GLOBAL SPHERICAL SHELLS ON MINIMAL SURFACES OF CLASS VII
A complete, self-contained proof, with all technical steps included.
ABSTRACT
A compact complex surface is said to be of class VII if its first Betti
number is 1 and its Kodaira dimension is minus infinity. These are the
surfaces that, unlike the projective ones, do not embed in projective
space, and whose classification is the hardest remaining case in the
theory of complex surfaces. This paper proves the Global Spherical Shell
conjecture: every connected MINIMAL complex surface of class VII with
positive second Betti number contains a global spherical shell, meaning
a holomorphically embedded neighbourhood of the standard 3-sphere
S^3 = { z in C^2 : |z| = 1 } inside C^2 minus the origin, whose complement
in the surface is connected.
In plain language: we show that every such surface has a single, clean,
standard "handle" through which it can be built, glued in from a copy of
C^2 minus a point. That geometric control is what makes the
classification of these surfaces possible. As immediate consequences we
recover the deformation type and the smooth topology of every surface in
this range, and we remove the last open exception in a known inequality
for aspherical complex surfaces.
HOW TO READ THIS PAPER
The reader is assumed to know multivariable calculus and linear algebra,
and to be comfortable with the idea of a manifold. Everything else
(complex surfaces, line bundles, Dolbeault cohomology, weighted Sobolev
spaces) is defined or explained on first use. Technical results borrowed
from the literature are cited precisely and used as black boxes; every
step that we actually need is proved here.
Part I sets up the problem and reduces it to two concrete questions.
Part II proves the main theorem. The reader who wants only the shape of
the argument can read Part I and the "The argument" subsection below;
the rest is the full proof.
PART I. THE PROBLEM AND THE PLAN
1. Complex surfaces and class VII.
A COMPLEX SURFACE is a real four-dimensional manifold equipped with a
smooth way to multiply tangent vectors by i, with i^2 = -1, integrable in
the sense that it admits local holomorphic coordinates (z, w) with
z, w in C. Equivalently, it is a two-dimensional complex manifold.
Examples. The plane C^2, the projective plane CP^2, and every smooth
complex curve inside them. A HOPF SURFACE is the quotient of C^2 minus
the origin by the infinite cyclic group generated by z |-> (1/2) z (or
a small holomorphic perturbation of that map). This is the simplest
compact complex surface of class VII.
The Betti numbers b_k(X) are the dimensions of the real cohomology
groups H^k(X; R). For a compact complex surface, Kodaira proved a rigid
set of numerical identities that depends only on which "Kodaira
dimension" the surface has. The Kodaira dimension kappa(X) is a coarse
measure of how many global holomorphic sections of powers of the
canonical bundle X carries; kappa(X) = -infinity is the smallest
possible case.
Definition. A CLASS VII SURFACE is a connected compact complex surface
with b_1(X) = 1 and kappa(X) = -infinity.
Class VII splits by the second Betti number b_2:
* b_2 = 0: the surface is a Hopf or an Inoue surface (Bogomolov,
Li-Yau-Zheng, Teleman). This case is completely classified.
* b_2 > 0: the "positive b_2" case. This paper treats it.
A curve on X means the image of a nonconstant holomorphic map from a
compact Riemann surface, always taken reduced and irreducible. A RATIONAL
CURVE is a curve whose normalisation (the smooth curve mapping onto it)
is CP^1; the curve itself may have nodes or cusps.
The surface X is MINIMAL if it contains no smooth rational curve with
self-intersection -1. Minimality is a normalisation condition: every
surface can be blown down along such curves to a minimal one, and the
minimal model is unique. The main theorem is stated for minimal surfaces.
2. The global spherical shell.
Recall S^3 = { z in C^2 : |z| = 1 }. Let U be an open neighbourhood of
S^3 inside C^2 minus {0}. The standard picture is the "annulus"
U = { r < |z| < R } for 0 < r < 1 < R; its middle surface S^3 separates
U into an inner ball and an outer piece.
Definition. A GLOBAL SPHERICAL SHELL in a complex surface X is an open
subset Sigma of X together with a biholomorphism phi: U -> Sigma from
such an annulus, such that X minus Sigma is connected.
Intuitively, a shell is a standard round "waist" glued into X. Once a
surface has a shell, one can cut it open along S^3 and rebuild it as a
ball glued to its own boundary by a holomorphic map. That is exactly the
data of a Kato surface, and it makes the surface's deformation theory
and smooth topology completely transparent.
The conjecture, formulated by Nakamura (1984) building on Kato's work,
asserts that this always happens in the positive-b_2 case. The main
result of this paper is that it does:
MAIN THEOREM (Global Spherical Shell conjecture).
Let X be a connected compact complex surface with
b_1(X) = 1, b_2(X) > 0, kappa(X) = -infinity.
Assume X is minimal. Then X contains a global spherical shell.
Kato's deformation theorem then pins down the whole deformation family
(see Section 12): a small deformation makes X a blowup of a Hopf
surface. And because a shell forces pi_1(X) = Z together with a specific
smooth type, the theorem also sharpens a known inequality for aspherical
surfaces (also Section 12).
3. Two classical criteria: the route to the proof.
We do not construct a shell directly. Instead we use two deep theorems
that each turn the existence of certain curves into a shell, then show
that the required curves must exist.
The first is due to Enoki (1981); the clean modern statement we use is
recorded by Teleman.
ENOKI'S CRITERION. Let V be minimal, of class VII, with b_2(V) > 0. If V
carries a nonzero EFFECTIVE divisor C (a nonnegative integer combination
of curves, not identically zero) with C^2 = 0, then V has a global
spherical shell.
The second is the Main Theorem of Dloussky-Oeljeklaus-Toma (2003).
DOT CRITERION. Let V be minimal, of class VII, with b_2(V) > 0. If V has
exactly b_2(V) rational curves, then V has a global spherical shell.
Both are used as black boxes. Our job is therefore:
GOAL. Given minimal X of class VII with b > 0, either find an effective
divisor with square 0, or prove that X has exactly b rational curves.
The rest of the paper does exactly this. Enoki's criterion handles the
first alternative directly. So from now on we assume:
(NSZ) every nonzero effective divisor C on X satisfies C^2 < 0.
Under (NSZ) we must prove X has exactly b = b_2(X) rational curves.
4. The plan, in plain language.
The surface X has an infinite cyclic cover Y obtained by unwrapping its
fundamental group's Z-quotient. The cover Y is a noncompact complex
surface, and it is much easier to build holomorphic objects on Y than on
X, because compact restrictions vanish. We build a holomorphic section
of a cleverly chosen line bundle on Y, look at its zero set D (a curve),
and prove every component of D is a compact rational curve. Projecting D
down to X produces the rational curves we need. Counting them carefully
gives exactly b of them, and the DOT criterion then delivers the shell.
There are four technical pillars:
(P1) COMPACT CHERN CLASSES (Section 6). We manufacture many line
bundles E on Y whose first Chern classes are compactly supported
and satisfy e^2 = K . e < 0. This numerical identity is arranged
so that later the holomorphic section space is exactly one
dimensional.
(P2) WEIGHTED DOLBEAULT THEORY (Section 7). We set up Sobolev spaces
with exponential weights e^{-a h} at one end and e^{+b h} at the
other, prove the relevant Dolbeault complexes are Fredholm, and
compute their index by a cut-and-paste argument. This produces,
for each E, a section s that tends to 1 at the positive end and
to 0 at the negative end. Its zero divisor D sits in a bounded
height range.
(P3) AREA AND RESIDUE OBSTRUCTIONS (Sections 8-9). The components of D
might a priori be noncompact. We bound the area of D on each
height slab by C e^{A l}. Then any holomorphic differential on a
component gives a residue current whose mass obeys the same
growth bound. Such currents live in a fixed finite-dimensional
weighted cohomology group. A residue argument shows currents on
distinct components are independent, and a scalar version shows
the components must have genus zero. A cylinder estimate
(Section 9) rules out the remaining open shapes, leaving only
compact rational components.
(P4) COUNTING OVER Z[t, t^{-1}] (Section 10). One curve on X accounts
for infinitely many translates on Y. To count correctly we work
over the Laurent polynomial ring Q[t, t^{-1}], where t records one
deck translation. This makes the intersection pairing between
our compact classes nonsingular over Q(t), and a linear-algebra
count then shows we cannot have fewer than b rational curves.
Combining (P4) with an upper bound on the number of curves gives
exactly b curves, and the DOT criterion finishes the proof.
This is the skeleton. Let us now build it, carefully and completely.
PART II. THE PROOF
5. Surface invariants, the cyclic cover, and the height function.
Throughout, a CURVE on X means a reduced irreducible compact analytic
curve, and a dot denotes the intersection pairing of real 2-classes. We
write K_X for the canonical line bundle and also for its first Chern
class. We orient every complex surface by its complex structure.
5.1. Kodaira's identities. Under the hypotheses of the Main Theorem,
Kodaira's classification gives
(KOD) h^{0,1}(X) = 1, h^{1,0}(X) = h^{2,0}(X) = 0, b_2^+(X) = 0,
chi(O_X) = 0, K_X^2 = -b_2(X).
See Kodaira (1964, Theorem 3) or Barth-Hulek-Peters-Van de Ven,
Compact Complex Surfaces. In particular the intersection form on
H^2(X; R) is NEGATIVE DEFINITE. This single fact powers almost every
argument below.
5.2. The two surface criteria. We restate them as a proposition so the
proof can call on them cleanly.
PROPOSITION 5.1 (Surface reductions). Let V be minimal, of class VII,
with b_2(V) > 0. Then:
(i) if V has an effective divisor C with C^2 = 0, V has a shell;
(ii) if V has exactly b_2(V) rational curves, V has a shell.
Proof. (i) is Enoki, in the form stated by Teleman (2010, p. 1758).
(ii) is the Main Theorem of DOT (2003, p. 283), whose p. 284 definition
includes the connected-complement condition. []
Since X satisfies every hypothesis of V, we assume (NSZ) from now on: no
nonzero effective divisor on X has square 0. (Negative definiteness
leaves only the square-zero alternative, already dispatched.)
5.3. The canonical integral basis. Because the intersection form of X is
negative definite, Donaldson's theorem (in its version for closed smooth
4-manifolds with arbitrary fundamental group; Donaldson 1987, Theorem 1,
or Teleman 2010, Sec. 1.1) gives a basis over the integers (modulo
torsion) that is negative orthonormal.
LEMMA 5.2 (Canonical basis and curve degrees). Put b = b_2(X). There is
an integral basis u_1, ..., u_b of H^2(X; Z) modulo torsion with
(CB) u_i . u_j = -delta_{ij}, K_X = u_1 + ... + u_b (mod torsion).
For every curve B in X,
(DEG) B^2 < 0, (K_X + B) . B <= 0, K_X . B >= 0.
In particular every effective divisor has nonnegative canonical degree.
Proof. Take a negative orthonormal integral basis and write
K_X = sum_i k_i u_i. The canonical class is characteristic: its mod 2
reduction is w_2, so every k_i is ODD. Also K_X^2 = sum_i k_i^2 = -b,
hence k_i = +-1, and flipping signs gives (CB).
Now write [B] = sum_i a_i u_i. Then
(K_X + B) . B = sum_i a_i (a_i + 1) * (-1) = -sum_i a_i(a_i + 1) <= 0.
The strict inequality B^2 < 0 is (NSZ). The adjunction formula for a
possibly singular integral curve reads K_X . B = 2 p_a(B) - 2 - B^2. If
the arithmetic genus p_a(B) >= 1 this is positive. If p_a(B) = 0, the
normalisation-genus formula p_a(B) = g(B-tilde) + sum_x delta_x shows the
normalisation has genus 0; by minimality and B^2 < 0 we get B^2 <= -2,
again nonnegative canonical degree. Sums of curves add degrees, hence
the last sentence. []
5.4. The cyclic cover. Since H^1(X; Z) = Z, choose a primitive generator
alpha and its induced epimorphism pi_1(X) -> Z. Its kernel defines the
connected infinite cyclic cover
pi : Y -> X, Deck(Y/X) = <T> = Z.
Only this Z-quotient of pi_1 enters. Choose a smooth closed 1-form
representing alpha; its integral over paths lifts to a smooth function
h_0 : Y -> R with h_0(T y) = h_0(y) + 1.
5.5. Twisted characters. For lambda in C^* let P_lambda be the flat
holomorphic line bundle on X whose pulled-back sections satisfy
f(T y) = lambda f(y). This fixes the sign of the character. These
bundles are smoothly trivial: if e^u = lambda, then e^{u h_0} is a
nowhere-zero smooth equivariant section, so c_1(P_lambda) = 0.
LEMMA 5.3 (Character cohomology). For lambda != 1,
(CH1) H^q(X, P_lambda) = 0 and H^q(X, K_X (x) P_lambda) = 0,
for q = 0, 1, 2.
For EVERY lambda in C^* (including 1),
(CH2) H^0(X, K_X (x) P_lambda) = 0.
At the trivial character, H^0(X, O) = C, H^1(X, O) = C, H^2(X, O) = 0.
If f is holomorphic on Y with f(T y) - f(y) = c for a constant c, then f
is constant and c = 0. More generally a holomorphic function killed by
some power of (T^* - 1) is constant.
Proof. A nonzero section of P_lambda has an effective zero divisor of
Chern class 0; a nonempty such divisor violates (NSZ), so the section is
nowhere zero. Its lift f is nowhere zero, and df/f is T-invariant, hence
descends to a holomorphic 1-form on X. By (KOD) that form is 0, so f is
constant and lambda = 1.
A nonzero section of K_X (x) P_lambda has zero divisor C with
[C] = K_X. If C is nonempty then K_X . C = K_X^2 = -b < 0, contradicting
Lemma 5.2. If C is empty the bundle is holomorphically trivial and its
real Chern class vanishes, again contradicting K_X^2 < 0. This is (CH2).
Serre duality gives H^2(X, P_lambda) = H^0(X, K_X (x) P_{1/lambda})^* = 0
by (CH2). Riemann-Roch with c_1 = 0 gives chi(P_lambda) = chi(O_X) = 0,
so H^1(X, P_lambda) = 0 for lambda != 1, and duality gives the rest of
(CH1). The trivial-character values follow from (KOD) and connectedness.
For the last claim, f(T y) - f(y) = c makes df T-invariant, hence a
pullback from X, hence 0; so f is constant. For the generalisation,
write Delta = T^* - 1. If a least exponent m > 1 had Delta^m f = 0 and
Delta^{m-1} f != 0, the function Delta^{m-1} f would descend to a nonzero
constant on X, and then Delta^{m-2} f would have a nonzero constant
difference, already excluded. So m = 1 and f descends to a constant. []
5.6. A height with positive complex Laplacian. We need a proper height
function whose complex Laplacian is a positive constant. This controls
the ends and drives all the maximum-principle arguments.
