METRIC DESCENT AND RANK-PRESERVING CONTRACTIONS
A clear rewrite for students
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0. WHAT THIS IS ABOUT
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This is advanced geometry, but the main idea is simple enough to state.
We have a smooth complex projective variety X. Its anticanonical bundle is
-K_X = det(T_X). A Hermitian metric on -K_X is written locally as
|e|^2 = exp(-phi)
where e is a local frame and phi is a real function. The metric is
semipositive if phi is plurisubharmonic (psh). It has bounded weights if
phi is bounded above and below on small coordinate charts.
The question is: when does positivity of -K_X force a nonzero section of
some positive multiple -mK_X? Equivalently, when does a line bundle with a
positive metric have enough holomorphic sections?
The paper proves several reusable statements. The most important are:
Theorem A (toroidal volume metric).
Pushforward of a smooth semipositive anticanonical volume along an
equidimensional toroidal fibration gives an integrable semipositive metric
on a corrected anticanonical line of the base. The correction is explicit
and disappears on return to X.
Theorem B (two-metric transfer).
A finite-volume metric on -K_X+D and a bounded semipositive metric on a
line L can be pushed to a lower-dimensional base. Sections on the base
come back to X with poles only on D.
Theorem C (rank-preserving contraction).
If a generic fiber has an invariant section and its adjoint section space
has rank one, then either a global invariant section already exists, or
the base can be contracted to a smaller base while keeping both properties.
Theorem D (allowed-pole nonvanishing).
If X has no positive-degree holomorphic forms and -K_X+D has finite
volume, then for every line L with bounded semipositive weights there are
integers a>0 and c>=0 with H^0(X, aL+cD) nonzero.
Theorem E (invariant anticanonical section).
If -K_X has a smooth semipositive metric and a torus acts naturally, then
some positive anticanonical multiple has a nonzero invariant section.
The proof is a dimension induction. At each step we either find a section
immediately, or we shrink the base dimension while preserving exactly the
data needed for the next step.
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1. NOTATION AND BASIC FACTS
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All varieties are smooth connected projective varieties over C unless we
explicitly say "normal model" or "resolution".
Line bundles are written additively. Thus L+M means L tensor M, and mL
means the m-th tensor power. The canonical bundle is K_X. The
anticanonical bundle is -K_X.
A divisor is a finite formal sum of prime divisors with integer or rational
coefficients. D >= 0 means D is effective. A section of a line bundle has
a divisor of zeros and poles. "Poles only on D" means that after multiplying
by a section of a sufficiently large multiple of D, the section becomes
regular.
A singular Hermitian metric on a line bundle has local squared frame norm
exp(-phi). It is semipositive if phi is psh. It has bounded weights if
phi is locally bounded above and below. If A=-K_X+D, the canonical section
s_D is an A-valued top form. The measure
mu = |s_D|_h^2
is the distinguished adjoint measure. The finite-volume assumption is
integral_X mu < infinity.
This does not follow from semipositivity alone. It is a real hypothesis.
If f: W -> Y is a fibration and L is a line bundle on W, the pushforward
f_*(K_{W/Y}+L) is the adjoint direct image. Its rank is the number of
independent adjoint sections on a generic fiber. Rank one is the key
hypothesis that turns an integral over fibers into a squared norm of a
single line.
Two standard facts are used repeatedly.
(1) Direct image positivity (Berndtsson; Paun-Takayama).
If the coefficients have semipositive curvature and the adjoint direct
image has rank one on a smooth family, then minus the logarithm of its
squared norm is psh.
(2) Psh extension.
A psh function that is locally bounded above outside a proper analytic
set extends across that set. Integrability is checked by computing
exponents in normal-crossing coordinates.
We also use the following convention. A rational line bundle is handled by
clearing denominators: choose a positive integer q such that qL is an
honest line bundle, work with qL, and divide the metric weight by q.
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2. THE TOROIDAL VOLUME METRIC
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This is Theorem A. It is the cleanest place to see why pushforward volume
produces a metric.
Set-up. Let X and Y be smooth projective, with dim Y = d > 0. Let
L = -K_X carry a smooth semipositive metric h, with curvature alpha. Let
p: W -> X, f: W -> Y
be projective morphisms. W is normal and integral. p is birational.
f is surjective, equidimensional, and toroidal for strict toroidal
embeddings. Choose a smooth projective resolution rho: Z -> W, and put
pi = p rho, g = f rho, E_Z = K_{Z/X}.
Assume the full generic adjoint rank at exponent zero is one:
rank g_* O_Z(E_Z) = 1.
For a prime divisor Q in Y and a prime E in W dominating Q, define
a_E = ord_E K_{W/X},
e_E = ord_E f^*Q.
Here K_{W/X} = K_W - p^*K_X is the relative canonical divisor. The
number a_E is the Jacobian order over X. The number e_E is the pullback
multiplicity of the base prime Q.
Define the correction divisor
a_Q = max( 0, min_{f(E)=Q} ( (1+a_E)/e_E - 1 ) ),
A_0 = sum_Q a_Q Q.
Theorem A. With this notation:
(i) A_0 is an effective rational divisor with finite support.
(ii) f^*A_0 is p-exceptional, so it disappears on pushing forward to X.
(iii) The rational line bundle -K_Y + A_0 has a semipositive singular
metric whose squared frame norms are locally integrable, without
multiplying by any extra vanishing divisor.
(iv) The metric is smooth on a nonempty Zariski open subset of Y.
(v) If on one ordinary open patch the coefficient curvature dominates
a positive pulled-back base form, then the metric is strictly
positively curved on a nonempty ordinary open subset of Y.
Proof of (i) and (ii). At a general point of a prime of W, the variety is
smooth and the Jacobian is holomorphic, so a_E >= 0. If E is not
exceptional over X, then p is an isomorphism at the general point of E,
so a_E = 0.
An equidimensional morphism cannot send a divisor into a set of
codimension at least two. Hence every vertical prime of W dominates a
prime of Y. If a_Q > 0, then the minimum in its definition is positive.
Thus every prime E over Q has a_E > 0, so E is p-exceptional. Therefore
f^*A_0 is p-exceptional. Only finitely many p-exceptional primes exist,
so A_0 has finite support. Rationality is clear from the integer orders.
Proof of (iii): the discrepancy lemma. Put
Theta = -K_{W/X} + f^*A_0.
We first show that Theta <= 0 and that the pair (W,Theta) is sub-klt.
"Sub-klt" means that every coefficient of the crepant pullback of Theta
to a log resolution is less than one. Negative coefficients are allowed.
Let E dominate Q. If a_Q > 0, then
coeff_E Theta = -a_E + e_E a_Q
<= -a_E + e_E ( (1+a_E)/e_E - 1 )
= 1 - e_E
<= 0.
If a_Q = 0, then coeff_E Theta = -a_E <= 0. At a horizontal prime,
the coefficient is also -a_E <= 0. Thus Theta <= 0.
