frames are invariant under
the deck lattice (translations of C^a do not move eta_0; the action on F
is trivial, so it does not move eta_F). The third factor is invariant
under the closure of the deck action on Z, which is T. Hence the whole
product is invariant under every deck transformation. It therefore
descends to a nonzero holomorphic section of K_S^{-m}. By projectivity,
a holomorphic section is algebraic.
Step 4: the etale norm down to X. Lemma 3 (Part 2) says that a finite
etale cover of degree e sends every nonzero section of K_S^{-m} to a
nonzero section of K_X^{-m e} by the sheetwise product (the norm). Apply
it to the section from Step 3. The result is a nonzero section of
K_X^{-m e} = (-K_X)^{m e} , with m e > 0 .
That is exactly H^0(X, -M K_X) nonzero for the positive integer
M = m e. This proves Theorem 1. /End proof.
Notice what each part contributed:
- Part 2 (Theorems 2 and Lemma 3) split X into factors, recorded the
deck action, and set up the norm that returns a section to X;
- Part 3 stated the engine, and Part 4 proved it by induction;
- Part 5 supplied the transfer proposition used at the inductive
step of Part 4;
- Part 6 made the invariant version needed to beat the torus
monodromy, proving Theorem 12;
- this part glues them into Theorem 1.
7.2 GLOSSARY
-------------
Complex manifold: a space locally like an open piece of C^n, glued by
holomorphic maps.
Projective variety: a set cut out inside projective space P^N by
homogeneous polynomial equations. Compact and algebraic.
Smooth, connected: no corners or self-crossings; one piece.
Holomorphic line bundle L: a holomorphically varying family of
one-dimensional complex vector spaces over X. The trivial one is X x C.
Section of L: a holomorphic choice of one vector in the fiber above each
point. H^0(X, L) is the vector space of all of them.
Divisor: the zero set of a nonzero section of a line bundle; a formal
sum of hypersurfaces. Effective (D >= 0): a genuine zero set, not signed.
Canonical bundle K_X: the line bundle whose fiber at a point is the
space of holomorphic n-forms there, n = dim X.
Anticanonical bundle -K_X: its dual, equal to det T_X, the top exterior
power of the tangent bundle.
Semipositive metric of curvature: a way to measure lengths in the fibers
whose curvature form is nonnegative. It makes the Chern class lie in the
closure of the positive cone.
Nef: a numerical class that is a limit of positive (ample) classes;
equivalent to admitting metrics of semipositive curvature.
Kahler metric: a Hermitian metric with a closed, symmetric curvature
form. Every projective variety has plenty.
Ricci curvature: within a Kahler metric, the (1,1)-form measuring how
the metric's volume changes.
Yau's prescribed-Ricci theorem: any smooth closed positive (1,1)-form
representing c_1(X) is the Ricci form of some Kahler metric.
Universal cover X-tilde: the simply connected space that covers X, with
X = X-tilde / pi_1(X).
Deck group pi_1(X): the group of covering transformations of X-tilde
over X.
Etale (unramified) cover: a finite cover that is locally an isomorphism,
with distinct sheets.
Norm of a section (over an etale cover of degree e): over each small
open set, multiply the sheetwise coefficient functions. It turns a
section on the cover into a section of the e-th tensor power downstairs.
Torus (real): a compact connected abelian Lie group, like (S^1)^r.
Algebraic torus: a group isomorphic to (C^*)^r acting algebraically.
Zariski closure: the smallest algebraic subvariety containing a set;
here it turns the compact torus T into an algebraic torus G.
Rational quotient (Rosenlicht quotient): for a group acting on a
variety, the variety whose function field is the field of invariant
rational functions; its orbits are the group orbits on a dense open.
Movable curve class: an element of the cone dual to the pseudoeffective
divisor cone; think "a curve direction along which divisors can move".
Pseudoeffective cone: the cone of divisor classes that are limits of
classes of effective divisors.
Big: a class with a positive multiple whose sections give a generically
finite map to a variety of the same dimension; bigness is the interior
of the pseudoeffective cone.
Slope mu_alpha(E): c_1(E) . alpha divided by rank(E); the average
degree-per-rank in the direction alpha.
Harder-Narasimhan filtration: the canonical way to split a sheaf into
pieces of strictly decreasing slope.