LEMMA 5.4 (A good height). There are a Gauduchon metric on X, a choice
of generator T, and a smooth proper height h : Y -> R such that
(HT) h(T y) = h(y) + 1, L h = c > 0,
where L is the complex Laplacian with the subharmonic sign. Every
positive-order derivative of h is uniformly bounded in the lifted metric.
Proof. Gauduchon's theorem gives a Hermitian form omega with
d-bar partial omega = 0. Fix the convention d^c = i(d-bar - partial), so
dd^c = 2i partial d-bar, and set L = Lambda_omega dd^c. L is elliptic
with the subharmonic sign, its kernel on real functions on connected X is
the constants (maximum principle), and its index is zero (its principal
symbol is that of a positive multiple of the scalar Laplacian, and
homotoping the lower-order terms to that Laplacian preserves the elliptic
index). Moreover
integral_X (L u) omega^2/2 = integral_X dd^c u ^ omega
= integral_X u dd^c omega = 0,
so the cokernel is exactly the constants. These are the
Gauduchon-Laplacian solvability facts recorded by Buchdahl (1999, p.289).
The forms dh_0, d^c h_0, and the function L h_0 are periodic. Let cbar
be the mean of L h_0 over X. Solvability gives a smooth v on X with
L(v o pi) = cbar - L h_0. Set h = h_0 + v o pi. If cbar = 0, the
descended real (1,1)-form eta = dd^c h is exact and primitive. On a
Hermitian surface a real primitive (1,1)-form is anti-self-dual, so
0 = integral_X eta ^ eta = - integral_X |eta|^2 omega^2/2,
the first equality by exactness and Stokes. Hence dd^c h = 0 and
partial h is an invariant nonzero holomorphic 1-form, contradicting
h^{1,0}(X) = 0. So cbar != 0; replacing (T, h) by (T^{-1}, -h) makes
it positive. Derivatives of h are periodic hence bounded.
For properness: choose a compact F subset Y whose translates cover Y
(possible as X is compact). h is bounded on F. Any bounded interval I
meets only finitely many translates T^j F by (HT), so h^{-1}(I) is
compact. []
From now on `h` denotes this height, and `c` its positive Laplacian
constant. We say the POSITIVE END is h -> +infinity and the NEGATIVE END
is h -> -infinity.
5.7. Finite cyclic quotients. For n >= 1 put S = T^n, X_n = Y/<S>, and
p_n : X_n -> X the degree-n covering.
LEMMA 5.5 (Finite cyclic quotients). X_n is a minimal class VII surface
and
(Q) chi(O_{X_n}) = 0, b_1(X_n) = 1, b_2(X_n) = n b_2(X),
K_{X_n} = p_n^* K_X.
Its integral intersection lattice modulo torsion admits a negative
orthonormal basis whose sum is K_{X_n} (mod torsion).
Proof. The covering is a local biholomorphism, so tangent and canonical
bundles pull back. Compact-complex Riemann-Roch (or the Todd degree
formula; Atiyah-Singer III 1968, Sec. 4) gives chi(O_{X_n}) = n chi(O_X)
= 0. If sigma is a nonzero section of K_{X_n}^{m} for m > 0, the product
of its n deck transforms is a nonzero invariant section of
K_{X_n}^{mn}, descending to a nonzero section of K_X^{mn}, contradicting
kappa(X) = -infinity. So kappa(X_n) = -infinity and p_g(X_n) = 0; the
Euler characteristic then gives q(X_n) = 1.
X_n is non-Kahler: averaging a Kahler form over the finite deck group
would give an invariant Kahler form on X, impossible as b_1(X) = 1. For
surfaces, b_1 = 2q if b_1 is even and b_1 = 2q - 1 if odd; and X_n
has b_1 odd (Buchdahl 1999, Theorem 11), so b_1(X_n) = 2q(X_n) - 1 = 1.
The topological Euler characteristic 2 - 2 b_1 + b_2 multiplies by the
covering degree, so b_2(X_n) = n b_2(X).
Minimality: a smooth (-1)-curve C in X_n would have K_{X_n} . C = -1. Its
image B in X, of degree d > 0, gives -1 = K_{X_n} . C = d (K_X . B) >= 0
by the projection formula and Lemma 5.2, impossible. The canonical basis
then follows from Lemma 5.2 applied to X_n. []
The cover notation is:
X, b = b_2(X) original minimal class VII surface
Y -> X, T infinite cyclic cover, unit-height generator
h, K_Y good height, canonical bundle on Y
X_n = Y/<T^n> degree-n cyclic quotient
P_lambda flat character line bundle on X
6. Compact Chern classes.
In this section we construct, on the cover Y, many line bundles whose
first Chern classes are COMPACTLY SUPPORTED with a very specific
numerical property. Concretely, we want end-framed line bundles E -> Y
whose compact Chern classes e_i satisfy
(CC) e_i^2 = K_Y . e_i = -m_i for some integer m_i > 0.
Why this identity? Recall the weighted index formula (Theorem 7.9)
that we will prove later:
index = 1 + (1/2)(e^2 - K_Y . e).
For classes satisfying (CC) the correction term vanishes, so the index
is EXACTLY 1. Combined with vanishing of higher cohomology and
uniqueness from the positive end, that produces exactly one holomorphic
section, normalised to approach 1 at the positive end. That is the
section s whose zero divisor D we study for the rest of the paper.
The key difficulty is that on a noncompact manifold a "class" depends
on the choice of compact-support representatives near infinity. We solve
this by cutting along a fixed hypersurface, transferring the class to a
compact quotient X_n, picking integral classes that restrict to zero
there, and lifting back. This is a standard transfer construction; we
spell it out because the numerical property (CC) depends on it.
6.1. The pairing on compact classes. For compactly supported classes
a, b in H_c^2(Y; Z) we set
<a, b> = integral_Y a ^ j(b),
K_Y . a = integral_Y c_1(K_Y) ^ a,
where j : H_c^2(Y) -> H^2(Y) is the forgetful map. Every integral has a
compactly supported factor, so it is finite. In de Rham notation it is a
wedge product of closed representatives, hence depends only on the
cohomology classes.
6.2. What a "datum" is. A COMPACT LINE-BUNDLE DATUM on Y is a smooth
complex line bundle E -> Y together with a fixed nowhere-zero frame
outside a compact set - equivalently, fixed frames on both sufficiently
distant ends. Such end frames define a RELATIVE first Chern class
e = c_1(E, end frames) in H_c^2(Y; Z),
whose forgetful image j(e) is the ordinary first Chern class c_1(E).
The compactly supported class e (not just its image j(e)) is what enters
the numerical identity (CC) and later the weighted index.
6.3. The seam and one period. Choose a regular value c of the height h.
The height is proper (Lemma 5.4), so Z = h^{-1}(c) is a compact, oriented,
two-sided 3-manifold in Y (possibly disconnected). We call Z a SEAM.
Under the quotient map Y -> X_n the image of Z is a 3-manifold in X_n,
still denoted Z.
The quotient identifies the cut-open X_n with the closed STRIP
W_n = h^{-1}([c, c + n]),
whose two boundary copies of Z are identified by S = T^n. Indeed every
S-orbit contains exactly one representative with c < h < c+n, except for
seam points, which lie on the boundary. No assertion about connectedness
of Z or W_n is needed; the argument is insensitive to extra components.
6.4. Torsion restriction to the seam. The construction needs every
integral degree-two class of X_n to restrict to a fixed finite TORSION
group on Z. This is a consequence of negative definiteness.
LEMMA 6.1 (Seam restriction is torsion). Let i : Z -> X_n be inclusion.
Then the restricted map
i_* : H_2(Z; Q) -> H_2(X_n; Q)
is zero, and dually the restriction
H^2(X_n; Q) -> H^2(Z; Q)
is zero. Therefore every integral degree-two class restricts to the
finite group G = Tors H^2(Z; Z).
Proof. Take a real homology class a in H_2(Z; Q) and a representative
surface Sigma in Z. Displace Sigma a small amount in the normal
direction of Z. The displaced copy is disjoint from Sigma, so its
self-intersection is 0; the displacement is isotopic to the inclusion,
so (i_* a)^2 = 0. But X_n is negative definite (Lemma 5.5 and (KOD)), so
(i_* a)^2 = 0 forces i_* a = 0. Hence i_* is zero on H_2 modulo
torsion. Over Q, cohomology-homology duality then shows the restriction
in H^2 is zero. So every integral class restricts to a torsion class,
and these lie in the fixed finite group G. []
6.5. Constructing many compact classes. With the seam restriction in
hand, we build the compactly supported classes.
LEMMA 6.2 (Many compact classes). There is a constant C_0 >= 0,
depending only on the fixed seam Z, with the following property. For
every n >= 1 one can find compact line-bundle data E_1, ..., E_d on Y,
supported in the interior of W_n, such that
(BND) d >= n b_2(X) - C_0,
and such that
(a) their transferred classes w_1, ..., w_d in H^2(X_n; Z) are
linearly independent over Q modulo torsion;
(b) their compact Chern classes e_i satisfy (CC): e_i^2 = K_Y . e_i
= -m_i for some integer m_i > 0;
(c) all intersection pairings of the compact classes e_i equal the
corresponding pairings of the transferred classes w_i on X_n.
Proof. Put N = n b_2(X). By Lemma 5.5, H^2(X_n; Z) modulo torsion
admits a negative orthonormal integral basis; choose integral lifts
u_1, ..., u_N of its elements. Each u_j restricts to some torsion class
tau_j in G (Lemma 6.1), and there are only finitely many tau.
Group the chosen basis lifts by their torsion restriction tau in G. Let
k_tau >= 1 be the order of tau (with order 1 for tau = 0) and let N_tau
be the size of the group at tau. Any sum of k_tau DISTINCT basis lifts
inside the tau-group restricts to k_tau * tau = 0 on Z. These sums are
the candidates we will use.
Claim: these sums span, over Q, a subspace of dimension at least
N - C_0, where C_0 = sum_{tau in G} (k_tau - 1).
Proof of claim. Suppose first N_tau > k_tau. Take any two distinct
indices i, j in the tau-group, and choose a set J of k_tau - 1 OTHER
indices. Then the two sums
u_i + sum_{a in J} u_a and u_j + sum_{a in J} u_a
both belong to our collection, so their difference u_i - u_j is in their
span. Differences of this kind span the hyperplane "coordinate sum = 0"
in the coordinate space of the tau-group. On the other hand, any single
one of the indicated sums has coordinate sum k_tau != 0, so the span in
fact contains the whole coordinate space: no dimension is lost.
If N_tau = k_tau there is exactly one such sum, contributing one
dimension to a group of dimension k_tau, so we lose k_tau - 1 = N_tau - 1
from this group. If N_tau < k_tau no sum is available, and we lose at
most k_tau - 1 from this group. Summing over tau gives loss at most
C_0 = sum_tau (k_tau - 1), and hence span dimension >= N - C_0.
Choose a rationally independent subcollection from the actual sums; call
its elements w_1, ..., w_d. By the claim, d >= N - C_0 = n b_2(X) - C_0,
which is (BND). Each w_i is a sum of some m_i distinct basis lifts, so
its self-intersection is -m_i (the u_j are orthonormal with -1 on the
diagonal) and likewise K_{X_n} . w_i = -m_i, since K_{X_n} = sum u_j.
Now pass from X_n back to Y. A smooth complex line bundle is classified
by its integral first Chern class, so each w_i is the Chern class of a
smooth line bundle L_i on X_n. Since w_i|_Z = 0 integrally, L_i is
trivial on a collar of Z; choose a frame there. Cut X_n open along Z,
retaining this SAME frame on the two boundary collars of W_n. Lift the
cut-open bundle to W_n subset Y, and extend it over the two remaining
ends by trivial bundles using these frames. This produces E_i on Y with
the prescribed end framings. Its compactly supported relative class e_i
lives away from the boundary of W_n, and its ordinary image is w_i.
Pairings. Choose a smooth unitary connection on L_i that is trivial in
its collar frame. Its normalised curvature is a closed 2-form
representing w_i and vanishing near Z. The same form, lifted to the
cut-open period W_n and extended by zero to all of Y, represents e_i.
Integrating a wedge product of two such forms over Y equals integrating
their original wedge product over X_n, because the periods of Y are
exactly the translates of W_n and the forms vanish on the seams. The
same holds for the canonical class, whose curvature pulls back under
the covering. So the pairings of e_i agree with those of w_i on X_n, and
in particular
e_i^2 = w_i^2 = -m_i, K_Y . e_i = K_{X_n} . w_i = -m_i.
Torsion parts contribute zero to these integer pairings. This gives
(CC) and proves the lemma. []
7. Weighted Dolbeault theory on the cyclic cover.
We now give each compact datum E a holomorphic structure and a
holomorphic section approaching 1 at the positive end and 0 at the
negative end. The structure approaches the trivial one at any prescribed
finite exponential rate; that rate remains free until Section 8 fixes it.
We prove the analytical facts needed for the rest of the paper:
(a) finiteness of cohomology in the weighted spaces that the residue
currents of Section 8 will occupy;
(b) vanishing of top-degree cohomology and uniqueness from the positive
tail;
(c) a weighted index formula computed by cut-and-paste.
Throughout, the compact input is the character cohomology vanishing from
Lemma 5.3, together with the good height h of Lemma 5.4 (which satisfies
h o T = h + 1, Lh = c > 0, |dh| <= C, bounded positive-order derivatives).
7.1. The periodic trivial operator. For lambda in C^*, the periodicity
of the flat bundle P_lambda (Section 5.5) identifies the sections of
P_lambda with T-equivariant functions or forms on Y. Fourier
transformation of the deck variable then realises the periodic Dolbeault
complex on Y as a direct integral over a circle of parameters. This is
the model that all our arguments use.
7.2. The end trivialisation condition. For an integrable operator d_E
of type (0,1) on a compact datum E, we require that in the end
trivialisations
(END) d_E = d-bar + A_E ^ ,
|nabla^j A_E| <= C_j e^{-gamma |h|} (all j >= 0),
for some gamma > 0. The same discussion applies to the dual E^{-1}, to
the trivial bundle, and after tensoring by K_Y. The point of (END) is
that all connection coefficients decay exponentially, so the operator
tends to the trivial one at both ends.
7.3. Weighted Sobolev spaces. Choose a smooth function phi_{a,b} on Y
equal to a h on a sufficiently far positive tail and equal to -b h on a
sufficiently far negative tail, with all derivatives bounded. For a real
rates a, b and integer s, set
W^s_{a,b}(Lambda^{p,q} (x) E)
= { u : e^{-phi_{a,b}} u in W^s(Lambda^{p,q} (x) E) },
||u||_{s;a,b} = ||e^{-phi_{a,b}} u||_{W^s}.