Let D_W be the full reduced toroidal boundary. The toroidal pair
(W,D_W) is log canonical, K_W+D_W is rationally Cartier, and W-D_W is
smooth. Also
K_W + Theta = p^*K_X + f^*A_0.
Therefore
B = D_W - Theta
is an effective rational Cartier divisor whose support contains D_W.
For every divisorial valuation v over W, log discrepancies satisfy
A_{W,Theta}(v) = A_{W,D_W}(v) + v(B).
The first term is nonnegative because (W,D_W) is log canonical. The
second term is nonnegative because B is effective. If the first term is
zero, then the center of v lies in D_W. Since the support of B contains
D_W, the second term is then strictly positive. Hence every discrepancy
is strictly positive, so (W,Theta) is sub-klt.
Now take a smooth model Z' -> W on which K_{Z'/X} and the pullback of A_0
have simple normal crossings. The crepant pullback of Theta is
Theta_{Z'} = (f tau)^* A_0 - K_{Z'/X}.
The sub-klt statement says exactly that every coefficient of Theta_{Z'}
is less than one. Equivalently,
coeff_F( K_{Z'/X} - (f tau)^* A_0 ) > -1
at every prime F of Z'. This is the integrability exponent estimate.
Proof of (iii), continued: positivity and integrability. On a nonempty
Zariski open U where g is smooth and base change holds, the canonical
section s_{E_Z} gives an isomorphism
g_*(K_{Z/Y} + pi^*L)|_U = -K_Y|_U.
Indeed, the left side is g_*O_Z(E_Z) tensor -K_Y, and the rank is one.
Let Omega_h be the smooth positive volume form on X determined by h.
Write
g_* pi^* Omega_h = H(y) dV_y
in local coordinates y on U. By the identification above, H is the
squared L^2 norm of a frame of -K_Y. Berndtsson's direct image theorem
says that -log H is psh on U. Thus -K_Y has a semipositive metric there.
If the horizontal patch hypothesis holds, the same curvature formula
shows that -log H is strictly psh at one point, hence on an ordinary
neighborhood.
It remains to cross the boundary and to prove integrability. Let Q be a
prime component of Y-U. Choose a prime E over Q attaining the minimum
in the definition of a_Q. At general points of E and Q, the map is locally
(u_1,...,u_n) -> (u_1^{e_E}, u_2,...,u_d),
with Q = {y_1 = 0}. The pullback volume is bounded below by a positive
constant times |u_1|^{2a_E} times ordinary volume. Integrating over a
fixed polydisc in the fiber variables and changing variables
y_1 = u_1^{e_E} gives
H(y) >= C_Q |y_1|^{2 b_Q},
where
b_Q = min_{f(E)=Q} ( (1+a_E)/e_E - 1 ).
By construction a_Q = max(0,b_Q). Hence b_Q - a_Q <= 0. Therefore
Phi = -log H + sum_Q a_Q log|t_Q|^2
is bounded above near a general point of Q. This is exactly the psh
extension condition. Extending across all boundary primes and then across
the remaining codimension-two set gives a semipositive metric on
-K_Y+A_0.
For integrability, use the projection formula on the smooth locus:
exp(-Phi) dV_y
= g_* ( prod_Q |t_Q o g|^{-2a_Q} pi^* Omega_h ).
On a normal-crossing resolution Z' -> W, the right-hand measure has local
form
positive bounded factor * prod_j |z_j|^{2 gamma_j} dV_z,
with
gamma_j = coeff_{z_j=0}( K_{Z'/X} - (g')^*A_0 ) > -1.
Each one-variable integral integral_0^eps r^{2 gamma_j+1} dr is finite.
Thus the measure is locally integrable on Z'. Properness makes its
pushforward locally finite on Y. Therefore exp(-Phi) is locally
integrable. This proves Theorem A.
Corollary A.1. For the correction A_0, kappa(Y,A_0)=0. More precisely,
for every positive integer m with mA_0 integral,
H^0(Y, O_Y(mA_0)) = C s_{mA_0}.
Proof. Divide a section by the canonical section s_{mA_0}. The quotient
is a rational function q on Y with poles bounded by mA_0. Pull it back
to W. Its only possible poles are p-exceptional. Since C(W)=C(X) and X
is smooth, a rational function with no pole along any prime of X is
regular on X. It is constant because X is connected and projective.
Dominance of f makes q constant too. Hence the only section is the
canonical one.
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3. TWO-METRIC TRANSFER
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This is Theorem B. It explains how a finite-volume metric and a bounded
metric move to a smaller base. The two metrics play different roles:
- the finite-volume metric controls the allowed poles;
- the bounded metric controls the section we want to return to X.
Set-up. Let V be smooth connected projective. Let D >= 0 be an integral
divisor and put A = -K_V + D. Suppose A has a semipositive singular metric
h_A with
integral_V |s_D|_{h_A}^2 < infinity.
Let L be a line bundle with a semipositive metric h_L whose weights are
locally bounded. Fix a dominant rational map from V with connected generic
fibers. Use a normal equidimensional model
p: W -> V, f: W -> Y,
with W normal and Y smooth projective, and choose a smooth resolution if
needed. Let rho be a nonzero rational section of kL, k>0, with no
horizontal poles on the model.
On a dense base open set define
g_0(y) d(lambda_y) = f_* p^* mu,
S_rho(y) = ess sup_{W_y} |p^*rho|_{h_L^k}^2,
where mu = |s_D|^2. Assume that -log g_0 and -log S_rho have psh
representatives on this open set.
Lemma 3.1 (Transfer with controlled poles). Under these hypotheses, after
clearing a positive integer denominator, there are:
- a line bundle B on Y with a bounded semipositive metric;
- an effective integral divisor D_Y on Y;
- a positive integer b;
- a regular nonzero map
f^*B -> bk p^*L.
The line bundle -K_Y + D_Y has a semipositive metric whose distinguished
adjoint measure has finite mass. Every nonzero section of aB + cD_Y,
where a>0 and c>=0 are integers, produces a nonzero rational section of
abkL with poles only on D. If the data and rho are invariant for a torus
acting trivially on Y, then the produced section is invariant.
Proof. The proof has two separate normalizations.
First normalize the section. For a prime T of Y put
d_T = min_{Q -> T} ord_Q(p^*rho) / ord_Q(f^*T),
B_0 = sum_T d_T T.
The orders are taken on the normal model. Only finitely many d_T are
nonzero. Equidimensionality and the absence of horizontal poles give
div(rho) - f^*B_0 >= 0.
This is an effective rational Cartier divisor. Choose b>0 clearing the
coefficients of B_0 and set B = bB_0. Normality extends the normalized
section in codimension one to a regular morphism
f^*B -> bk p^*L.
Over each base prime T, a component attaining the minimum has no vanishing
there. This is what makes the normalized norm bounded below.
Second construct the bounded metric on B. Let l be a local meromorphic
equation of B. The weight
-b log S_rho + log|l|^2
is psh on the original open set. Its negative exponential is the fiber
supremum of the squared norm of the regular normalized section
(p^*rho)^b / f^*l.