Foliation: a bracket-closed subsheaf of the tangent bundle; a
holomorphically varying family of tangent subspaces. Algebraically
integrable when it comes from a rational fibration.
Relative canonical bundle K_{Y/S}: K_Y minus f^*K_S; on smooth fibers it
is the fiber's canonical bundle.
Plurisubharmonic (psh) function: a function convex in the complex sense;
locally an upper-semicontinuous limit of logarithms of absolute values
of holomorphic functions. The weight of a semipositive singular metric.
Singular metric: a metric whose weight may fail to be smooth but is psh.
Multiplier ideal: the ideal of germs that are locally L^2 for a singular
metric; trivial when the weight is locally bounded.
Adjoint measure: the positive measure |s_D|^2_{h_J} obtained by pairing
the J-valued top form s_D with its conjugate via h_J.
Berndtsson-Paun relative Bergman theorem: the relative Bergman kernel of
a fibration with a semipositive twisting metric is semipositive.
Berndtsson's theorem (smooth direct image): for a smooth family with
smooth semipositive twisting metric, the L^2 metric on the direct image
of the relative canonical times the twist is semipositive.
Hard Lefschetz with multiplier ideals: wedging with a Kahler form
surjects twisted holomorphic forms onto cohomology, provided the
multiplier ideal is trivial.
Riemann-Roch: chi(E) is a polynomial determined by Chern characters and
Todd classes; in one variable it is a degree-n polynomial in m.
Hodge symmetry: H^q(Z, Omega^p) = H^p(Z, Omega^q); it equates vanishing
of forms with vanishing of structure-sheaf cohomology on a compact
Kahler manifold.
Holomorphic Euler characteristic: chi(O_Z) = sum_q (-1)^q dim H^q(Z,O_Z).
7.3 WHERE THE IDEAS COME FROM
------------------------------
The geometric reduction (split the universal cover, record the deck
action) is the Demailly-Peternell-Schneider structure theorem, refined by
Campana-Demailly-Peternell for rational connectedness. Yau's
prescribed-Ricci theorem turns the anticanonical metric into a
nonnegatively curved Kahler metric. The positive-multiple form of the
question is Yau's Problem 75.
The slope argument is the foliation strategy of Campana-Paun and the
movable-slope calculus of Greb-Kebekus-Peternell, following Ou and
Lazic-Matsumura-Peternell-Tsakanikas-Xie. Hard Lefschetz with multiplier
ideals is due to Demailly-Peternell-Schneider. The transfer of the two
metrics uses the relative Bergman theorem of Berndtsson-Paun (singular
version, for the ordinary induction) and Berndtsson's smooth
direct-image positivity (for the invariant version). The rational
quotient is Rosenlicht's. The etale norm is the standard norm for
sections of line bundles over finite covers.
The statement proved here is the smooth-semipositive, projective,
positive-multiple anticanonical nonvanishing theorem. Its two smallest
witnesses are the Enriques surface E (m = 2 needed, because -K_E is a
nonzero two-torsion class) and the product E x P^1 (m = 2, with a
non-torsion anticanonical class). Both are checked in Part 1.
7.4 WHY THE PROOF IS SHAPED THIS WAY
-------------------------------------
A first attempt might hope that a smoothly semipositive anticanonical
bundle is itself effective (m = 1). The Enriques example forbids it.
A second attempt might hope that a finite cover splits X as a product,
so the section is just a product of factor sections. Infinite monodromy
forbids that too: the finite cover S need not be a product, and a
section descends only when it is invariant under the deck action.
So the proof separates the two difficulties. The geometric part (Part 2)
records the action precisely and supplies the two canonical frames plus
the norm. The analytic part (Parts 3-6) produces a section that is
invariant under the residual torus, by running a finite-volume
induction on a rational quotient. The induction is the engine; the
transfer (Part 5) is how the engine passes data to a smaller base; the
slope bound (Part 4.2) is what makes the boundary behave; and the
invariant refinement (Part 6) is what lets the section survive the deck
group.
Every ingredient is necessary: drop smoothness of the anticanonical
metric and the finite-volume measure may blow up; drop projectivity and
the slope and Lefschetz tools are unavailable; drop the no-forms
hypothesis and the Euler characteristic is not one. With all of them,
the section exists after some positive power, which is Theorem 1.
================================================================
END OF DOCUMENT
================================================================