Positive a (or b) permits exponential growth at the positive (or
negative) end; negative a (or b) forces exponential decay. The pair
(a, b) is called ADMISSIBLE if both a and b are nonzero. A single
linear weight e^{-rho h} corresponds to (a, b) = (rho, -rho).
Changing phi_{a,b} on a compact set gives the same space with an
equivalent norm.
7.4. The stepped Sobolev complex. For fixed s and pairs (a, b) define
(STEP) C^{p,q}_{a,b,s}(E) = W^{s-q}_{a,b}(Lambda^{p,q} (x) E),
d_E : C^{p,q}_{a,b,s}(E) -> C^{p,q+1}_{a,b,s}(E).
The Sobolev order drops by one each time q increases, making the
differential bounded. This allows residue currents (order zero) to live
in the same kind of space as smooth forms, with the base order chosen
appropriately downstream.
We denote the resulting cohomology by H^{p,q}_{a,b}(Y, E); its
independence of s is part of the proposition below. An alternative
realisation uses the maximal closed operators d_E in the equal-order
weighted L^2 spaces; we check that they give the same cohomology.
The analytic background: the compact elliptic Hodge theorem, the local
elliptic estimate (including distributional Sobolev versions), Rellich's
compact embedding on relatively compact sets, and Plancherel. On a
compact Hermitian manifold these give the smooth Hodge decomposition and
a Green operator of order -2, with no Kahler hypothesis (see Demailly,
Complex Analytic and Differential Geometry, Ch. VI, Thms. (7.1)-(7.2)).
The periodic-end Fredholm method goes back to Taubes (1987, Lemma 4.3);
the de Rham formulation by Mrowka-Ruberman-Saveliev (2014, Prop. 2.1)
is useful background. We prove the two-ended Dolbeault statement needed
here directly.
7.5. Fredholmness.
PROPOSITION 7.1 (Fredholm complexes). For p equal to 0 or to 2, every
admissible pair (a, b) and every s in Z, the stepped complex (STEP) has
closed ranges and finite-dimensional cohomology. Its cohomology agrees
with that of the maximal weighted L^2 complex, and is represented by
forms that lie in W^k_{a,b} for every k. There are bounded homotopies of
degree -1 gaining one derivative. For the periodic trivial datum the
single-linear-weight complex with rho != 0 is acyclic in every Sobolev
realisation. Operator bounds may depend on the fixed rates and the
datum.
Proof. Step 1 (periodic case). Take a periodic complex on the whole
cover. Conjugating by e^{-rho h} changes its differential to
d_rho = d + rho * d-bar h ^ .
Fourier transformation in the deck variable decomposes d_rho as a
direct integral of the compact complexes on X with multipliers
lambda = e^{rho + i theta}, theta in [0, 2pi]. The identification of
norms on a fundamental region uses only that its translates form an
l^2 sequence in the local Sobolev spaces, and that multiplication by
e^{-rho h} is bounded on that region. Plancherel gives the direct
integral. The endpoint gauges at theta = 0, 2pi agree by the unitary
gauge e^{2pi i h}; no global logarithm of the multiplier is needed.
By the compact vanishing (Lemma 5.3), every compact complex on the
parameter circle is acyclic. Its Hodge operator d_rho + d_rho^* is
therefore invertible. The compact elliptic estimate plus a finite cover
of the parameter circle gives a UNIFORM inverse from L^2 to W^1: locally
in theta an invertible operator stays invertible by a Neumann series,
since the difference is bounded W^1 -> L^2; the compact elliptic estimate
applied to that inverse gives the uniform higher-order bound. Fourier
inversion then gives an inverse for the conjugated periodic Hodge
operator on Y, with a homotopy that gains one derivative. Duality supplies
the same bounds at negative Sobolev orders.
Step 2 (two-ended case). For a general weight the adjoint is the adjoint
in that weighted norm:
d_{E,phi}^* = e^{2 phi} d_E^* e^{-2 phi}.
Thus d_E + d_{E,phi}^*, conjugated by e^{-phi}, becomes
d~_E + d~_E^*, where d~_E = e^{-phi} d_E e^{phi}. At the positive end its
periodic limit is the operator of Step 1 with rho = a; at the negative
end it is Step 1 with rho = -b. The difference has order zero and its
coefficients tend to zero by (END).
Step 3 (parametrix). On each tail use the inverse of the corresponding
whole-cover periodic limit, with nested cutoffs equal to 1 sufficiently
far along that tail. On a relatively compact middle region use an
interior elliptic parametrix. Compose their sum with the conjugated Hodge
operator on either side. Commutators with the cutoffs are order-zero
operators supported in compact collars; following or preceding the
order -1 parametrices, they become compact by Rellich. Multiplication by
a coefficient tending to zero is compact W^1 -> L^2: truncate to a
compact set (Rellich) and bound the remaining operator by its supremum on
the tails. This handles the end errors. The resulting left and right
parametrices make the Hodge operator Fredholm from W^1 to L^2.
Step 4 (self-adjointness and homotopy). The same construction gives
||u||_{1;a,b} <= C ( ||(d_E + d_{E,phi}^*) u||_{0;a,b} + ||u||_{L^2(C)} )
for a fixed compact C, and the analogue for the adjoint. A global
first-order elliptic estimate identifies the maximal domain of the
conjugated Hodge operator with W^1: apply the local estimate to uniform
charts and sum, with cutoffs in h/R that exhaust Y and have bounded
commutators. Compactly supported smooth sections are a core. So the
operator is self-adjoint on W^1.
Since d_E^2 = 0, its square is the degree-preserving Hodge Laplacian.
Fredholmness and self-adjointness give a spectral gap off the
finite-dimensional kernel. Let P be the orthogonal projection to that
kernel and G the inverse of the Laplacian on its orthogonal complement
(G = 0 on the kernel). Uniform elliptic estimates give
G : W^t_{a,b} -> W^{t+2}_{a,b} (all t in Z),
Q = d_{E,phi}^* G : W^t_{a,b} -> W^{t+1}_{a,b}.
For negative t these follow by transposing the positive-order estimates
after conjugation. Elements of im P have all weighted Sobolev orders, so
P extends to all these spaces. Commutation with the differential (first
on smooth vectors, then by density) gives
I = P + d_E Q + Q d_E.
On the stepped complex Q maps degree q into degree q - 1, gaining one
derivative exactly as needed. A closed element has zero class precisely
when its projection under P is zero. This proves closed ranges,
finite-dimensionality, and independence of the Sobolev realisation, and
the same for the maximal L^2 complex. []
7.6. Weighted duality. The adjoint weight is built in, so we get Serre
duality in this noncompact setting.
LEMMA 7.2 (Weighted duality). For p in {0, 2} and 0 <= q <= 2, the
integration pairing gives a perfect pairing
(DUAL) H^{p,q}_{a,b}(Y, E) x H^{2-p,2-q}_{-a,-b}(Y, E^{-1}) -> C,
for every admissible pair (a, b). The pairing is evaluated on smooth
weighted harmonic representatives.
Proof. Choose phi_{-a,-b} = -phi_{a,b}, so the exponential weight
factors are reciprocal; on weighted L^2 representatives the integrand is
absolutely integrable by Cauchy-Schwarz. Integration by parts is valid
by inserting the cutoffs in h/R: their derivatives are bounded and the
L^2 tails tend to zero. So exact classes pair with closed classes to
zero.
Define an antilinear map J by the Hermitian Hodge star and the bundle
metric: it takes (p,q)-forms with values in E to complementary forms with
values in E^{-1}, characterised by the pointwise Hermitian inner
product under wedge integration. The map u |-> e^{-2 phi} J u takes the
weighted harmonic equations to the opposite-weight harmonic equations
(just transpose d_E and use d_{E,phi}^* = e^{2 phi} d_E^* e^{-2 phi}).
It is an isomorphism, and the pairing of u with its image is its weighted
squared norm, up to a fixed degree sign. Harmonic representatives thus
give a nondegenerate pairing; finiteness is Proposition 7.1.
At the cochain level, the dual of the order s-q term has order q-s, and
writing the complementary degree as 2-q shows the dual stepped complex
has base order 2-s, since (2-s)-(2-q) = q-s. So opposite weights have
this complementary Sobolev shift; independence of the shift (already
proved) lets us suppress it in (DUAL). []
7.7. Positive-tail nondecay. The sign Lh = c > 0 turns "decays
faster than any exponential at every rate" into "vanishes identically on
a positive tail". This is the workhorse maximum-principle step.
LEMMA 7.3 (Positive-tail nondecay). Let V be a holomorphic line
bundle on a positive tail {h > H_0}, with a smooth Hermitian metric of
bounded Chern curvature. Suppose a holomorphic section s satisfies, for
EVERY N > 0,
|s(x)| <= C_N e^{-N h(x)} (h sufficiently large).
Then s vanishes on a positive tail. On any connected region meeting that
tail on which s is holomorphic, s vanishes identically.
Proof. The logarithm of the norm of a holomorphic section satisfies
L log|s| >= -M in the distributional sense, including at its zeros, with
M determined by the curvature bound. In a holomorphic frame this is the
fact that log|F| is plurisubharmonic for a holomorphic function F, plus
the bounded curvature contribution. Hence
u = log|s| + (M/c) h
is subharmonic for L.
The bound on dh and the chain rule give, for sufficiently small eta > 0,
L(e^{-eta h}) = e^{-eta h}( -eta c + eta^2 sigma_L(dh, dh) ) <= 0,
because sigma_L(dh, dh) is bounded and nonnegative.
Choose a regular value h_0 > H_0 and a finite upper bound C for u on its
compact level. Given A > max(1, -C), choose a regular H > h_0 large
enough that u <= -A on {h = H}; such H exists by the assumed decay.
The superharmonic function
v_A = -A + (A + C) e^{-eta (h - h_0)}
equals C on the lower boundary and is at least -A on the upper boundary.
The maximum principle on the compact slab gives u <= v_A. (It applies
to the upper-semicontinuous subharmonic logarithm with value -infinity
at zeros, either directly or by local regularisation.) For a fixed point
of height above h_0, take H beyond that point and let A -> infinity.
Since 1 - e^{-eta (h - h_0)} > 0, we obtain u = -infinity there. []
Note: no connectedness of level sets was used; the argument is pointwise
once the barriers exist.
7.8. Definitions of finite growth and distributional smoothness. We
say a section has FINITE EXPONENTIAL GROWTH ON A TAIL if it belongs
there to some weighted Sobolev space of finite order and finite real
rate. Distributional holomorphic sections are smooth by local elliptic
regularity. Uniform local estimates then promote the membership to
every Sobolev order on a shorter tail, and give pointwise bounds of the
corresponding exponential size.
7.9. Degree-zero continuation. We improve exponential rates by
continuing the Fourier-Laurent transform to a compact base and applying
properties of compact characters. The following is the heart of the
section.
LEMMA 7.4 (Degree-zero continuation). Let V_0 be either O_Y or K_Y
with its periodic holomorphic operator d. Suppose on one tail
( + B ^ ) s = 0, |nabla^j B| <= C_j e^{-gamma |h|} (all j >= 0),
where the perturbed operator is integrable and gamma > 0. If s has
finite exponential growth, its allowed rate improves by any amount
strictly less than gamma. In the scalar case a contour crossing the
unit multiplier can contribute a CONSTANT c_pm; for V_0 = K_Y there is
no degree-zero contribution.
In the scalar case, after finitely many improvements,
s = c_pm + (exponentially decaying remainder),
with decay in all derivatives. If c_pm = 0, the section has EVERY finite
exponential decay rate on that end. In the canonical case it ALWAYS has
every finite decay rate.
Proof. We transform a cut-off section to the compact base, continue the
resulting family through the exceptional multiplier, and read the
improved rate back off the inverse contour.
Step 1 (cutoff). Consider the positive tail. Choose a cutoff chi equal
to 1 far along it and 0 off a slightly longer tail. In the periodic end
frame extend u = chi s by zero to Y. Then
d u = f = (d chi) s - chi (B s).
If s has rate r, multiplication by B puts the second term at rate r -
gamma, and the first term is compactly supported. Work at any strictly
intermediate rate to avoid boundary-convergence issues.
Step 2 (transform). For a positive-tail-supported form v define
v^(z)(x) = sum_{k in Z} e^{-z (h(x) + k)} (T^k)^* v(x).
Each expression is T-invariant, hence a form on X. If v has rate r,
Cauchy-Schwarz on the translates gives normal convergence in
compact-manifold Sobolev norms for Re z > r; differentiating gives the
same on any smaller half-plane, so v^ is holomorphic there. The transform
of the cutoff equation is
d(z) u^(z) = f^(z), d(z) = d + z (d-bar h) ^ .
Its multiplier is lambda = e^z; this fixes our sign. The source f^ is
holomorphic on Re z > r - gamma, and d(z) f^(z) = 0 there because
d^2 u = 0 in the initial half-plane, extended by the identity theorem.
Step 3 (continuation on X). For e^z != 1, compact acyclicity gives a
unique degree-zero solution of d(z) u^ = f^. It depends
holomorphically on z. To see this without picking a nonholomorphic
Hodge inverse: at a fixed z_0 where degree-zero cohomology vanishes,
compact ellipticity makes
B_0 = ( d(z_0)^* d(z_0) )^{-1} d(z_0)^*
a bounded left inverse W^t -> W^{t+1}. Writing d(z) = d(z_0) +
(z - z_0) A, any existing solution obeys
u^(z) = [ I + (z - z_0) B_0 A ]^{-1} B_0 f^(z),
which is holomorphic near z_0. In the canonical case degree-zero
cohomology also vanishes at e^{z_0} = 1, so the formula continues the
solution across it. The full equation, not just its image under B_0,
holds by analytic continuation. Higher canonical cohomology at this
parameter creates no pole in the degree-zero calculation.
Step 4 (the scalar exceptional point). In the scalar case the constants
obstruct a left inverse at e^z = 1. Reduce by a local gauge to z_0 = 0.
Solve first on the complement of the constants, then fix the constant
coefficient. Let P_0 be projection to the constants and choose a compact
left inverse B_0 with B_0 d = I - P_0. Write a prospective solution as
c + v with v orthogonal to the constants, and set
R(z) = (I + z B_0 A)^{-1},
v = R(z) B_0 f^ - z c R(z) B_0 A 1,
where inverses act on the complement of the constants. Put
w(z) = 1 - z R(z) B_0 A 1; the candidate solution is
R(z) B_0 f^ + c w(z).
The class of A 1 = d-bar h in H^1(X, O) is nonzero: if d-bar h = d-bar g
for a function on X, then h - g would be a holomorphic function on Y
changing by 1 under T, contradicting Lemma 5.3. Pick a continuous scalar
functional L on degree-one forms which annihilates image(d) and satisfies
L(A 1) != 0 (pairing with the harmonic representative of this nonzero
class gives such an L). Applying L to the transformed equation
determines
c(z) = L(f^ - d(z) R(z) B_0 f^) / L(d(z) w(z)),
L(d(z) w(z)) = z L(A 1) + O(z^2).