Properness and boundedness of h_L bound this norm above locally over every
base point. Hence the weight is bounded below even near the omitted locus.
At a general point of an omitted base prime T, choose a minimizing
component. In transverse coordinates the map is
y_1 = z_1^e, y_i = z_i (2 <= i <= dim Y).
Every nearby fiber meets a patch where the normalized section is nonzero
and its norm is bounded below. Since the fiber measure has positive
density almost everywhere on that patch, the essential supremum has a
fixed positive lower bound. Thus the weight is bounded above near that
prime. Psh removable singularities extend the weight across the prime,
and then across the remaining codimension-two set. The weights glue and
give the bounded metric on B.
Third construct the finite-volume metric on -K_Y+D_Y. Call a base prime T
invisible if every component of f^*T is exceptional over V or maps into D.
There are only finitely many invisible primes, since they are images of
the finitely many exceptional or boundary primes.
Choose a component Q above an invisible prime T and write
e = ord_Q(f^*T), a = ord_Q(J + p^*D),
where J = K_{W/V}. The psh weight of h_A is locally bounded above, so its
negative exponential is bounded below by a positive constant. The
Jacobian and the boundary section therefore bound the pulled-back measure
below by
constant * |z_1|^{2a}
times coordinate volume. Integrate over a fixed disc in the free
variables and use y_1 = z_1^e. This gives
g_0(y) >= C |y_1|^{2(a-e+1)/e}
almost everywhere near the general base point.
If T is visible, choose a component with a = 0. Then -log g_0 is bounded
above there. For the invisible primes, choose sufficiently large integral
coefficients of D_Y. The displayed lower bound makes
-log g_0 + log|s_{D_Y}|^2
locally bounded above at their general points. This extends as a psh
weight in codimension one and then across codimension two, so it defines a
metric on -K_Y+D_Y. Pairing with s_{D_Y} cancels the added factor. Its
adjoint measure is g_0 d(lambda_y) = f_* p^* mu almost everywhere, hence
finite.
Finally return sections. A section of aB+cD_Y is a rational section of aB
with poles only on D_Y. Pull it back and compose with the a-th power of
the normalized map. On the model, every new pole lies above an invisible
prime, hence is exceptional over V or maps into D. The normal intermediate
model excludes nonexceptional divisors over base subsets of codimension at
least two. Birational pushdown therefore leaves poles only on D. A large
multiple of D clears them. The section remains nonzero. The
normalization uses only base factors, so invariance is preserved.
Lemma 3.2 (Rank-one sufficient condition). In the setting of Lemma 3.1,
suppose that on the generic fiber of f,
h^0( W_eta, (J + p^*D + mk p^*L)|_{W_eta} ) = 1
for every m >= 0. Then the two psh assumptions in Lemma 3.1 hold. There
is an invariant version: take D=0, smooth invariant input metrics, and an
invariant rho; it suffices that the invariant dimensions in the display
equal one.
Proof. On a base open set where rho is regular, define
g_m(y) d(lambda_y)
= f_* ( |p^*rho|_{h_L^k}^{2m} p^*mu ).
These are finite positive densities almost everywhere. Give
p^*(A+mkL) the coefficient metric h_m. Consider the integrable direct
image
F_m = f_* ( O_W(K_{W/Y}+p^*(A+mkL)) tensor I(h_m) ).
For a local canonical frame eta_Y on the base, the section
u_m = s_{J+p^*D} (p^*rho)^m / f^*eta_Y
is locally integrable on the total space and belongs to F_m. It is
nonzero. The rank-one hypothesis makes it a generic generator, so F_m has
rank one. On a dense open set where u_m is its frame, the squared
direct-image norm is g_m.
Singular direct-image positivity gives a psh representative of -log g_m.
In the invariant case, smooth direct-image positivity and averaging over
the compact torus give a holomorphic orthogonal projection onto the
invariant line. Its subspace metric equals the quotient metric, hence is
semipositive.
There is a fixed dense open set where f is a submersion and every fiber has
a patch avoiding the zeros of s_{J+p^*D} and rho. Integrating over such a
patch gives
g_m >= C c^m
locally, with C>0 and c>0. Hence the psh representatives of
-m^{-1} log g_m have uniform local upper bounds. Psh extension fills any
base-change exceptions on the fixed open set. Finite fiber measure and
positive density imply
-m^{-1} log g_m(y) -> -log ess sup_{W_y} |p^*rho|_{h_L^k}^2
almost everywhere. The limit is finite by boundedness of h_L and
regularity of rho. Local compactness of psh functions gives the required
representative. The direct-image argument at m=0 supplies -log g_0.
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4. DIVISORS FROM TWISTED FORMS
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This determinant lemma is the engine that makes unbounded twists.
Lemma 4.1. Let L have bounded semipositive weights on a smooth projective
n-fold V with H^0(V, Omega_V^j)=0 for every j>0. Then there are a line
bundle M mapping nontrivially into a tensor power of Omega_V^1 and effective
integral divisors
N_i ~ M + m_i L, m_i -> +infinity.
The integers m_i may be chosen strictly increasing and divisible by any
prescribed positive integer.
Proof. Replacing L by a prescribed positive multiple gives the
divisibility assertion, so it is enough to prove the statement without it.
Hodge symmetry gives chi(O_V)=1. Serre duality gives chi(K_V)=(-1)^n.
Riemann-Roch makes chi(K_V+mL) a polynomial in m with nonzero constant
term. Therefore it is nonzero for all sufficiently large integers.
A fixed cohomological degree q therefore has
H^q(V, K_V+mL) nonzero
for unbounded m>0. The metric on mL has trivial multiplier ideal, because
its weights are bounded. Hard Lefschetz with pseudoeffective coefficients
gives nonzero sections of
Omega_V^{n-q} tensor mL
at those exponents.
Take the saturated subsheaf E of Omega_V^{n-q} spanned by their untwisted
generic images. Its determinant
M = ( exterior^{rank E} E )^{**}
is a line bundle and maps to a cotangent tensor power. This map is first
defined outside codimension two and extends because the ambient tensor
bundle is locally free.
Every nonzero vector from the family can be completed to a generic basis
using vectors of the same family. Wedging that basis gives a nonzero
section of M twisted by the sum of its positive exponents. These sums are
unbounded because one basis vector can have arbitrarily large exponent.
Saturation ensures the maps factor through E. Extension across
codimension two gives global determinant sections. Their zero divisors
are the required N_i, after taking a strictly increasing subsequence.
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5. SLOPE ESTIMATES
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The contraction theorem needs one numerical fact: certain cotangent bundles
cannot have positive degree on a certain movable curve class. This section
proves that fact in the forms we need.
First, a word about movable classes. A movable curve class is a limit of
classes of curves that move in a family. It is the dual notion to a
pseudoeffective divisor: a divisor class is pseudoeffective exactly when it
has nonnegative pairing with every movable curve class. We use this
duality without further comment.
