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ANTICANONICAL NONVANISHING FROM SMOOTH SEMIPOSITIVITY
A plain-language, complete version of the paper
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WHAT THIS DOCUMENT IS
---------------------
This is a rewrite of a research paper of the same title. The paper
proves one clean statement about the shapes called "complex
projective varieties" and the holomorphic sections that can live on
them. The original is written for specialists: it assumes you already
know every word. This version keeps every real idea and result but
explains each one, so a careful reader starting from scratch can
follow the whole argument.
Everything below is genuine mathematics. A few long computations are
shortened to their punchline and marked [sketch]; the full technical
version of each is in the original paper. Nothing essential is
omitted: you will see the statement of every theorem and lemma, why
each is needed, and how they join into a single proof.
HOW TO READ IT
--------------
Read Part 1 first; it fixes the vocabulary. Every unusual word is
defined when it appears and again in the glossary (Part 7). On a first
pass, aim for the picture. The proof is a machine with a few big parts
(structure theory, a finite-volume theorem, a transfer step, an
induction), and the machine is described before it is assembled.
CONTENTS
--------
1. The question, the answer, and two examples
2. The shape of the answer: structure theory and the universal cover
3. The finite-volume theorem (the engine of the induction)
4. How the engine turns: slope, determinants, and the smaller base
5. The transfer step: moving a section across a fibration
6. Invariant sections and the torus action
7. Descent to X, glossary, and notes
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.
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PART 1. THE QUESTION, THE ANSWER, AND TWO EXAMPLES
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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 Parts 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. Plural subharmonic 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.
Parts 4 and 5 expand the pieces of that paragraph: Part 4 the slope
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