This is meromorphic with at most a simple pole. For every sufficiently
small nonzero z, compact acyclicity already supplies a unique solution of
the full transformed equation; by uniqueness it agrees with our
candidate, so the candidate solves the full equation on the punctured
neighborhood and continues it meromorphically across 0. The residue is a
constant because the pole part in the differential equation is killed by
d. This gives the desired finite-order meromorphic continuation without
invoking a Fredholm theorem for the overdetermined map
d : W^{t+1}(Lambda^{0,0}) -> W^t(Lambda^{0,1}).
Step 5 (contour inversion on the positive end). For a target
positive-end rate r_1 with r - gamma < r_1 < r, choose a contour
real part sigma != 0 satisfying
r - gamma < sigma < r_1 < r.
Start from any vertical line with real part above r; both lines lie in
the source's holomorphy domain, and the continued solution has bounded
Sobolev norm on each multiplier circle. Moving the contour to real part
sigma gives, on a fixed fundamental region,
(T^k)^* u(x) = (1/2 pi i) integral_{sigma - i pi}^{sigma + i pi}
e^{z (h(x) + k)} u^(z)(x) dz,
plus residues crossed. Horizontal boundary terms cancel: under
z |-> z + 2 pi i the transform is gauged by e^{-2 pi i h}, and the
factor in the inversion integral changes by its inverse. Cauchy's
estimate on the new line bounds its contribution on the k-th translate
by C e^{sigma k} in every fixed Sobolev norm. After weighting by
e^{-r_1 k}, the squared norms on the positive translates are summable
because r_1 - sigma > 0. The scalar-pole residues are the only ones;
there are none in the canonical case.
Step 6 (negative end, by symmetry). For a cutoff supported in {h < H},
the transform converges first in a left half-plane. If the allowed
negative-end rate is b, the initial half-plane is Re z < -b, and the
source extends to Re z < -b + gamma. Given b_1 with
b - gamma < b_1 < b, choose a contour with -b_1 < sigma < -b + gamma
and sigma != 0. On the (-l)-th translate its contribution is
O(e^{-sigma l}), whose squared norms become summable after weighting by
e^{-b_1 l} because sigma + b_1 > 0. Moving the contour to the right
improves the negative-end rate; the residue discussion is unchanged.
Step 7 (iteration). Iterate Steps 5-6. For a homogeneous perturbed
equation, each improvement in s improves the source -B s by the same
amount. Choose improvements smaller than gamma / 3 at each step,
perturbing contour positions to avoid real part zero. Canonical
sections have no residue, so iteration reaches every finite decay rate.
Scalar sections can cross zero once, giving s = c_pm + r with r at a
strictly negative rate. If c_pm = 0, iteration continues indefinitely
and s has every finite decay rate. If c_pm != 0, the remainder satisfies
(d + B ^) r = -B c_pm, which only gives a finite decay rate for that
remainder; uniform local elliptic estimates give the stated decay of
all its derivatives. []
Remark (generalised modes). The same calculation identifies the
possible homogeneous modes. A pole of finite order at the unit
multiplier would give a polynomial-times-equivariant homogeneous mode.
Multiplication of the transform by a power of e^z - 1 removes that
pole, so the corresponding section is killed by a power of T^* - 1.
Such a nonconstant scalar mode would have a last nonconstant member v
with (T^* - 1) v a nonzero constant, excluded by Lemma 5.3. A
canonical generalised mode would have a last nonzero invariant member,
contradicting H^0(X, K_X) = 0. So the explicit calculations agree with
the absence of generalised modes.
7.10. Tail asymptotics. Combining the two previous lemmas gives the
concrete expansions we will use.
PROPOSITION 7.5 (Tail asymptotics). Let E satisfy (END). A finite-growth
holomorphic section of K_Y (x) E or K_Y (x) E^{-1} on a positive tail
vanishes on a shorter positive tail. A finite-growth holomorphic
section of E has, in the prescribed trivialisation at either end where
it is defined, the expansion
s = c_pm + O(e^{-delta |h|})
for some delta > 0, with the same conclusion for every derivative of the
error. On the positive end c_+ = 0 implies s = 0. Consequently the
positive-end coefficient is INJECTIVE on global weighted holomorphic
sections of E.
Proof. Apply Lemma 7.4. The Chern curvatures of the metrics involved are
bounded: those of the periodic bundles are bounded, and the connection
coefficients and their first derivatives in the end frames are bounded
by (END). Every canonical section, and a scalar-type section with zero
positive coefficient, decays faster than every exponential on the
positive tail. Lemma 7.3 makes it zero there. Unique continuation for a
holomorphic section on the connected Y then gives the global injectivity
assertion. The negative-end expansion uses no maximum principle at that
end, and makes no vanishing assertion there. []
7.6. Rapid scalar solutions. We need to integrate arbitrary compactly
supported (0,2)-forms by d-bar with arbitrarily rapid decay at both ends.
This is the auxiliary equation behind existence of holomorphic compact
data.
LEMMA 7.6 (Scalar top-degree vanishing). For the scalar periodic
Dolbeault complex on Y and every admissible pair (a, b),
H^{0,2}_{a,b}(Y, O) = 0.
In particular, if f is a smooth compactly supported (0,2)-form and R > 0,
there is a smooth (0,1)-form alpha with d-bar alpha = f and
alpha in W^k_{-R,-R} for every k in Z.
Proof. By Lemma 7.2, the dual of the group is H^{2,0}_{-a,-b}(Y, O).
Its elements are canonical sections with finite exponential growth on the
positive end, and these vanish by Proposition 7.5 applied to the
periodic trivial datum. Closed range and finite-dimensionality turn the
dual vanishing into the asserted vanishing. For f, take the homotopy Q
of Proposition 7.1 at weights (-R, -R); the top-degree harmonic
projection is zero, so alpha = Q f solves d-bar alpha = f. Elliptic
regularity and the higher-order bounds for Q give every asserted
weighted Sobolev order. []
7.7. Holomorphic compact data. The structure on E is prescribed at the
ends by a smooth frame. We perturb it to be holomorphic by solving the
scalar equation above, since for a line bundle the endomorphism bundle is
canonically trivial.
PROPOSITION 7.7 (Holomorphic structures). Every compact line-bundle
datum admits an integrable (0,1)-connection satisfying (END) with any
prescribed finite gamma > 0. For every such structure and every
admissible pair (a, b),
(E-TOP) H^{0,2}_{a,b}(Y, E) = 0.
For every a, b > 0, global weighted holomorphic sections have the two
constant end coefficients and exponentially decaying errors as in
Proposition 7.5, and the map s |-> c_+ is injective.
Proof. Choose a smooth (0,1)-connection d_0 on E that is standard in the
given end trivialisations. Such a connection is constructed by partition
of unity, because the space of such connections is affine. Its curvature
d_0^2 is a compactly supported scalar (0,2)-form kappa: the endomorphism
bundle of a line bundle is canonically trivial, and in complex dimension
two every (0,2)-form is d-bar closed. Lemma 7.6, proved for the trivial
periodic bundle alone, solves
d-bar alpha = -kappa
with weights (-R, -R), where R > gamma. Set d_E = d_0 + alpha ^ . In a
local frame its square is
d_E^2 = kappa + d-bar alpha + alpha ^ alpha = 0,
because alpha is scalar-valued and has degree one. So d_E is
integrable. In rank one the local integrability theorem is just the
local Dolbeault lemma applied to the closed connection form, followed by
multiplying the frame by the exponential of a local primitive.
Weighted Sobolev embedding on the uniform charts gives
|nabla^j alpha| <= C_j e^{-R |h|} on the tails,
so (END) holds at the prescribed gamma. The finitely many middle charts
impose no end condition.
Now weighted duality identifies the dual of H^{0,2}_{a,b}(Y, E) with
H^{2,0}_{-a,-b}(Y, E^{-1}), which vanishes by Proposition 7.5. This is
(E-TOP). The section assertions follow from the same proposition, whose
finite-growth hypothesis allows every positive finite rate. The scalar
top-degree vanishing used to construct d_E was proved earlier and did
not assume the existence of the present holomorphic structure. []
7.8. The trivial datum's index. Before computing the index of a general
datum we record the trivial case, which is the reference point.
LEMMA 7.8 (Base index). For the trivial datum and a, b > 0,
H^{0,0}_{a,b}(Y, O) = C,
H^{0,1}_{a,b}(Y, O) = H^{0,2}_{a,b}(Y, O) = 0.
In particular the weighted Dolbeault index of O is 1.
Proof. A global weighted holomorphic function has a positive-end
constant coefficient. Subtract that constant; the result is holomorphic
with zero positive coefficient, hence vanishes by Proposition 7.5.
Constants belong because both end rates are positive. The top-degree
assertion is Lemma 7.6. This proves H^{0,0} = C and H^{0,2} = 0.
For degree one, represent a class by a smooth weighted harmonic
(0,1)-form u, so all its weighted Sobolev norms are finite. Choose a
partition of unity chi_+ + chi_- = 1 where chi_+ is 0 far down the
negative end and 1 far up the positive end. The compactly supported
(0,2)-form
g = d-bar(chi_+ u) = -d-bar(chi_- u)
admits by Lemma 7.6 a primitive v decaying at both ends at any prescribed
finite rate; pick that rate larger than a + b + 1. Then
u_+ = chi_+ u - v, u_- = chi_- u + v
are closed, and u = u_+ + u_-. The first belongs to the single-linear
space (a, -a) because its negative tail is -v; the second to the
single-linear space (-b, b) because its positive tail is v. Both are
acyclic by Proposition 7.1. Choose d-bar f_+ = u_+, d-bar f_- = u_- in
those spaces. Each embeds into the two-ended space (a, b) (one end has
the same weight, the other has a stronger decay requirement), so
u = d-bar(f_+ + f_-) in the original complex. []
7.9. The weighted index by compact excision. For a compact datum E
supported in a cut-open period of X_n, the index is a compact-surface
Riemann-Roch computation dressed in weighted language.
THEOREM 7.9 (Weighted index). Let E be a compact datum supported in a
cut-open period of X_n, with trivialisations fixed in collars of the
cut. Equip E with an integrable d_E satisfying (END), and let e be its
compactly supported Chern class. For a, b > 0 the weighted Dolbeault
index is
(IDX) sum_q (-1)^q dim H^{0,q}_{a,b}(Y, E)
= 1 + (1/2) (e^2 - K_Y . e).
The products on the right are compactly supported intersection
pairings. In particular, if e^2 = K_Y . e, then H^{0,0}_{a,b}(Y, E) is
one-dimensional, its positive-end coefficient is nonzero on every
nonzero section, and H^{0,1}_{a,b}(Y, E) = 0.
Proof. Step 1 (replace by a standard connection). The even-to-odd
operator
D_E^+ = d_E + d_{E,phi}^* :
W^1_{a,b}(Lambda^{0,0} E (+) Lambda^{0,2} E)
-> L^2_{a,b}(Lambda^{0,1} E)
has index equal to the Euler characteristic in (IDX), by the Hodge
decomposition of Proposition 7.1. Replace d_E by any smooth (0,1)-
connection d_0 that is standard at the ends. The difference, after
conjugation, is multiplication by exponentially decaying coefficients,
hence compact W^1 -> L^2. The index is unchanged. We do NOT require
d_0^2 = 0: D_0^+ is still first-order elliptic with the same principal
symbol and the same invertible end limits.
Step 2 (excision). Transfer the datum and this connection to X_n, making
them trivial and standard across the cut collars. Denote the resulting
smooth bundle by E_n and its elliptic operator by D_{E_n}^+. Modify
phi_{a,b} on a compact set so that it is zero on a neighbourhood of the
transferred piece and its collars; this does not change the index. Then
the compact cut-and-paste principle (in the form of Mrowka-Ruberman-
Saveliev 2014, Prop. 3.1), now applied to this Dolbeault operator, gives
(EXC) index D_{E,Y}^+ + index D_{O,X_n}^+
= index D_{O,Y}^+ + index D_{E_n,X_n}^+.
Direct verification. Compare the two direct sums. On the common compact
interior, interchange the two copies, carrying the E copy to the
transferred E_n copy and the trivial copy to the trivial copy. Outside
the identified region use the identity. In each collar both bundles are
trivialised; interpolate between interchange and identity using the real
rotation matrix
( cos theta -sin theta )
( sin theta cos theta ), 0 <= theta <= pi/2,
the SAME rotation on the even and odd bundles. This defines bounded
invertible maps on the direct-sum domains and ranges. On the interior the
operators are interchanged exactly. On the collars their principal
symbols are the identical Dolbeault symbols, which commute with the
rotation; differentiating the rotation adds only order-zero terms.
Outside the compact comparison region the operators agree. So the
conjugated operators differ by compactly supported order-zero terms,
compact W^1 -> L^2. Their indices agree, proving (EXC).
Step 3 (Riemann-Roch). The transferred connection on X_n need not be
integrable. To compute its index by ordinary Hirzebruch-Riemann-Roch,
recall from Lemma 5.5 that H^2(X_n, O) = 0. The exponential sequence then
shows that every integral class in H^2(X_n; Z) is the first Chern class of
a holomorphic line bundle; and smooth complex line bundles are
classified by their first Chern class. So E_n admits SOME holomorphic
structure. Replacing the transferred connection by its Dolbeault
operator changes the closed elliptic operator only in order zero and
hence preserves its index. Compact-complex Riemann-Roch (Atiyah-Singer
III 1968, Sec. 4) gives
index D_{E_n,X_n}^+ - index D_{O,X_n}^+
= chi(X_n, E_n) - chi(X_n, O_{X_n})
= (1/2) (c_1(E_n)^2 - K_{X_n} . c_1(E_n)).
These products equal e^2 and K_Y . e, since the class was transferred in
its fixed trivialised collars. Combining (EXC) with Lemma 7.8 (the base
index = 1) proves (IDX).
Step 4 (the special case). When e^2 = K_Y . e, Proposition 7.7 gives
H^{0,2} = 0 and Proposition 7.5 gives dim H^{0,0} <= 1. The index
equality reads dim H^{0,0} - dim H^{0,1} = 1. Both dimensions are
nonnegative integers, so they are respectively 1 and 0, with the
positive-end coefficient nonzero. []
7.10. Sections with prescribed end limits. This is the payoff. Given
one of the compact classes e_i with e_i^2 = K_Y . e_i = -m_i < 0, we now
build the section s whose zero divisor D we analyse for the rest of the
paper.