Lemma 5.1 (Weighted sub-klt approximation). Let Y be smooth projective,
of dimension d>0. Let J be a rational line bundle with a semipositive
metric of weight psi. Let P >= 0 be a rational divisor. Suppose that in
local divisor frames,
|s_P|^2 exp(-psi) is locally integrable.
Then there is a fixed ample Cartier divisor A such that for every
sufficiently large divisible integer m there is an effective rational
divisor
Delta_m ~ J + A/m
for which the pair (Y, Delta_m - P) is sub-klt. When P=0, the pair
(Y, Delta_m) is klt.
Proof. Choose a very ample H and an ample A such that A-K_Y-iH is ample
for 1 <= i <= d. Nadel vanishing and Castelnuovo-Mumford regularity give
global generation of
O_Y(mJ+A) tensor I(m psi).
Indeed, its twist by -iH is an adjoint sheaf with strictly positive
curvature supplied by A-K_Y-iH, and multiplier ideal I(m psi). The ideal
is coherent and nonzero; local point extension where psi is finite gives a
nonzero integrable germ.
Let u: Y' -> Y resolve this ideal and P. Write
I(m psi) O_{Y'} = O_{Y'}(-F_m).
For a prime E on Y' put
b_E = ord_E F_m,
p_E = ord_E u^*P,
a_E = ord_E K_{Y'/Y}.
We claim
b_E/m - p_E < 1 + a_E.
The local point-extension estimate for m psi supplies holomorphic functions
v_x with |v_x(x)|^2 = exp(m psi(x)) and uniformly bounded L^2(exp(-m psi))
norm on fixed nested coordinate balls. The local upper bound for psi
bounds their ordinary norms, hence their suprema on a smaller ball. Each
v_x belongs to I(m psi). At a general point of E=(z_1=0), Cauchy estimates
after division by z_1^{b_E} give
exp(m psi o u) <= C_m |z_1|^{2 b_E}.
The change-of-variables formula bounds the pulled-back integrable density
below by a positive constant times
|z_1|^{2(p_E+a_E-b_E/m)}.
Its transverse integral is finite only if the displayed inequality holds.
Global generation identifies the fixed divisor of the pulled-back system
with F_m. Choose a general member of the free residual system and divide
the corresponding divisor downstairs by m. The crepant boundary of
Delta_m-P has coefficient b_E/m-p_E-a_E < 1 on each fixed component, and
coefficient 1/m < 1 on new moving components. Bertini, applied also to
the strata of the resolved divisor, makes their union simple normal
crossing. Decreasing coefficients from the positive part preserves
positive log discrepancies, proving sub-klt on every higher model.
Lemma 5.2 (From foliation determinants to tensor bounds). Let Y be smooth
projective and P >= 0 rational. Suppose that -K_Y+P is pseudoeffective and
that
K_F - K_Y + P
is pseudoeffective for every algebraically integrable saturated foliation
F inside T_Y of positive rank and corank. Then
hP - c_1(M)
is pseudoeffective for every line subsheaf M of (Omega_Y^1)^{tensor h},
h >= 0.
Proof. The case h=0 follows from the inclusion M subset O_Y. For positive
dimension, fix a nonzero movable class alpha and put t = alpha * P >= 0.
The total degree of T_Y is at least -t. If its minimum Harder-Narasimhan
slope were less than -t, its positive-slope part F would be nonzero and
proper. The quotient T_Y/F has only nonpositive slopes and degree less
than -t. In particular
mu_{alpha,min}(F) > 0,
2 mu_{alpha,min}(F) > mu_{alpha,max}(T_Y/F).
The bracket map vanishes by the slope inequality, and the Campana-Paun
algebraicity theorem makes F algebraically integrable. But
c_1(T_Y/F) = K_F - K_Y
has degree less than -t, contradicting the hypothesis. Hence
mu_{alpha,min}(T_Y) >= -t. Duality and the tensor-product rule for movable
slopes give c_1(M) * alpha <= h t. Movable-cone duality proves the
assertion.
Proposition 5.3 (Finite-volume cotangent estimate). Let Y be smooth
projective and P >= 0 rational. Suppose J = -K_Y+P has a semipositive
metric with
|s_P|^2 exp(-psi) locally integrable.
Then hP-c_1(M) is pseudoeffective for every line subsheaf M of
(Omega_Y^1)^{tensor h}. If dim Y>0 and alpha is a nonzero movable class,
then
mu_{alpha,min}(T_Y) >= -P * alpha.
In particular, if P * alpha = 0, then mu_{alpha,min}(T_Y) >= 0 and
c_1(M) * alpha <= 0.
Proof. The point case is immediate. Suppose dim Y>0. Lemma 5.1 gives
effective
Delta_m ~ -K_Y + P + A/m
with (Y, Delta_m-P) sub-klt. Resolve a first-integral map for an algebraic
foliation F together with that pair by u: Y' -> Y. Write its crepant
boundary as B_m^+ - B_m^- with disjoint effective parts. Then
K_{Y'} + B_m^+ ~ u^*A/m + B_m^-,
0 <= u_* B_m^- <= P.
The positive boundary is klt. A general fiber of the resolved
first-integral map meets the isomorphism locus of u, so u^*A restricts to
a big divisor there. A divisible positive multiple of K_{Y'}+B_m^+
therefore has a nonzero section on that fiber.
Druel's relative pseudoeffectivity theorem applies to the effective
boundary and its log canonical general fiber. It yields pseudoeffectivity
of K_{F'} + B_m^+. Push down, add P-u_*B_m^- >= 0, and let m tend to
infinity. This gives K_F - K_Y + P pseudoeffective. The metric already
gives -K_Y+P pseudoeffective, so Lemma 5.2 applies.
Proposition 5.4 (Signed-boundary cotangent estimate). Let Y be smooth
projective and let B be a rational divisor with simple normal crossing
support and coefficients less than one. Write B = B^+ - B^- in disjoint
positive and negative parts. Suppose -(K_Y+B) is nef. Then
h B^- - c_1(M)
is pseudoeffective for every line subsheaf M of (Omega_Y^1)^{tensor h},
h >= 0. If alpha is a nonzero movable class with alpha * B^- = 0, then
mu_{alpha,min}(T_Y) >= 0 and c_1(M) * alpha <= 0.
Proof. The class
-K_Y + B^- = -(K_Y+B) + B^+
is pseudoeffective. For an algebraically integrable foliation F of
positive rank and corank, resolve its rational first-integral map together
with B by u: Y' -> Y. Write
K_{Y'} + C = u^*(K_Y+B), C = C^+ - C^-.
Every coefficient of C is less than one: the positive part of the original
simple normal crossing boundary is klt, and decreasing coefficients can
only increase discrepancies. In particular (Y',C^+) is klt.
For an ample Cartier H on Y' and rational delta>0, choose a general
effective rational divisor
A_delta ~ -u^*(K_Y+B) + delta H
such that (Y', C^+ + A_delta) is klt. The class is ample, so a sufficiently
divisible multiple supplies such a member by Bertini, also on a general
fiber of the resolved map. We have
K_{Y'} + C^+ + A_delta ~ C^- + delta H.
This is big on that fiber and hence has a nonzero section after taking a
positive divisible multiple. Druel's relative pseudoeffectivity theorem
gives pseudoeffectivity of K_{F'} + C^+ + A_delta. Its pushforward is
numerically
K_F - K_Y + B^- + delta u_*H.
Letting delta decrease to zero proves K_F-K_Y+B^- pseudoeffective.
Lemma 5.2 now gives the assertion for every movable class, and therefore
in the pseudoeffective cone.
===============================================================================
6. METRIC CONCLUSIONS ON THE PREPARED BASE
===============================================================================
Now return to the contraction set-up. We have X smooth projective with no
positive-degree holomorphic forms, L=-K_X smoothly semipositive, a torus T
acting with its natural linearization, and a dominant invariant rational
map f: X -> Z to a smooth projective base whose function field is
relatively algebraically closed in C(X).
Resolve the map and prepare a normal equidimensional toroidal model
p: W -> X, f: W -> Z.
Let s be the invariant generic section supplied by the section condition.
Normalize its divisor by vertical minima:
1/p div(s) = H + f^*V,
where V is a rational divisor on Z and H is effective. Choose r>0
sufficiently divisible. Then there is an invariant rational section
sigma of r pi^*L - f^*(rV), div(sigma) = rH.
Let G be the reduced sum of primes P in Z for which every component of
f^*P having coefficient zero in H is exceptional over X. Then G is
finite. For small rational delta>0,
(H - delta f^*G)^-
is exceptional over X.
Write the base boundary as sum_i P_i. For a component D over P_i put
b_{iD} = mult_D(f^*P_i),
e_D = coeff_D(K_{Y_0} - pi_0^*K_X).
Define
c_i = min_{D -> P_i} (1+e_D)/b_{iD} > 0,
B = sum_i (1-c_i) P_i = B^+ - B^-.
If c_i>1, every component above P_i is exceptional, so P_i is a component
of G. Thus B^- is supported on G. The coefficients of B^+ are strictly
less than one.
Lemma 6.1 (Toroidal density bounds). Near an arbitrary base point, with
y_i=0 the local boundary branches and eta>0 arbitrary,
rho(y) <= C_eta prod_i |y_i|^{-2(1-c_i+eta)}
on the general smooth locus. Near a general point of P_i,
rho(y) >= C |y_i|^{-2(1-c_i)}, C>0.
At a general point of a prime outside the base boundary, rho also has a
positive constant lower bound.
Proof. The divisor
K_{Y_0} + Sigma - pi_0^*K_X - sum_i c_i f_0^*P_i
is effective and Q-Cartier. At a component above P_i its coefficient is
1+e_D-c_i b_{iD} >= 0. At other boundary components it is 1+e_D, and away
from the boundary it is e_D. The toroidal boundary pulls back log
crepantly under toroidal subdivisions. Hence on a smooth toroidal
resolution, if x_j=0 are local source boundary branches, then
a_j := 1+e_j > 0, a_j >= sum_i c_i b_{ij},
where e_j is the Jacobian order over X and b_{ij} is the order of y_i
along x_j=0.
Formal toroidal charts can be replaced by analytic monomial charts: write
f^*y_i = prod_j x_j^{A_{ij}}, x_j = X_j U_j, y_i = Y_i V_i,
with analytic boundary equations and formal units. Dominance makes A full
row rank. The unit equations
prod_j (U_j')^{A_{ij}} = R_i f^*V_i'
become linear equations after taking logarithms near 1. A linear right
inverse gives a holomorphic solution; a kernel correction matches the
prescribed first jets. Thus the monomial identities hold analytically.
Now estimate by coarea in logarithmic coordinates. Relative to Lebesgue
measure in logarithmic variables, the pullback of d(mu_X) is bounded above
by a constant times prod_j |x_j|^{2a_j} over vanishing coordinates. The
analogous change on the base introduces prod_i |y_i|^2, up to bounded
positive factors. To estimate
rho prod_i |y_i|^{2(1-c_i+eta)},
divide the logarithmic pushforward density by prod_i |y_i|^{2(c_i-eta)}.
On an affine fiber this factor is constant. The resulting exponent on a
vanishing source coordinate is
2( a_j - sum_i (c_i-eta) b_{ij} ) > 0.
If some b_{ij}>0, strict positivity follows from the order inequality and
eta>0; otherwise it follows from a_j>0. The integrand therefore decays
exponentially as the real logarithms of vanishing coordinates tend to
minus infinity. Affine plane integrals are uniformly bounded, and
properness gives finitely many chart pieces. This proves the upper bound.
For the lower bound, take a general point of a component realizing the
minimum in c_i. Use coordinates
y_1 = x_1^{b_{iD}}, y_j = x_j (2 <= j <= d).
The pulled-back volume is comparable to |x_1|^{2e_D} times coordinate
volume. Changing y_1 = x_1^{b_{iD}} and integrating over a fixed patch in
the remaining fiber variables contributes
|y_1|^{2((1+e_D)/b_{iD}-1)} = |y_1|^{-2(1-c_i)}.
This is a lower bound for the full pushforward density. At a general point
of an excluded prime outside the base boundary there is a nonexceptional
component, where the Jacobian order is zero; the same power-coordinate
calculation gives a positive constant lower bound.
Lemma 6.2 (Metric conclusions). In the preceding setup, V has a
semipositive metric with locally bounded weights. For the effective
rational divisors B^+, B^- defined above, there is a nef rational divisor M
such that
K_Z + B^+ + M ~ B^-.
The support of B^+ has simple normal crossings and all its coefficients
are less than one. The divisor B^- is supported on primes all of whose
components on Y_0 are exceptional over X; in particular it is supported on
G.
Proof. Match the data with Lemmas 3.1 and 3.2. The original variety is X,
the allowed-pole divisor is zero, and both input bundles are L=-K_X with
the compact-invariant smooth metric. The finite measure is d(mu_X). Take
exponent r and the rational section whose pullback has divisor rH+f_0^*(rV).
It is invariant and has no horizontal poles. The invariant adjoint ranks
equal one for every multiple of r, including zero. Thus Lemmas 3.2 and 3.1
apply.
The zeroth density is rho. Hence -log rho is psh on the smooth base-change
locus. The normalized base divisor in Lemma 3.1 is rV. Its bounded
metric, divided by r, gives the asserted metric on V. More concretely, for
a local frame v of rV,
|v|^2 = ( max_{f_0^{-1}(z)} |sigma f_0^*v|^{2/r} )^r.
The essential supremum is this maximum on a smooth connected general fiber.
For the signed metric, put M = -K_Z-B. On the general open its local
weight is
phi_M = -log rho - sum_i (1-c_i) log|y_i|^2.
This has the transition law of M. It is psh away from the boundary. The
lower density bounds give the upper bound needed for extension across a
dense open of every excluded divisor. Hartogs extends across the remaining
codimension-two set. The upper density estimate gives
phi_M >= eta sum_i log|y_i|^2 - O_eta(1)
for every eta>0. In particular the metric has zero Lelong numbers: for
every fixed divisible k>0 choose eta<1/k. Normal-crossing integrability
shows I(k phi_M)=O_Z for every such k.