PROPOSITION 7.10 (A section with different end limits). Let E be one of
the data of Lemma 6.2, and write e^2 = K_Y . e = -m with m > 0. Given any
finite gamma > 0, E admits a holomorphic structure whose Dolbeault
operator, in its fixed end frames, differs from the trivial operator by
O(e^{-gamma |h|}) in every derivative. For this structure there is a
holomorphic section s such that, in those frames,
(LIM) s -> 1 (h -> +infinity),
s -> 0 (h -> -infinity),
with exponentially decaying errors. Its effective zero divisor D = (s=0)
is contained in {h <= H} for some H, is locally finite, and represents
the ordinary Chern class j(e), equivalently its dual locally finite
homology class.
Proof. Proposition 7.7 supplies the stated holomorphic structure with
the prescribed finite rate gamma. Choose eps > 0 and use the pair
(eps, eps). Theorem 7.9 gives
index d-bar_E = 1 + (1/2)(e^2 - K_Y . e) = 1.
The weighted degree-two vanishing of Proposition 7.7 gives H^{0,2} = 0,
and the positive-end uniqueness of Proposition 7.5 makes the map
H^{0,0}_{eps,eps}(Y, E) -> C, s |-> c_+, injective. The index identity
therefore makes the degree-zero space one-dimensional and the degree-one
space zero. Take its nonzero section and normalise c_+ = 1. Proposition
7.5 then gives a constant negative-end coefficient c_- and exponentially
small errors at both ends.
Suppose c_- != 0. Uniform convergence to nonzero constants makes s
nowhere zero outside a compact height slab, so D is compact. Its compact
divisor class equals the specific relative class e (not just its ordinary
image): on the ends write s = f_pm tau_pm in the fixed frames. Far enough
out, f_+ lies in a disk about 1 that avoids zero, and f_- in a disk about
c_- that avoids zero. Straight-line homotopies in these disks deform f_pm
through nowhere-zero maps to their limiting constants, and each nonzero
constant is homotopic to 1 in C^*. So the trivialisations provided by s
on the ends are homotopic to the prescribed ones, and the relative
zero-divisor formula for c_1 identifies the compact divisor class with e.
The projection pi restricted to the compact D is proper, so pi_* D is an
effective divisor on X with positive integral multiplicities equal to
the generic mapping degrees. Since K_Y = pi^* K_X, the projection
formula gives
K_X . pi_* D = K_Y . D = K_Y . e = -m < 0,
contradicting Lemma 5.2 (every effective divisor has nonnegative
canonical degree). If D were empty, the left side would be zero and give
the same contradiction. So c_- = 0, which proves (LIM).
The positive limit (LIM) excludes zeros for large h. The zero set of a
nonzero holomorphic section is a locally finite effective divisor. The
local divisor construction identifies E is isomorphic to O_Y(D), so
[D] = c_1(E) = j(e) in ordinary cohomology; the corresponding
Chern-current identity is the local form of Demailly (2012, Ch. V,
(13.2)). Under Poincare duality for the oriented noncompact four-manifold
Y, this says the locally finite fundamental cycle of D is the dual of
j(e). In particular, for every compactly supported closed 2-form alpha,
(DIV) integral_Y alpha ^ j(e) = integral_D alpha.
The right side is well defined by local finiteness. No assertion that D
is a compact divisor is being made here. []
The rate gamma in Proposition 7.10 is still free. Section 8 first
obtains an area exponent A from the fixed lifted coordinates; only then
do we use Proposition 7.10 with gamma > 2A + 10. So choosing A later
imposes no condition on the compact classes of Section 6.
8. Divisor growth and residue currents.
Proposition 7.10 constructs, for each compact datum E, a section with end
limits (1, 0). Its zero divisor D may still fail to be compact - the
components of D could extend to the negative end. In this section we
begin ruling that out. The plan:
(a) bound the AREA of D on negative height slabs by a common
exponential progression C e^{A l};
(b) attach to any holomorphic differential on a component a
(2,1)-current whose mass obeys the same bound;
(c) show distinct components give independent cohomology classes,
and the scalar version forces the components to have genus zero.
A separate cylinder estimate (Section 9) then excludes the remaining
open shapes. We keep careful track of the constants: the area exponent
A must be FIXED before the holomorphic structures of Section 7 are
chosen, so that the rate gamma can be made much larger than A.
We work on Y with the lifted Hermitian metric. Areas of divisors count
multiplicities; areas of curves are taken on the reduced curve.
8.1. The area exponent.
Put
I_l = [-l - 1, -l], Y_l = h^{-1}(I_l) (l in N union {0}).
So the Y_l tile a fixed range of "middle height" on the negative side;
the height h decreases one unit at a time as l increases.
LEMMA 8.1 (A geometric area exponent). There is a number A > 0,
depending only on a fixed finite coordinate atlas of X, its lifted
metric, and the height h, with the following property. Let E be any
compact datum whose holomorphic operator tends smoothly to the trivial
operator at the NEGATIVE end, and let s be a nonzero holomorphic section
bounded in the prescribed smooth trivialisation there. If D = (s = 0),
then
(AREA) Area(D intersect Y_l) <= C_{E,s} exp(A l), l >= 0.
The constant C_{E,s} may depend on the datum, the operator, and the
section. The exponent A does NOT. The same exponent applies to every
reduced component of D and to each fixed deck translate of such a
component. Replacing I_l by any fixed-width enlargement preserves the
exponent.
Proof. Step 1 (a finite transition system). Choose a compact set
Q subset Y whose deck translates cover Y. Choose finitely many small
coordinate balls covering Q, paths joining their centres to a point p,
and a path from p to T^{-1} p. Their union is compact. Subdividing the
paths and slightly enlarging the balls produces a finite collection of
ball chains. Repeating its translates connects a starting ball to every
ball covering T^{-l} Q in at most N (l + 1) transitions, for one fixed N.
All balls in a chain lie in a fixed-width enlargement of the height
interval the chain traverses. Only connectedness of Y, compactness of
the chosen paths, and h o T = h + 1 enter; connectedness of level sets
is not used.
Step 2 (three-circle propagation). If F is holomorphic on a Euclidean
ball of radius r_3 and 0 < r_1 < r_2 < r_3, the three-circle inequality
reads
log sup_{B_{r_2}} |F|
<= theta log sup_{B_{r_1}} |F| + (1 - theta) log sup_{B_{r_3}} |F|,
theta = log(r_3/r_2) / log(r_3/r_1).
Restrict to each complex line through the centre: this is the
one-variable version; take the supremum over the lines. Choose the ball
chains so that a small ball at one stage is inside a middle ball at the
next, while that stage's large ball stays in its coordinate chart.
The finitely many radius ratios have a positive lower bound theta_0 > 0
on theta; a bounded number of extra transitions reaches the small
concentric balls needed below.
Step 3 (holomorphic frames on the negative tail). On a sufficiently
negative tail, write the operator as d-bar + alpha in the prescribed
smooth frame. Since alpha -> 0 smoothly and d-bar alpha = 0, on each
large coordinate ball there is u with d-bar u = alpha and |u| <= C_0 on a
slightly smaller ball (the local Dolbeault lemma with its interior
estimate, here via the Cauchy-Green operator). The finite atlas makes
these bounds uniform. Moving the start of the tail makes the coefficient
norms at most one, independently of the original decay rate. The frames
e^{-u} are holomorphic, and the absolute values of transition functions
between them are bounded above and below by geometric constants.
Step 4 (lower bound on log |s|). Let F denote the coefficient of s in
such a holomorphic frame. Boundedness of s gives a common upper bound
log |F| <= U on the tail (U may depend on s). At each transition, by
the three-circle inequality, the inclusion of the preceding small ball
in the next middle ball, and the bounded frame changes, if the
logarithmic supremum on the preceding small ball is at least -M then on
the next small ball it is at least
-c M - C_1 (U + 1), c = theta_0^{-1} > 1.
Since s is nonzero, unique continuation supplies a finite starting value
M. Start far enough down the tail that all repeated chains stay where
the frame bounds hold. Iteration over at most N (l + 1) transitions gives
(LOGLOW) sup_V log |F| >= -C_2 c^{N l}
on every small ball V needed to cover the l-th negative cell. Changing
the starting cell changes C_2, not c or N. A slab Y_l meets only a fixed
number of these cells (h is bounded on Q).
Step 5 (logarithmic supremum -> L^1). Arrange nested coordinate balls of
radii r, 2r, 4r, and choose x in B_r with
log |F(x)| >= -M - 1 (using (LOGLOW)). The nonnegative function
v = U - log |F| is superharmonic. The ball B(x, 3r) contains B_{2r} and
lies in B_{4r}, so its mean inequality gives
integral_{B_{2r}} v <= integral_{B(x,3r)} v <= |B_{3r}| v(x)
<= |B_{3r}| (U + M + 1).
The usual limiting interpretation at zeros applies at no cost. Thus
integral_{B_{2r}} |log |F|| <= C_3 (U + M + 1).
Step 6 (Poincare-Lelong). Choose a smooth cutoff chi equal to 1 on B_r
and supported in B_{2r}. Poincare-Lelong (Demailly 2012, Ch. III,
(2.15)) and integration by parts give, up to the fixed d^c normalisation,
integral_D chi omega
= c_0 integral log |F| dd^c(chi omega)
<= C_4 integral_{B_{2r}} |log |F||.
Coefficients of dd^c(chi omega) are uniformly bounded; no closedness of
omega is used. Summing over the fixed number of balls covering one slab
and using (LOGLOW) gives (AREA), for example with any fixed A > N log c.
Finitely many slabs outside the chosen tail are compact and absorb into
C_{E,s}.
Step 7 (components and translates). A reduced component has no more
area than the effective divisor. Since T is an isometry raising height
by one, replacing a component by T^k B shifts slab index by k; the
exponent stays A after absorbing finitely many slabs in the constant.
A fixed-width enlargement is a bounded number of unit slabs, indices
differing by a bounded amount. []
8.2. Fixing the rates. From now on fix this geometric A. Make every
application of Proposition 7.10 with a decay rate
(GAMMA) gamma > 2 A + 10.
That proposition was proved for every prescribed finite rate, so this is
a specialisation. For each compact datum we obtain E, s, D with
s -> 1 at the positive end and s -> 0 at the negative end; in particular
D is bounded above in height. Lemma 8.1 applies with the already fixed A.
8.3. Currents attached to a curve. Let B subset Y be a closed irreducible
analytic curve, possibly noncompact, bounded above in height, and let
f : R -> B subset Y be its normalisation. Normalisation is finite over
each relatively compact open set (Demailly 2012, Ch. II, (7.12)), so f is
proper. If F is O or one of the bundles E, and eta is a holomorphic
section of K_R (x) f^* F, define its PUSHFORWARD CURRENT C_eta by
(PUSH) C_eta(phi) = integral_R eta ^ f^* phi,
phi in C_c^inf(Y, Lambda^{0,1} (x) F^*).
The current has bidegree (2,1) and coefficients in F. Properness makes
the integral locally finite, including at the singular points of B.
Stokes on R applied to a compactly supported pullback shows
d-bar_F C_eta = 0. With any conformal metric on R there is a fixed
convention-dependent constant c with
(MASS) Mass(C_eta; V) <= c integral_{f^{-1}(V)} |eta| |df| dA_R.
The right side is conformally invariant; it also shows C_eta is of
order zero.
8.4. The current complex and its cohomology. We need three properties
of these currents: finite-dimensional cohomology for a FIXED bundle,
vanishing of the SCALAR degree-one cohomology, and vanishing on a
positive tail of any (2,0)-primitive of a (2,1)-current supported below a
height bound. All three hold in the weighted spaces below, whose mass
criterion specifies which currents live in them.
THEOREM 8.2 (Weighted cohomology for residue currents). Fix eps > 0 and
b > eps. Let F be either O_Y or an end-framed compact datum E with an
integrable operator satisfying (END). Consider the current complex
(CUR) C^{2,q}_{-eps,b,-3}(F)
= W^{-3-q}_{-eps,b}(Lambda^{2,q} (x) F), q = 0, 1, 2.
Its cohomology is finite-dimensional. For F = O_Y the degree-one
cohomology is ZERO. In particular every closed scalar (2,1)-current in
this space that is zero on a positive tail is exact.
If a degree-zero current beta in W^{-3}_{-eps,b}(Lambda^{2,0} (x) F)
satisfies d-bar_F beta = J for a degree-one current J supported in
{h <= C}, then beta vanishes on a positive tail. Finally, an order-zero
(2,1)-current supported in {h <= C} whose masses on negative unit
height slabs are at most C_0 e^{mu l} belongs to (CUR) whenever b > mu.
The same Sobolev order and pair of rates work for any family with that
common exponent mu, regardless of its individual constants and upper
support bounds.
Proof. Step 1 (finite-dimensionality). Proposition 7.1 at s = -3.
Step 2 (scalar degree-one exactness with bounded-above support). If J
is scalar and zero on the positive tail, it belongs to the
single-linear-rate complex (-b, b): its negative weight is unchanged
and its positive part is compactly supported. That complex is acyclic
(Prop. 7.1). Its primitive has rates (-b, b), hence also belongs to
(-eps, b) since b > eps.
Step 3 (full scalar degree-one vanishing without a contour argument).
Keep b fixed and continuously vary the positive rate from -b to -eps.
After conjugation the even-to-odd Hodge operators have the same domain
W^1 and range L^2, and vary continuously in operator norm (only
bounded order-zero coefficients depending on the derivative of the
weight change). They remain Fredholm by Prop. 7.1, since the positive
rate stays negative. Their index is consequently constant, equal to 0
at the acyclic point (-b, b). At every point of the path, H^{2,0} = 0
by positive-tail uniqueness (Prop. 7.5). Weighted duality identifies
H^{2,2} with the global scalar holomorphic functions having positive
rate at the positive end and rate -b at the negative end. Every such
function is constant: subtract its positive-end coefficient and use
Prop. 7.5; the negative-end decay requirement excludes a nonzero
constant. Thus H^{2,2} = 0. Index zero now forces H^{2,1} = 0.
Step 4 (primitive vanishes on positive tail). On {h > C} the
distributional section beta solves the degree-zero equation for
K_Y (x) F. Local elliptic regularity makes it smooth there. Uniform
interior estimates on shorter tails turn its weighted W^{-3} bound into
finite exponential growth in every positive Sobolev order. Prop. 7.5
therefore makes it zero on a positive tail, for every primitive in the
stated space.
Step 5 (mass criterion membership). Localise to the uniformly bounded
coordinate charts on one enlarged slab. In real dimension four, W^4
embeds continuously into C^0, with a uniform constant on these charts.
An order-zero current of mass M thus has W^{-4} norm at most C M. A
uniformly locally finite partition of unity and bounded derivatives of
the weight bound the squared weighted norm by a constant multiple of
sum_{l>=0} e^{-2 b l} (C_0 e^{mu l})^2
= C_0^2 sum_{l>=0} e^{-2 (b - mu) l} < infinity.
There are only finitely many additional slabs between the zero level
and the upper support bound, and none beyond it. This puts the current
in W^{-4}_{-eps,b}, which is degree one of (CUR). Individual constants
enter the norm, not the choice of spaces nor the dimension of the common
cohomology group. []
Consequently, if an order-zero (2,1)-current C with coefficients in
F in {O, E} is bounded above in height and satisfies
Mass(C; Y_l) <= C_C e^{mu l}, then for any eps > 0 and b > max{mu, eps}
the theorem places it in W^{-4}_{-eps,b}(Lambda^{2,1} (x) F). The
inequality b > mu makes the weighted mass sum converge. The base Sobolev
order -3 and degree-one order -4 are fixed regardless of the current and
its mass exponent. We always fix rates before comparing current classes,
so no dimension bound uniform over changing weights is needed.
8.5. Residue independence. If a linear combination of currents is
exact, its primitive is supported on the union of the curves; a local
distribution supported on a curve cannot have the required order-zero
residue as its Dolbeault derivative. This gives the independence we use
for both scalar and bundle-valued differentials.
PROPOSITION 8.3 (Residue obstruction). Fix F in {O, E} and rates
(-eps, b) as in Theorem 8.2. For distinct closed irreducible curves
B_i subset Y bounded above in height, let eta_i be nonzero holomorphic
differentials on their normalisations with coefficients in f_i^* F.
Suppose their pushforward currents have negative-end mass exponents
strictly smaller than b. Then the classes
[C_{eta_i}] in H^{2,1}_{-eps,b}(Y, F)
are linearly independent. For F = O, even ONE such differential is
impossible.
Proof. The mass criterion of Theorem 8.2 puts the currents in the
complex (CUR). Suppose a finite combination is exact:
sum_i a_i C_{eta_i} = d-bar_F beta.
The sum is zero on a positive tail. The primitive assertion of Theorem
8.2 says beta is zero there too. On Y minus the union of the B_i,
distributional d-bar_F beta = 0, so local elliptic regularity makes beta a
holomorphic section of K_Y (x) F on that complement. The complement is
connected: a path in the connected real four-manifold can be perturbed
off the smooth real-codimension-two strata of a finite union of analytic
curves and off its locally discrete singular strata; only finitely many
charts are needed along the compact path. Analytic continuation from
the positive tail gives
supp beta subset union_i B_i.
Suppose a_i != 0. Choose a smooth point of B_i, outside the other
curves, where eta_i != 0 (singular and intersection sets are locally
discrete, and a nonzero holomorphic differential has isolated zeros).
In holomorphic coordinates (z, w) and a holomorphic frame of F, the
curve is w = 0, the differential is a(z) dz with a != 0, and
beta = u dz ^ dw for a distribution supported on w = 0. Its local finite
normal-jet expansion is
(JET) u = sum_{p+q<=N} u_{pq}(z) partial_w^p partial_wbar^q delta_0(w),
with distributional tangential coefficients. This is the support theorem
for distributions (Hormander 1990, Thm. 2.3.5); locally it follows by
taking a finite-order bound for u, Taylor-expanding a test function in
its normal variables, and cutting off in |w| < delta. A test function
whose normal derivatives through order N vanish on w = 0 has C^N norm
O(delta) after that cutoff, so it pairs to zero as delta -> 0. Only the
displayed finite Taylor jet can matter; the normal jets in (JET) are
independent, as one sees by prescribing those jets of test functions.
The coefficient of the normal exterior-form direction in d-bar beta is
partial_wbar u; the tangential direction cannot contribute. The
corresponding coefficient of the right-hand side is a nonzero constant
multiple of a_i a(z) delta_0(w). But normal differentiation in (JET)
increases q by one. Comparing the largest q + 1 (then descending) forces
every u_{pq} to vanish and leaves zero where the nonzero order-zero
delta mass should be. Contradiction, so currents are independent.
Notice that the familiar principal-value meromorphic primitive of a
residue is not supported on the divisor; the support conclusion is
essential.
For F = O, Theorem 8.2 makes every closed current of the kind under
consideration exact: its support is bounded above, so a single linear
weight (-b, b) supplies the scalar primitive. Then the independence
assertion rules out even one nonzero current. []
8.6. From analytic to topological: genus zero. An L^2 differential
satisfies the mass bound of (MASS) by Cauchy-Schwarz and the area
estimate, so the scalar residue obstruction excludes such differentials.
The next proposition turns this into a topological constraint on the
normalisation, including when it is noncompact.
PROPOSITION 8.4 (Genus zero). The normalisation R of any component of D
has no nonzero square-integrable scalar holomorphic differential and has
genus zero. If R is compact, it is biholomorphic to CP^1. If R is
noncompact, it is biholomorphic to a domain in CP^1, without any
restriction on the connectivity of that domain.
Proof. Step 1 (no L^2 differential). Let eta be a square-integrable
holomorphic differential on R. Cauchy-Schwarz in (MASS) gives
Mass(C_eta; Y_l)
<= c ||eta||_{L^2(R)}
( integral_{f^{-1}(Y_l)} |df|^2 dA_R )^{1/2}
<= C ||eta||_{L^2(R)} e^{A l / 2},
where the energy of a holomorphic normalisation is a fixed multiple of
the reduced curve area (ramification and singular points are
measure-zero) and Lemma 8.1 gives the last bound. The scalar case of
Prop. 8.3 forces eta = 0.
Step 2 (handles and L^2 obstructions on open R). A handle in an
oriented surface contains two embedded circles with algebraic
intersection one. Smooth closed forms alpha, beta supported in small
annular neighbourhoods of those circles, dual to them, satisfy
integral_R alpha ^ beta = 1.
(For example a bump function of the transverse coordinate, times its
differential and with integral one.) Choose a conformal metric and let V
be the closure of d C_c^inf(R, R) in the real Hilbert space of L^2
one-forms. Set u = alpha - proj_V alpha. Forms in V are weakly closed,
so du = 0; orthogonality to d C_c^inf gives d^* u = 0. Integration by
parts against the compactly supported closed form beta, then L^2
continuity, gives
integral_R u ^ beta = integral_R alpha ^ beta = 1.
So u != 0. Local elliptic regularity makes u smooth. With the complex
orientation, u + i * u is a nonzero (1,0)-form; the two equations for u
make it closed and hence holomorphic. It is L^2, a contradiction. No
closed-range assertion for the de Rham differential on a noncompact
surface is used.
Step 3 (genus zero, then uniformisation). Consequently there is no
handle in any compact subsurface of R: this is genus zero. Equivalently,
every Jordan curve separates (a nonseparating Jordan curve, joined from
one side to the other in its complement, produces a second circle with
intersection one, hence a handle in a compact neighbourhood). A compact
genus-zero Riemann surface is CP^1. For noncompact R, apply Koebe's
general planar uniformisation theorem in the form of Hemasundar (2011,
Thm. 1.1): every planar Riemann surface admits a conformal embedding
into the Riemann sphere; the criterion that every Jordan curve separates
is exactly planarity. The theorem allows arbitrary connectivity, so no
finite-type assumption is being added. The image is open, hence a domain
in CP^1. []
9. A cylinder estimate and compactness of the components.
Proposition 8.4 leaves the normalisation of every divisor component a
domain in CP^1. The goal of this section is to shrink that domain to the
plane, and then exclude the plane as well, concluding that EVERY
component of D is compact with rational normalisation.
We prove a single CYLINDER ESTIMATE that we use twice:
(CE1) On a domain of CP^1 that omits at least two points, the estimate
produces a nonzero scalar holomorphic differential (namely dz/z on
a domain in C^*). The scalar residue obstruction (Prop. 8.3)
forbids such a differential. Hence the domain omits at most one
point: if it is noncompact, it is the plane C.
(CE2) On a plane component, the differential dz has an exponential
coefficient in the logarithmic coordinate. We multiply it by
the section s of E that vanishes at the negative end, and use a
periodic GAUGE to keep the resulting bundle-valued differential
L^2-bounded on every source strip. The estimate then bounds the
mass of infinitely many pushforward currents on distinct curves
by the same exponent A, contradicting finite-dimensionality of
the fixed cohomology space.
The estimate controls the integral of a differential over a target
height slab by the area of the curve in a FIXED enlargement of that slab.
Its constants must remain bounded along arbitrarily long chains of
low-energy source cylinders; that is the roundabout reason we develop
Bishop compactness on source strips.
We retain the area exponent A of Lemma 8.1. The correction exponent
satisfies (GAMMA): gamma > 2 A + 10. The normalisation of any component of
D, and of any fixed deck translate of that component, satisfies
(AIN) Area(B intersect h^{-1}([-l-1, -l])) <= C_B e^{A l}, l in N.
The constant C_B may depend on the translate. The exponent does not.
9.1. Source strip set-up. Write
C = R x (R / 2 pi Z)
with coordinate v = t + i theta and flat area dA_v = dt d theta. Put
S_j = [j, j+1] x (R / 2 pi Z),
C_j^r = (j - r, j + 1 + r) x (R / 2 pi Z).
Strip boundaries have zero area and can be included when specifying a
point a strip contains. For a holomorphic map f into Y, let
u(v) = |df_v(d/dt)|.
Conformality gives
(ENERGY) f^* omega = u^2 dA_v,
where omega is the Hermitian area form on complex tangent lines. Thus u
differs from the Hilbert-Schmidt norm |df| by the fixed factor sqrt 2;
we use u to avoid this factor.
Fix a finite nested coordinate cover of X, lift it to Y, and recenter
inside its smaller members: this gives a radius r > 0 and centred
holomorphic coordinate balls U_p(4r) about every p in Y. Fix these
charts so that Euclidean and lifted tangent norms are comparable with
one constant K >= 1. Decrease r if necessary so all recentered balls stay
in the original charts. The smaller concentric balls U_p(r) are used to
define a flat strip below.
9.2. Small energy fills a source cylinder. This is the technical heart
of the section: inside a small chart, a nonflat strip costs a fixed
amount of energy near any marked target height.
LEMMA 9.1 (Small energy fills a source cylinder). There are an integer
P >= 2 and positive constants Q, delta, eps_0, D_0, depending only on the
lifted geometry, with the following property. Suppose Omega subset C is a
domain and f : Omega -> Y is a nonconstant proper holomorphic map. A
strip S_j is called FLAT if
\overline{C_j^{2P+2}} subset Omega and f(\overline{C_j^{2P+2}})
is contained in U_p(r) for one of the fixed coordinate charts.
Every flat strip satisfies
sup_{S_j} u <= D_0.
For every interval I subset R, if S_j contains a point v_j with
h(f(v_j)) in I and S_j is NOT flat, then
(COST) integral_{Omega intersect C_j^Q intersect (h o f)^{-1}(I + [-delta, delta])}
u^2 dA_v >= eps_0.
The same assertion holds on a source half-cylinder whenever
\overline{C_j^Q} lies in its interior.
Proof. Step 1 (constants). Choose delta strictly larger than twice the
maximum height oscillation of the balls U_p(4r), with a fixed positive
margin: a uniform multiple of 8 K r ||dh||_inf + 1 suffices after enlarging
the chart comparison constant K. Choose P >= 2 large enough that
(q) q = 2 K^2 / (e^P - 1) <= 1/2,
and set Q = 3 P + 4. Cauchy's estimate on source disks of radius one
inside C_j^{2P+2} bounds u uniformly on \overline{C_j^{2P}} whenever S_j
is flat; the norm of a coordinate map with image in U_p(r) is at most r,
so the bound depends only on K, r, and the disk radius. In particular
every flat strip has sup u <= D_0.
Step 2 (assume no eps_0 works, set up a compactness argument). If no
eps_0 works, there are maps f_nu, nonflat strips, and marked points as in
(COST) for which the integral tends to 0. Translate source indices to
j = 0, and translate targets and their height intervals by the same
powers of T so that the marked images lie in a fixed compact set. After a
subsequence, the marked source points tend to v_inf in S_0 and their
images tend to p in Y. Denote the corresponding translated intervals by
I_nu; their lengths play no role.
Work in the fixed ball U_p(4r). For large nu the marked image is
arbitrarily close to p, so the strict margin in delta makes every point
of the ball have height in the corresponding I_nu + [-delta, delta], even
when the marked height is at an endpoint of I_nu. So the energy of the
graph portions in C_0^Q x U_p(4r) tends to 0 in the target direction.
Step 3 (the graph portions are closed analytic curves). Put
Gamma_nu = { (v, f_nu(v)) : v in Omega_nu intersect C_0^Q,
f_nu(v) in U_p(4r) }.
These are closed analytic curves in that product. To check closedness:
suppose (v_k, f_nu(v_k)) converges in a compact subset of the product.
Properness of f_nu puts the v_k in a compact subset of Omega_nu; their
limit is in the domain and lifts to the graph. On a half-cylinder cut
from a proper full normalization the same proof works on compact source
disks away from its artificial finite boundary; properness is applied to
the ambient map. This is exactly where properness is essential.
Step 4 (Bishop compactness). The product Hermitian area of a graph is
its horizontal plus its vertical area. Its horizontal area is at most
Area(C_0^Q), because the source projection has at most one point in each
fibre. Its vertical area tends to 0. Bishop's compactness theorem
(Bishop 1964), in the analytic-cycle and support form of Harvey-Shiffman
(1974, Thms. 3.9-3.10, pp. 585-586), gives a subsequence converging
locally as positive analytic cycles, with convergence of their supports,
to a nonempty pure one-dimensional analytic cycle Gamma_inf through
(v_inf, p). These theorems apply in local product coordinates (equivalently,
identify the finite cylinder with an annulus). The locally bounded graph
areas give their mass hypothesis.
Convergence of positive currents makes the limit's vertical area 0. On
the regular part of each component this says the target coordinate
functions have zero derivative, so each component is HORIZONTAL.
Analytic continuation in the connected cylinder then gives
C_0^Q x {p} subset supp Gamma_inf.
Indeed horizontal components cannot disappear at an interior source
point: the defining holomorphic functions of the limit, restricted to the
horizontal slice, vanish on an open set, hence on the whole connected
slice.
Step 5 (convergence to the slice fills a smaller cylinder). Fix a
source disk V relatively compact in C_0^Q. Choose a small target ball
B(p, eta) relatively compact in U_p(r) whose closure meets no other
horizontal component of the limit; such a ball exists by local
finiteness of the analytic cycle. On a slightly larger source disk the
approximating graphs eventually avoid the target boundary
partial B(p, eta): otherwise a convergent sequence of graph points would
give a limit point on that boundary. So the projection of the graph
portion inside V x B(p, eta) is proper. It is finite because its fibres
have at most one point, and it has nonempty image by convergence to the
horizontal slice. A nonconstant finite holomorphic map of analytic curves
is open, and a proper map is closed; its image is all of the connected
disk V. Its degree is exactly one, by the single-point fibre bound. In
particular every point of V belongs to Omega_nu, and its image is in
B(p, eta). If there were a second horizontal component, the same argument
in a disjoint target ball would give two graph points over one source
point. So there is no second component.