Choose a very ample divisor A on Z. Nadel vanishing, after adding the
smooth positive metric on (d+1-i)A, gives
H^i(Z, K_Z+kM+(d+1-i)A)=0 (1 <= i <= d).
Castelnuovo-Mumford regularity makes K_Z+kM+(d+1)A globally generated.
Dividing its class by k and letting k tend to infinity proves M nef. The
same argument for the bounded metric on V proves V nef. The relation
K_Z+B^++M ~ B^- follows from the definition of M.
===============================================================================
7. THE RANK-PRESERVING CONTRACTION
===============================================================================
We can now prove Theorem C. This is the central dimension-reduction step.
Theorem 7.1 (Rank-preserving contraction). Let X be smooth connected
projective with H^0(X, Omega_X^q)=0 for q>0. Suppose L=-K_X has a smooth
semipositive Hermitian metric. Let an algebraic torus T act on X with its
natural action on L. Let f: X -> Z be a dominant T-invariant rational map
to a smooth connected projective variety Z, where C(Z) is relatively
algebraically closed in C(X). On a smooth projective resolution
pi: Y -> X, f_Y: Y -> Z,
put L_Y = pi^*L and E_Y = K_{Y/X}. Let eta = Spec C(Z). Suppose:
(1) H^0(Y_eta, pL_Y|_{Y_eta})^T is nonzero for some integer p>0;
(2) there is an integer a>0 such that
dim_{C(Z)} H^0(Y_eta, (E_Y+mL_Y)|_{Y_eta})^T = 1
for every positive multiple m of a.
Then either H^0(X, rL)^T is nonzero for some r>0, or there is a dominant
rational map h: Z -> W to a smooth connected projective variety with
dim W < dim Z such that h o f satisfies the same two conditions, possibly
with different positive integers. The field C(W) is relatively
algebraically closed in C(X).
Proof. Average the given metric over a maximal compact subgroup of T.
The resulting smooth semipositive metric is invariant. Via the identity
section of K_X+L, it defines a smooth positive volume form d(mu_X).
Call the two hypotheses the section condition and the rank condition.
Both are unchanged by resolving further or by replacing the base
birationally. Indeed, if nu: Y' -> Y is a further smooth modification,
then
E_{Y'} = nu^*E_Y + K_{Y'/Y},
nu_* O_{Y'}(K_{Y'/Y}) = O_Y.
Multiplication by the relative Jacobian section compares the adjoint
spaces. The degree-zero rank follows from both hypotheses: multiply the
section in (1) by a suitable power to inject into a positive-degree
invariant space of dimension one, while the Jacobian section gives a
nonzero element at degree zero.
If dim Z=0, the section condition gives a global section on Y, hence on X.
We may therefore assume d=dim Z>0. Every smooth base encountered below has
no positive-degree holomorphic forms: pullback under a dominant map is
injective, and holomorphic forms are invariant under smooth projective
birational modification. Consequently chi(Z,O_Z)=1.
Step 1: preparation and the first effectivity test. Apply weak
toroidalization to a resolution of the rational map, prescribing the
complement of an isomorphism open over X as part of the source boundary.
Then apply the equidimensional modification by projective subdivisions.
After renaming the base we have
pi_0: Y_0 -> X, f_0: Y_0 -> Z,
where Y_0 is normal and toroidal, Z is smooth projective, pi_0 is
projective birational, and f_0 is projective, surjective, equidimensional,
and toroidal. The reduced source boundary contains every pi_0-exceptional
divisor. The base boundary has simple normal crossings. This step is
birational: it does not assert that the source is smooth or that fibers
are reduced.
Let s be the rational section supplied by the section condition. On Y_0
its horizontal divisor is effective. Write
1/p div(s) = H + f_0^*V,
where the coefficient of V at a prime P in Z is the minimum of
coeff_D(div(s)/p) / mult_D(f_0^*P)
over components D of f_0^*P. Equidimensionality ensures that every
vertical prime maps onto a prime of Z. Thus V is a rational divisor and H
is effective and Q-Cartier.
Choose r>0 sufficiently divisible. There is an invariant rational section
sigma in H^0(Y_0, r pi_0^*L - f_0^*(rV)),
div(sigma) = rH.
Its pullback to every dominating smooth model is regular. Take r also
divisible by the integer in the rank condition. For every prime of the
base there is at least one component above it with coefficient zero in H.
Let G be the reduced sum of primes P in Z for which every component of
f_0^*P having coefficient zero in H is exceptional over X. This is a
finite sum, since each such prime is the image of an exceptional divisor of
pi_0. Every nonexceptional component above G has strictly positive
coefficient in H. Hence for every sufficiently small rational delta>0,
(H - delta f_0^*G)^-
is exceptional over X.
If r(V+delta G) is rationally effective, take a nonzero rational section q
of r(V+delta G). Then sigma f_0^*q is a rational section of rL with
divisor at least r(H-delta f_0^*G). Its negative part is exceptional over
X. Pushdown removes it, and smoothness of X extends the resulting
invariant section globally. This gives a nonzero invariant section of a
positive multiple of L. Thus it suffices either to prove this effectivity
or to contract the base.
Step 2: a supporting movable class. From now on assume that
V+delta G is not rationally effective for any small delta>0. In
particular V is pseudoeffective and not big. Use Lemma 6.2: V has a
bounded semipositive metric, and there is a nef rational divisor M with
K_Z + B^+ + M ~ B^-,
where B=B^+-B^- is the signed boundary defined there. The negative part
B^- is supported on G.
If V+G were big, then for 0<delta<=1,
V + delta G = delta(V+G) + (1-delta)V
would be big, because V is pseudoeffective. This contradicts the failure
of effectivity. Hence V+G is not big. Cone duality gives a nonzero
movable curve class beta with
(V+G) * beta = 0.
Since V and G are pseudoeffective, we obtain
V * beta = G * beta = 0.
Since B^- is supported on G, also B^- * beta = 0.
Step 3: cotangent slope on beta. We claim that every line subsheaf N of
(Omega_Z^1)^{tensor s}, s>=0, satisfies
c_1(N) * beta <= 0.
For s=0 this follows from N subset O_Z. For s>0, recall that
K_Z+B^+ = B^- - M. Hence
-(K_Z+B) = -(K_Z+B^+-B^-) = -K_Z-B^++B^- = M.
Thus -(K_Z+B) is nef. Apply Proposition 5.4 with this signed boundary B.
We get
s B^- - c_1(N)
pseudoeffective. Pairing with beta and using B^- * beta = 0 gives
c_1(N) * beta <= 0.
Step 4: effective divisors of zero degree. Choose b>0 with bV Cartier.
Lemma 6.2 gives bV bounded semipositive weights. Apply Lemma 4.1 to bV,
prescribing any further divisibility needed. We obtain a fixed cotangent
line N and effective divisors
D_j ~ N + m_j V, m_j -> +infinity,
with m_j strictly increasing and sufficiently divisible. By Step 3,
N * beta <= 0. Since V * beta = 0, effectivity gives D_j * beta >= 0, and
therefore D_j * beta = 0 for every j.
Step 5: choose a maximal zero-degree system. Choose a linear system of
divisor class D with D * beta = 0 whose rational map has maximal possible
image dimension among all such systems. The constant map is allowed. This
maximal dimension is smaller than d: a generically finite system makes its
divisor big, and a nonzero supporting functional cannot vanish on a big
class.
Resolve this rational map, take its Stein factorization, and then a smooth
projective resolution of the Stein base. We obtain
h: Zhat -> Z, ghat: Zhat -> W, dim W < d,
where the general fibers of ghat are connected.
Lemma 7.2 (Maximal-field property). Every ratio of two nonzero sections
in any one linear system whose divisor class has degree zero against beta
belongs to C(W) subset C(Z).
Proof. Multiply the sections of that system by those defining the chosen
maximal system. Their products form a system of degree zero, and its map
is the product of the two maps followed by the Segre embedding. By
maximality it has the same image dimension as the chosen map. The added
section ratios are therefore algebraic over the field of that image. The
Stein factorization identifies C(W) with the relative algebraic closure of
the image field in C(Z). This proves the lemma.
Lemma 7.3 (Horizontal slope). The rational divisor h^*V is rationally
effective on the complete generic fiber of ghat.
Proof. Use the sequence D_j = N+m_j V. For j<k<l, linear equivalence
gives
(m_l-m_j)D_k
~ (m_l-m_k)D_j + (m_k-m_j)D_l.
Both sides are effective divisors in the same zero-degree system. By
Lemma 7.2, their ratio is a rational function from W. Its pullback has no
horizontal divisor over W. Thus, after pullback by h, the displayed
relation is an equality of actual horizontal divisors.
Fix the first two indices. For every horizontal prime Q on Zhat, the
coefficients of h^*D_j have the form a_Q + m_j b_Q. The preceding identity
proves this simultaneously for every index. Since all D_j are effective
and m_j -> infinity, every b_Q is nonnegative. These slopes have finite
support: they are the coefficients of the horizontal part of
h^*(D_2-D_1) / (m_2-m_1) ~ h^*V.
This representative of h^*V has effective horizontal part. Its vertical
components disappear on the generic fiber, yielding linear effectivity
there.
Step 6: preserve the section. Compose X -> Z with Z -> W. The
composition is invariant and has connected general fibers, because the
inclusions of function fields remain relatively algebraically closed.
Choose a smooth projective resolution pi': Y' -> X dominating Y_0 and the
map to Zhat, and write
f': Y' -> Zhat, u = ghat o f'.
Multiplication of the pullback of sigma by the generic section of h^*V
from Lemma 7.3 gives an invariant section of a positive multiple of L_{Y'}
on the generic fiber of u. Thus the section condition is preserved.
Step 7: preserve the rank. Fix a sufficiently divisible m>0 and take any
two nonzero invariant sections of E_{Y'}+mL_{Y'} over the generic fiber of
u. They can be extended simultaneously to global sections of
E_{Y'} + mL_{Y'} + k u^*A_W
for a sufficiently large integer k, where A_W is very ample. To see this,
note that every pole divisor has image in a proper closed subset of W. A
high multiple of a very ample bundle has a section in any prescribed power
of the ideal of this finite union of images. Pullback then vanishes to at
least the required orders on all the finitely many pole divisors. The same
section clears the poles of both rational sections. This also handles
images of codimension greater than one. It is a base section, so it
preserves invariance and does not change their ratio.
Divide the two sections by the Jacobian section and by sigma^{m/r}. The
old rank condition says that the results come from rational sections on
Zhat of
m h^*V + k ghat^*A_W.
Push their divisors down to Z. A pole outside G is impossible. Indeed,
for a prime P not contained in G, choose a component above P on Y_0 with
coefficient zero in H and nonexceptional over X. Its strict transform on
Y' has coefficient zero in both the pulled-back H and the Jacobian divisor
E_{Y'}. Since h is an isomorphism at the generic point of P, a pole
downstairs would remain a pole on this component, contradicting regularity
of the original global section.
After bounding the poles of both sections along G by a common integer ell,
their ratio is a ratio of sections in the single linear system
mV + k h_* ghat^*A_W + ell G.
We verify that its degree against beta is zero. The first and third terms
are zero because V * beta = G * beta = 0. Only the middle term needs
proof. If W is a point it is zero. Otherwise, let A_I be the pullback to
W of the hyperplane class of the image of the defining maximal system. The
map from W to that image is generically finite, so A_I is big. On a
sufficiently high resolution of the system,
h^*D ~ ghat^*A_I + F, F >= 0.
Some sufficiently large multiple of A_I minus A_W is rationally effective,
by bigness. Pulling back and pushing forward shows that a positive
multiple of D minus h_*ghat^*A_W is rationally effective. The latter
divisor is itself effective after choosing an effective representative of
A_W. Since D * beta = 0 and movable classes are nonnegative on effective
divisors, both inequalities force
(h_* ghat^*A_W) * beta = 0.
Together with V * beta = G * beta = 0, this proves that the rank-system has
degree zero against beta.
Lemma 7.2 now puts the ratio of the two original sections in C(W). Thus
the new invariant generic adjoint space has dimension at most one for every
positive multiple of a common integer. The new section condition and the
Jacobian supply a nonzero element in every sufficiently divisible degree,
giving dimension exactly one. Both conditions have therefore been
preserved while the base dimension has strictly decreased. This proves
Theorem 7.1.
===============================================================================
8. ALLOWED-POLE NONVANISHING
===============================================================================
Theorem 8.1 (Allowed-pole nonvanishing). Let V be a smooth connected
projective variety over C with H^0(V, Omega_V^j)=0 for every j>0. Let
D>=0 be an integral divisor. Suppose -K_V+D carries a semipositive
singular metric whose distinguished measure |s_D|^2 has finite mass. Let L
be a line bundle with a semipositive metric whose weights are locally
bounded. Then
H^0(V, aL+cD) is nonzero
for some integers a>0 and c>=0. The same statement holds for a rational
line bundle L, with a clearing its denominator.
Proof. Clear the denominator of L. We induct on dim V. A point is
immediate. Suppose the conclusion fails. Then L+D is pseudoeffective and
not big. Cone duality gives a nonzero movable curve class alpha with
L * alpha = 0, D * alpha = 0.
Apply Proposition 5.3 with P=D. Since -K_V+D has the required
finite-volume metric, every line subsheaf M of a cotangent tensor power
satisfies
h D - c_1(M) pseudoeffective.
Pairing with alpha and using D * alpha = 0 gives c_1(M) * alpha <= 0.
Apply Lemma 4.1 to L. We obtain a fixed cotangent line M and effective
divisors
N_i ~ M + m_i L, m_i -> +infinity.