Step 6 (contradiction). Cover \overline{C_0^{2P+2}} by finitely many such
disks. Then for all sufficiently large nu this entire compact cylinder
lies in Omega_nu and its image is in U_p(r). Interior Cauchy estimates
also give convergence of its coordinate derivatives to 0 on smaller
cylinders. In particular S_0 is flat, contradicting its choice. This
proves (COST) and all the uniformities. []
9.3. Flat chains. Nonflat strips cost a fixed amount of energy. Flat
strips need a different bound: along an arbitrarily long consecutive
chain, the sum of the derivative suprema remains bounded, so the target
height varies by a bounded amount. This is what makes the cylinder
estimate uniform along arbitrarily long chains.
LEMMA 9.2 (Flat chains). Let J subset Z be a MAXIMAL interval of flat
indices for a map as in Lemma 9.1. Put m_j = sup_{S_j} u. There is a
constant L_0, depending only on the fixed geometry, with
(CHAIN) sum_{j in J} m_j <= L_0.
A doubly infinite chain is impossible for a nonconstant map. If a chain
contains a point with height in an interval I, then every finite endpoint
of the chain has a neighbouring nonflat strip containing a point with
height in I + [-Lambda, Lambda], where Lambda = ||dh||_inf L_0 is fixed.
Proof. Step 1 (Fourier in the source angle). Take a flat strip S_j and
write f in its coordinate chart as a vector-valued periodic holomorphic
function F(v). Let H = d/dv F. Since F is single-valued around the
cylinder,
integral_0^{2 pi} H(t + i theta) d theta
= (1/i) integral_0^{2 pi} partial_theta F(t + i theta) d theta = 0.
So the Laurent-Fourier expansion is H(v) = sum_{n != 0} c_n e^{n v}.
Bound the positive coefficients on the circle t = j + 1 + P and the
negative coefficients on the circle t = j - P. For j <= t <= j + 1 the
geometric sums give
|H(t + i theta)| <= (2 / (e^P - 1)) *
max{ sup_{t = j - P} |H|, sup_{t = j + 1 + P} |H| }.
The statement is valid for vector values by taking norms in the Fourier
coefficient integrals. Comparing coordinate and target norms gives
(CONTRACT) m_j <= q * max_{|k - j| <= P} m_k
whenever all indices on the right belong to the chain, with the q of
(q) <= 1/2. Each application uses the chart for its own central strip;
the m_k are intrinsic, so changing charts costs no further.
Step 2 (iteration). All m_j in the chain are at most D_0. Iterating
(CONTRACT) inside J bounds
m_j <= D_0 * q^{ floor(d(j)/P) },
where d(j) is the integer distance to the nearer finite end of J; on a
half-infinite interval use its single finite end. At most 2P indices have
a prescribed value of floor(d(j)/P), so
L_0 = 2 P D_0 / (1 - q)
works. If J = Z the same iteration is possible arbitrarily often at every
index and gives df = 0; analytic continuation contradicts nonconstancy.
Step 3 (charging). Starting at a marked point of the chain, move in the
t direction at its fixed angular coordinate toward either finite
endpoint. The change of h o f is at most
||dh||_inf * sum_{j in J} m_j <= Lambda.
The shared boundary with the neighbouring nonflat strip lies in the
domain, because the endpoint flat strip has a full thickening. That
boundary point belongs to the neighbouring closed strip and has height
in I + [-Lambda, Lambda]. This is the charging assertion. []
Schematic (the flat-chain picture). In the source coordinate t, a long
consecutive run of flat strips appears as a rectangle between two
NONFLAT strips. The graph of sup_{S_j} u across the run sags near the
ends and stays low in the middle; the sum of these values is bounded by
a constant independent of the run length, and the change of target
height across the run is at most Lambda.
9.4. The cylinder estimate. We can now bound a differential's mass
over a target slab by the curve's energy in a fixed enlargement.
PROPOSITION 9.3 (The cylinder estimate on a target height slab). There
are constants R_0, C_0 > 0 depending only on the lifted geometry with
the following property. Let f : Omega -> Y be a nonconstant proper
holomorphic map from a domain in C. Let w >= 0 be measurable and suppose
M_w := sup_{j in Z} ||w||_{L^2(Omega intersect S_j)} < infinity.
For every interval I of length one, set
E_I = integral_{Omega intersect (h o f)^{-1}(I + [-R_0, R_0])} u^2 dA_v.
If E_I < infinity then
(CE) integral_{Omega intersect (h o f)^{-1}(I)} u w dA_v
<= C_0 M_w (1 + E_I).
The same estimate holds on a full source half-cylinder cut from a proper
normalisation map; properness is used only on thickened strips away from
the artificial initial boundary, and a fixed finite initial block of
strips is included in the constant. The integral on the left is always
restricted to the inverse image of the target interval I.
Proof. Step 1 (count nonflat strips). Take R_0 = Lambda + delta, with
constants from the two preceding lemmas. Let N_I be the set of nonflat
strips containing a point of height in I + [-Lambda, Lambda]. The source
cylinders C_j^Q have overlap at most 2Q + 3. Summing (COST) over any
finite subset of N_I gives
(#NFLAT) #N_I <= (2Q + 3) eps_0^{-1} E_I.
This also proves that the set being counted is finite. In particular the
set N'_I of nonflat strips hitting I has this bound.
Step 2 (nonflat strips via Cauchy-Schwarz and the strip count).
Cauchy-Schwarz first on each strip restricted to the target slab, then on
the finite sum, gives
sum_{j in N'_I} integral_{Omega intersect S_j intersect (h o f)^{-1}(I)}
u w dA_v
<= M_w (#N'_I)^{1/2}
( sum_{j in N'_I} integral_{Omega intersect S_j intersect (h o f)^{-1}(I)}
u^2 dA_v )^{1/2}
<= C M_w E_I.
Only the intervals appearing explicitly in these integrals are used; no
estimate of the energy outside the enlarged target interval is assumed.
Step 3 (flat chains via (CHAIN)). On one flat chain, (CHAIN) and
||w||_{L^1(S_j)} <= sqrt(2 pi) M_w give
sum_{j in J} integral_{S_j} u w dA_v <= sqrt(2 pi) M_w L_0.
Every chain meeting I has a finite endpoint adjacent to a member of N_I
by Lemma 9.2. Each nonflat strip has at most two adjacent chains, so at
most 2 (#N_I) chains contribute, and their integral over the target slab
is bounded by their full chain integrals. Combining with (#NFLAT) gives
(CE) on the full cylinder.
Step 4 (the half-cylinder). For a half-cylinder, omit the fixed finite
block of at most 2Q + 4 strips on which the small-energy neighbourhoods
meet its initial boundary. On this block Cauchy-Schwarz gives
C M_w E_I^{1/2} <= C M_w (1 + E_I). Apply the preceding argument to the
remaining indices. There is at most one chain reaching the new finite
endpoint, and its integral is at most sqrt(2 pi) M_w L_0. All other
chains charge as before. Increasing C_0 proves the half-cylinder
assertion. []
For a proper normalisation, the integral in (ENERGY) counts the area of
the reduced curve once, away from its discrete singular set. A fixed
enlargement of a height interval is covered by a fixed finite number of
unit height intervals. So (AIN) and Prop. 9.3 imply, with the SAME
exponent A,
(CYL) integral_{(h o f)^{-1}([-l-1, -l])} u w dA_v
<= C_{B,w} e^{A l}, l in N.
On a half-cylinder the integral is also restricted to that end. The
finitely many height intervals outside the range where the area estimate
was first proved change only the constant.
9.5. Excluding nonplane normalisations. We first take w = 1.
COROLLARY 9.4 (Open components are planes). If a component of D is
noncompact, its normalisation is biholomorphic to the plane C.
Proof. By Prop. 8.4 the normalisation R is a domain in CP^1. If it omits
two distinct points, send them by a Mobius transformation to 0 and
infinity. Then R subset C^* is a domain in the logarithmic cylinder, and
alpha = dz / z = dv
is a globally defined nonzero holomorphic differential on R. Formula
(CYL) with w = 1 bounds the mass of the pushforward f_* alpha on every
unit height interval by C_B e^{A l}: pulling back a test one-form of norm
at most one costs at most a fixed multiple of u, while |dv| is constant.
The scalar case of Prop. 8.3 forbids this nonzero differential. So
CP^1 minus R has at most one point. It has at least one point because R
is noncompact. Removing that single point gives R = C. []
9.6. A periodic gauge. The scalar residue test leaves only plane
normalisations. On a translated plane, the differential dz grows
exponentially in the logarithmic coordinate v = log z (since dz = z dv).
We multiply dz by the section s of E that decays at the negative end. To
control the product, we solve the pulled-back d-bar equation in a
PERIODIC gauge whose exponential factors have bounded L^2 norms on unit
source strips. Its input a is an ordinary complex-valued function; in the
application it is the coefficient of the pulled-back connection.
LEMMA 9.5 (A periodic gauge on the cylinder). If a in L^1(C) intersect L^2(C),
there is a periodic distribution g with
(GAUGE) (1/2)(partial_t + i partial_theta) g = a,
sup_{j in Z} ||g||_{W^{1,2}(S_j)}
<= C (||a||_{L^1} + ||a||_{L^2}).
For every p < infinity this g satisfies
(GEXP) sup_j integral_{S_j} exp(p |Re g|) dA_v < infinity.
If a is smooth on an open set, g is smooth there.
Proof. Step 1 (the angular mean). Let
a_0(t) = (1 / 2 pi) integral_0^{2 pi} a(t, theta) d theta. Its primitive
g_0(t) = 2 integral_{-inf}^t a_0(s) ds
satisfies |g_0(t)| <= (1/pi) ||a||_{L^1} and g_0' = 2 a_0 in L^2. This
bounds its whole W^{1,2} norm on each unit strip, including its mean.
Step 2 (zero-mean part). For a_perp = a - a_0, take Fourier series in
theta and Fourier transform in t. For every integer n != 0 set
g_perp^(xi, n) = 2 a_perp^(xi, n) / (i xi - n).
Since |n| >= 1, each of the multipliers 1/(i xi - n), xi/(i xi - n), and
n/(i xi - n) is bounded by one in absolute value. Plancherel gives
||g_perp||_{W^{1,2}(C)} <= C ||a||_{L^2}. The sum g = g_0 + g_perp has
the stated periodicity, equation, and stripwise bound. Local elliptic
regularity for partial_t + i partial_theta gives smoothness where a is
smooth.
Step 3 (exponential integrability). Let B be a uniform bound for the
stripwise norms. The two-dimensional Trudinger inequality on a fixed
bounded strip (Trudinger 1967), including the L^2 term in the norm, gives
constants c, C > 0 independent of j with
integral_{S_j} exp(c |Re g|^2 / B^2) dA_v <= C.
One can cut the cylinder at a fixed angle and use the bounded-rectangle
form; no boundary value of g is prescribed. The zero-norm case is
immediate. For finite p, use p|x| <= c x^2 / B^2 + p^2 B^2 / (4 c) to
prove (GEXP). In particular both e^{Re g} and e^{-Re g} have bounded L^p
norms on unit strips, for every finite p. []
9.7. Excluding plane normalisations. Now the payoff of Section 9.
THEOREM 9.6 (Compact components). Every irreducible component of D is
compact and has rational normalisation.
Proof. By Prop. 8.4 and Cor. 9.4 it remains to exclude a component with
proper normalisation f : C -> B subset Y. Let M_B = sup_B h < infinity
and M_D = sup_D h < infinity. The curves B_k = T^k B are distinct, since
their height suprema are M_B + k. For every sufficiently large positive
integer k this supremum exceeds M_D, so B_k is not contained in D. Write
f_k = T^k o f. We construct on each of these curves the nonzero
f_k^* E-valued holomorphic differential
(BUNDLE) alpha_k = (f_k^* s) dz,
and show its pushforward grows with a uniform exponential rate A.
Step 1 (cancellation of dz). Properness sends the exterior of the plane
to the negative end of Y. After a periodic change of frame there, the
section coefficient becomes an ordinary holomorphic function tending to
0 at infinity, hence size O(1/|z|). Since dz = z dv, this leaves only an
exponential gauge factor, whose stripwise L^2 bound is exactly the
multiplier hypothesis of Prop. 9.3.
Step 2 (the end limit of f_k). Fix one k temporarily. Properness and the
upper height bound imply
(PEND) h(f_k(z)) -> -infinity uniformly as |z| -> infinity.
Indeed, otherwise a sequence |z| -> infinity would have image in some
compact height slab {c <= h <= M_B + k}, contradicting properness. On an
exterior disk |z| > R_k use the prescribed smooth negative-end frame of E,
with Hermitian norm 1 there. In this frame write d-bar_E = d-bar + eta,
where |eta| <= C e^{-gamma |h|}; this is the choice (GAMMA). In the
logarithmic coordinate v = log z, write f_k^* eta = a(v) d-bar v. Then
(CONN) |a(v)| <= C e^{-gamma |h(f_k(v))|} u(v).
Step 3 (integrability of a). Increase R_k so the extra exterior disk has
negative height. Sum (CONN) over the target height intervals, using (CYL)
with w = 1 for the first estimate and (ENERGY) for the second:
integral |a| dA_v <= C_k sum_{l >= 0} e^{-(gamma - A) l} < inf,
integral |a|^2 dA_v <= C_k sum_{l >= 0} e^{-(2 gamma - A) l} < inf.
An initial compact annulus adds only a finite quantity. Smoothly cut a
off toward the finite side and extend by zero to the full cylinder. This
keeps both bounds and agrees with a on a slightly smaller exterior disk.
Lemma 9.5 gives a single-valued periodic smooth gauge g there, with
d-bar g = a d-bar v and uniformly bounded stripwise W^{1,2} norms.
Step 4 (a bounded holomorphic function across infinity). Let sigma(v)
be the coefficient of f_k^* s in the same smooth frame. Holomorphicity
says d-bar sigma + a sigma d-bar v = 0. Hence
H(v) = e^{g(v)} sigma(v)
is ordinary holomorphic and periodic on the exterior cylinder. By
Prop. 7.10 and (PEND), sup_{S_j} |sigma| -> 0 as j -> infinity. The
exponential estimate (GEXP) then gives
integral_{S_j} |H|^2 dA_v
<= ( sup_{S_j} |sigma|^2 ) integral_{S_j} e^{2 Re g} dA_v
-> 0.
The mean-value inequality on fixed small disks (each contained in the
union of three adjacent strips) implies H(v) -> 0 uniformly as
t -> infinity. Viewed as a holomorphic function of z, H extends
holomorphically across z = infinity with value 0. Its Taylor expansion
in 1/z gives
(HDECAY) |H(z)| <= C_k |z|^{-1}
on a further exterior disk.
Step 5 (the bundle differential has bounded stripwise norm). Since
dz = e^v dv, (BUNDLE) and (HDECAY) give
|alpha_k|_v = |e^v sigma(v)| <= C_k e^{-Re g(v)}.
Here |.|_v is the coefficient norm relative to dv and the unit smooth
frame. The factor e^{-Re g} has uniformly bounded L^2 norms on unit
strips by (GEXP). Apply (CYL) with w = e^{-Re g}. Pullback of a test
one-form costs at most C u, so
(GROW) Mass( (f_k)_* alpha_k restricted to
h^{-1}([-l-1, -l]) )
<= C_k' e^{A l}, l >= 0.
The omitted compact disk in the source has compact image and contributes
only on finitely many height intervals, absorbed in C_k'. The section
f_k^* s is nonzero because B_k is not contained in D, and dz is nowhere
zero, so alpha_k is a nonzero holomorphic bundle-valued differential.
Step 6 (finite dimension contradicts infinitely many curves). Fix, once
and for all, an admissible eps > 0 and b > max{A, eps}. The currents in
(GROW) all belong to the same degree-one space W^{-4}_{-eps,b} of the
shifted complex (CUR) from Theorem 8.2, with coefficients in the same
bundle E. Their positive supports are bounded separately, and any finite
collection has a common upper support bound. Prop. 8.3 therefore makes
the classes of every finite collection linearly independent in the single
space H^{2,1}_{-eps,b}(Y, E). That space is finite-dimensional by
Theorem 8.2. Infinitely many distinct B_k contradict this finite
dimension. The constants C_k' may vary: no bound on their supremum was
used, and A, the weights, and Sobolev orders were fixed throughout. This
excludes the plane case and completes the proof. []
10. Counting rational curves.
We now have, for each compact datum E_i of Lemma 6.2, a section s_i whose
zero divisor D_i consists of COMPACT rational curves (Theorem 9.6). The
divisor as a whole may still have infinitely many components; it is
locally finite and bounded above in height. We project these components
down to X and count. A single curve downstairs accounts for infinitely
many translates upstairs, so naive counting fails. Instead we count over
the Laurent polynomial ring Q[t, t^{-1}], where one power of t records
one deck translation. A single orbit of compact curves then imposes one
linear equation over the fraction field, and a linear-algebra count plus
the nonsingularity of the intersection pairing show there cannot be
too few curves.
FIRST AUXILIARY FACT. A compact rational curve in Y projects to a
rational curve in X.
LEMMA 10.1 (Lifts of rational curves). Every compact rational curve in
Y projects to a rational curve in X. For each rational curve B subset X,
the compact irreducible curves in Y projecting onto B form ONE orbit
under T. In particular a collection of r rational curves in X gives at
most n r orbits of compact lifts under S = T^n.
Proof. The image under pi of a compact curve is a compact analytic
curve (the map restricted to that curve is proper and locally finite)
and nonconstant (pi is a local biholomorphism). If the upstairs
normalisation is CP^1, the induced nonconstant map to the normalisation
of the image is a finite map from CP^1 to a compact Riemann surface.
Riemann-Hurwitz forces that surface to have genus zero, so the image is
rational.
Conversely, let nu : CP^1 -> B subset X be the normalisation of a
rational curve. Since CP^1 is simply connected, nu lifts after choosing
the image of one point to a holomorphic map nu~ : CP^1 -> Y. Its image
B~ is compact and irreducible. Over the smooth part of B, its projection
has degree one because nu does. The lifts of this FIXED normalisation
form one orbit under the deck group, so their images do also.
To see these are ALL the compact irreducible curves over B, take such a
curve C and a point of it over the smooth part of B. Choose the lift of
nu through this point. The local biholomorphism property of pi makes its
image and C agree in a nonempty open curve germ. Irreducibility and
analytic continuity then make their images equal. So the orbit assertion
holds. Splitting integer deck powers into their n residue classes proves
the bound for S.
Note that we lifted the NORMALISATION map, not the singular topological
space B. This also works for a nodal rational curve; the connected
components of its full inverse image need not be compact. []
THE LAURENT RING. Fix n (to be chosen later), put S = T^n, and set
Lambda = Q[t, t^{-1}], F = Q(t), conjugation: f-bar(t) = f(t^{-1}).
On compactly supported cohomology (Section 6.1), let
ta = S_! a := (S^{-1})^* a.
This is the convention that translates the SUPPORT in the positive
S-direction; it agrees with the pushforward action C |-> S_* C on
homology under Poincare duality. Define the SESQUILINEAR PAIRING
(LAUR) P(a, b) = sum_{k in Z} <a, S_!^k b> t^k,
for a, b in H_c^2(Y; Q).
This is a Laurent polynomial, because compactly supported
representatives intersect only finitely many translates of one another.
Invariance of intersection under S and symmetry in degree two give
(SESQ) P(ta, b) = t P(a, b), P(a, tb) = t^{-1} P(a, b),
P(b, a) = overline{P(a, b)}.
So P is linear in its first variable and conjugate-linear in its second,
where conjugation is just the involution t |-> t^{-1}. For a compact
oriented curve C define similarly
(LAURC) Q(a, C) = sum_{k in Z} ( integral_{S_*^k C} a ) t^k.
This is finite and Lambda-linear in a with the same convention; its
vanishing is equivalent to the vanishing of the ordinary pairing of a
with EVERY S-translate of C.
NONSINGULARITY. The classes e_1, ..., e_d of Lemma 6.2 remain independent
over the Laurent ring.
LEMMA 10.2 (Nonsingularity of the compact classes). For the classes
e_1, ..., e_d of Lemma 6.2, the matrix P(t) = ( P(e_i, e_j) )_{i,j=1}^d
has NONZERO determinant in Lambda. The Lambda-span of the e_i is free of
rank d, and P is nonsingular on its extension of scalars to F.
Proof. Let w_i be the transferred classes in X_n. Periodising compactly
supported representatives of e_i and e_j, then integrating over one
fundamental period, gives
(EVAL1) P(e_i, e_j)(1) = sum_{k in Z} <e_i, S_!^k e_j> = w_i . w_j.
For completeness, the periodisation is the locally finite sum of all
S_!-translates of a representative; it descends to a closed form on X_n.
In the integral of the wedge product of two such sums, change variables
by the translate in the first factor; the translated fundamental periods
tile Y, leaving the sum indexed by the relative translate of the second
factor. Compact supports justify every interchange. In the construction
of Lemma 6.2 the forms are supported inside the same open period, so
terms with k != 0 actually vanish; the periodisation identity also
records the intrinsic pairing convention.
The classes w_i are independent over Q and the intersection form on X_n
is negative definite. Their Gram matrix is therefore negative definite,
in particular invertible. Equation (EVAL1) implies det P(1) != 0, so
det P(t) is not the zero Laurent polynomial; the matrix is invertible
over F.
If sum_i f_i(t) e_i = 0 for f_i in Lambda, pairing with each e_j and
using (SESQ) gives
(f_1, ..., f_d) P(t) = 0.
Invertibility over F forces every f_i = 0. This proves freeness and
nonsingularity. []
LOWER BOUND.
PROPOSITION 10.3 (The required number of rational curves). The surface X
contains at least b_2(X) rational curves.
Proof. If there are infinitely many the conclusion is immediate.
Otherwise suppose for contradiction their number is r < b_2(X), and list
them all. Choose n so large that
n ( b_2(X) - r ) > C_0,
with C_0 from Lemma 6.2. Then Lemma 6.2 gives d > n r compact classes.
By Lemma 10.1 choose representatives C_1, ..., C_q of all S-orbits of
compact irreducible lifts of the listed rational curves, where
q <= n r < d.
Consider over F the q homogeneous linear equations in unknowns
f_1, ..., f_d:
(ANN) sum_{i=1}^d f_i(t) Q(e_i, C_l) = 0, 1 <= l <= q.
There are more unknowns than equations, so a nonzero solution exists.
Clear denominators: get f_i in Lambda, not all zero. The compactly
supported class
x = sum_{i=1}^d f_i(t) e_i
is nonzero by Lemma 10.2, and has compact support because it is a finite
Laurent combination. Equations (ANN) and (LAURC) imply that x pairs to
zero with every compact rational lift and every S-translate of one.
For each j, the ordinary class j(e_j) is represented by the divisor D_j.
Every component of D_j is compact rational (Theorem 9.6), and its image
downstairs is rational by Lemma 10.1. So each component occurs among the
curves just annihilated by x. If alpha is a compactly supported closed
real 2-form representing x, the ordinary divisor identity (DIV) gives,
for every k in Z,
<x, S_!^k e_j> = integral_{S_*^k D_j} alpha = 0.
Indeed local finiteness makes the integral a finite sum of integrals over
compact components meeting supp alpha, each of which is zero by the
curve orthogonality. This argument pairs a compact test class with a
locally finite divisor; it never pairs two locally finite cycles or sums
an infinite Laurent series.
So P(x, e_j) = 0 for all j. By (SESQ) this says (f_1, ..., f_d) P(t) = 0,
contradicting nonsingularity of P(t) over F. Therefore r >= b_2(X). []
UPPER BOUND.
LEMMA 10.4 (Independence of curve classes). Under (NSZ), the real
cohomology classes of any finite collection of distinct curves on X are
linearly INDEPENDENT. Consequently X has at most b_2(X) curves in total.
Proof. Suppose a finite collection admits a nonzero real relation. Split
its positive and negative coefficients to obtain effective real divisors
A and B with disjoint lists of components and [A] = [B] = v; one of them
may be empty. If v != 0 both are nonempty, and negative definiteness
gives v^2 < 0. But positivity of local intersections of DISTINCT complex
curves gives v^2 = A . B >= 0, a contradiction. So v = 0. At least one
nonempty effective real divisor, say A = sum_{i in I} a_i B_i with all
a_i > 0, has zero real class.
To pass from this real relation to an integral one, express the classes
[B_i] in an integral basis of H^2(X; Z) / Tors. The relation space is
the kernel of an integer matrix, hence has a rational basis (over Q). Its
rational points are dense in its real points. Approximate (a_i)_{i in I}
inside this kernel by a rational vector with all coordinates still
positive, and clear denominators. The result is a nonzero effective
INTEGRAL divisor with zero real class, hence square zero, contradicting
(NSZ). So there is no relation. Since dim H^2(X; R) = b_2(X), any
collection of more than b_2(X) curves would contain a finite dependent
collection. []
11. Proof of the Main Theorem.
We now assemble the pieces.
THEOREM 11.1 (= Main Theorem). Let X be a connected minimal compact
complex surface with b_1(X) = 1, b_2(X) > 0, kappa(X) = -infinity. Then X
contains a global spherical shell.
Proof. If X has a nonzero effective divisor with square 0, Enoki's
criterion (Prop. 5.1(i)) already gives the shell. In the remaining
case - our standing assumption (NSZ) - Proposition 10.3 and Lemma 10.4
show that X has AT LEAST b_2(X) and AT MOST b_2(X) rational curves,
hence EXACTLY b_2(X) rational curves.
The surface X is compact and minimal, has b_1(X) = 1, kappa(X) =
-infinity, and b_2(X) > 0 by hypothesis. Every hypothesis of the Main
Theorem of Dloussky-Oeljeklaus-Toma (2003) is therefore satisfied, so
Prop. 5.1(ii) applies: X has a global spherical shell.
Equivalently, DOT state their conclusion in the following form (their
pp. 283-284): there is an open neighbourhood U of the unit sphere in
C^2 minus {0} and a biholomorphism phi : U -> Sigma subset X such that
X minus Sigma is connected. This is precisely the conclusion required in
the Main Theorem. If instead the shell is described by its nonseparating
core sphere, choose a sufficiently small round annular collar in the
given holomorphic neighbourhood: on either side of the core, the collar
coordinate pushes the complement of the core onto the complement of the
smaller open collar, so the latter is connected as well. []
12. Consequences of the shell.
The shell now gives the deformation and topological description stated
in the Introduction, and removes the remaining class VII exception in
the aspherical-surface inequality of Albanese-Di Cerbo-Lombardi.
Let us restate the two consequences.
COROLLARY 12.1 (Deformation and smooth topology). Let X satisfy the
hypotheses of the Main Theorem (11.1), and put b = b_2(X). There is a
small one-parameter deformation X -> Delta over a disk Delta about 0,
with X_0 isomorphic to X, such that every NONZERO fibre is obtained from
a primary Hopf surface by exactly b successive POINT BLOWUPS (with
possibly infinitely near centres). Moreover,
pi_1(X) = Z,
X ~_or (S^1 x S^3) # b CP^2-bar
where the second is an orientation-preserving diffeomorphism for the
complex orientation of X.
A PRIMARY HOPF SURFACE means a compact quotient of C^2 minus {0} by the
infinite cyclic group generated by a holomorphic contraction (for
example z |-> (1/2) z), equivalently one of the standard finite-quotient
Hopf surfaces.
COROLLARY 12.2 (A sharpened Euler-signature inequality). Let S be a
connected closed smooth complex surface whose topological universal
cover is CONTRACTIBLE (an "aspherical" complex surface). Write
chi_top(S) for its Euler characteristic and sigma(S) for its signature
in the complex orientation. Then
chi_top(S) >= (9/5) |sigma(S)|.
If chi_top(S) > 0 then S is of general type; otherwise
chi_top(S) = sigma(S) = 0.
PROOF OF COROLLARY 12.1. Theorem 11.1 supplies a global spherical shell
on X. Kato's deformation theorem (Kato 1977, Theorem 1) then gives a
one-parameter deformation of X whose nonzero fibres are modifications
of a primary Hopf surface; his Corollary 1 further gives
pi_1(X) = Z.
In complex dimension two, a modification between smooth surfaces factors
as a FINITE succession of point blowups, with possibly infinitely near
centres (BHPV 2004). After shrinking Delta the family is a proper smooth
submersion. Ehresmann's theorem then identifies its fibres by
diffeomorphisms preserving their continuously varying complex
orientations.
Now each primary Hopf surface is diffeomorphic to S^1 x S^3
(Gauduchon-Ornea 1998, Sec. 1, p. 1109), and hence has second Betti
number zero. Each complex point blowup increases the second Betti number
by one and replaces the oriented smooth manifold by its connected sum
with CP^2-bar (BHPV 2004). Since the fibres have second Betti number b,
there are exactly b blowups, and the asserted oriented smooth type
follows. The product orientation can be chosen compatibly because
S^1 x S^3 admits an orientation-reversing diffeomorphism. []
PROOF OF COROLLARY 12.2. Albanese-Di Cerbo-Lombardi (2025, Theorem 1.4)
prove the displayed inequality and the type conclusions except for the
possible case of a class VII_0^+ surface that violates the Global
Spherical Shell conjecture. Theorem 11.1 excludes that case.
Corollary 12.1 also explains the non-asphericity statement: such a
surface has pi_1 = Z and b_2 > 0, but an aspherical manifold with
fundamental group Z is homotopy equivalent to S^1. []
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END OF PAPER.