Since c_1(M) * alpha <= 0 and L * alpha = 0, effectivity forces
N_i * alpha = 0
for every i.
No nonnegative integral combination of D,N_1,N_2,... is big. Indeed, a
big divisor is in the interior of the pseudoeffective cone, and a nonzero
movable functional is strictly positive there. But alpha annihilates every
member of the combination.
Choose such a nonnegative integral combination G of maximal Iitaka
dimension d. Since no combination is big, d<dim V. Take a multiple
realizing dimension d, resolve its rational map, and take the base Y with
connected fibers. Every ratio from the linear system of any allowed
combination lies in C(Y): adding systems and multiplying sections cannot
increase the Iitaka dimension d.
Let H be any base divisor. For sufficiently large a,
a p^*G - f^*H ~ E_H >= 0
for some effective E_H. This is because the resolved system contains the
pullback of the image hyperplane class, whose transform on Y is big.
We need a slightly stronger form of the ratio assertion. Suppose a
rational function on the model has horizontal poles bounded by
p^*G' + E,
where G' is an allowed combination and E is effective and p-exceptional.
Then the function belongs to C(Y). Its remaining poles are vertical.
Choose an effective sufficiently ample H whose pullback bounds them; a base
divisor can contain the images of all these finitely many poles, including
images of codimension at least two, to the necessary orders. By the
domination just proved, the function is a ratio of sections of
p^*(G'+aG) + E.
Normality gives p_*O_W(E)=O_V, so the exceptional term adds no sections.
The original ratio assertion proves the claim.
The three-index relations among the N_i now imply that the horizontal
multiplicity at every prime is affine in m_i. Let rho have divisor
N_j-N_i and exponent k=m_j-m_i>0. It has no horizontal poles, and its
horizontal zeros are bounded by p^*N_j. On a generic fiber,
s_{J+p^*D} rho^l
is a nonzero section of J+p^*D+lk p^*L. The ratio of any other section to
this one has horizontal poles bounded by
J + p^*D + l p^*N_j.
The strong ratio assertion makes that ratio a base function. Thus the
generic section space is one-dimensional for every l>=0, including l=0.
If d=0, then rho is already regular, and this gives the required section on
V.
If d>0, Lemmas 3.2 and 3.1 apply to the smaller base. They give the two
required metrics and the controlled boundary divisor on Y. The base has no
positive-degree holomorphic forms, because pullback by the dominant map is
injective and holomorphic forms are invariant under smooth projective
birational modification. Thus the theorem holds on Y by dimension
induction. Transfer gives a section on V with poles only on D. A large
integral multiple of D clears those finitely many poles. This contradicts
the assumption and proves Theorem 8.1.
===============================================================================
9. THE INVARIANT ANTICANONICAL SECTION
===============================================================================
Theorem 9.1 (Invariant anticanonical section). Let V be a smooth connected
projective variety over C with H^0(V, Omega_V^j)=0 for every j>0. Suppose
-K_V has a smooth Hermitian metric of semipositive curvature. Let an
algebraic torus T act on V with its natural linearization on -K_V. Then
there is an integer m>0 such that
H^0(V, -mK_V)^T is nonzero.
Proof. Average local weights over the compact form of T. This gives a
smooth semipositive metric invariant under that compact group. If the
action is trivial, Theorem 8.1 applies with D=0 and L=-K_V. It gives a
nonzero section of a positive anticanonical multiple, and the section is
invariant because the action is trivial.
Assume the action is nontrivial. Take the rational quotient with field
C(V)^T and a connected-fiber smooth projective model Y. The invariant
field is relatively algebraically closed in C(V): an element algebraic over
it has a finite orbit, and a connected torus acts trivially on a finite
orbit. General fibers contain a dense orbit.
Choose an equivariant model
p: W -> V, f: W -> Y,
by modifying the trivially acted-on base, normalizing the dominating graph,
and taking an equivariant resolution. Write J=K_{W/V}. Let r be the orbit
dimension. A wedge of r independent infinitesimal action fields is an
invariant regular relative anticanonical section on the generic fiber.
Commutativity of T gives invariance.
A rational frame of -K_Y and the canonical Jacobian section s_J put this
wedge in
p^*(-K_V) = -K_{W/Y} + f^*(-K_Y) + J.
It has no horizontal poles. Regard this section of p^*(-K_V)
birationally as a rational section rho on V. The Jacobian section is
invariant for the induced canonical actions.
The ratio of any two invariant sections of J+mp^*(-K_V) is a rational
invariant function, hence belongs to the base field C(Y). The sections
s_J (p^*rho)^m
supply the nonzero element at every m>=0, including m=0. Thus the
invariant rank-one condition holds.
If Y is a point, the full wedge of infinitesimal fields on V already gives
a global invariant anticanonical section. Otherwise the invariant case of
Lemma 3.2 and Lemma 3.1 gives a bounded semipositive base bundle and a
finite-volume donor metric on the base. The base has no holomorphic forms
by dominant pullback, and Theorem 8.1 gives a section with permitted base
poles.
The transferred section is invariant, and its only permitted poles on V lie
in D=0. It is therefore regular. This proves Theorem 9.1.
===============================================================================
10. WHAT HAS BEEN PROVED
===============================================================================
We have proved, from start to finish:
Theorem A. The toroidal pushforward volume gives an integrable
semipositive metric on -K_Y+A_0, with A_0 exceptional over X.
Theorem B. Two-metric transfer: a finite-volume metric and a bounded
metric descend to a base, and sections return with poles only on the
original allowed divisor.
Theorem C. Rank-preserving contraction: the generic section and adjoint
rank-one condition survive a strict decrease of base dimension, unless a
global invariant section already exists.
Theorem D. Allowed-pole nonvanishing: on a variety with no
positive-degree holomorphic forms, finite-volume -K_X+D and bounded L
force a section of aL+cD.
Theorem E. Invariant anticanonical nonvanishing: a smooth semipositive
anticanonical metric and a torus action force an invariant section of a
positive anticanonical multiple.
The logical dependencies are:
Theorem A -> metric conclusions on the prepared base (Lemma 6.2)
Lemmas 3.1, 3.2, 4.1, 5.1-5.4 -> Theorem C
Theorem C and Proposition 5.3 -> Theorem D
Theorem D and Lemmas 3.1, 3.2 -> Theorem E
Each step is a dimension induction. The only input from outside this
rewrite is the standard package:
- Berndtsson and Paun-Takayama positivity of direct images;
- Nadel vanishing and multiplier ideals;
- Hard Lefschetz with multiplier ideals;
- Campana-Paun orbifold cotangent semipositivity;
- Druel relative pseudoeffectivity for foliations;
- toroidalization and weak semistable reduction;
- psh extension and normal-crossing integration.
With those standard tools, the proofs above are complete. The original
paper contains additional alternative constructions (orbit integrals,
reduced-flat norms, multiplicative systems) that reach the same main
conclusions by different routes. This rewrite follows one complete route
for each main theorem, so a student can see the whole chain without
tracking every variant at once.