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A PROJECTIVE FOURFOLD WITH LARGE FUNDAMENTAL GROUP
AND NON-STEIN UNIVERSAL COVER
A complete proof
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This document proves that Shafarevich's holomorphic-convexity
conjecture fails even for projective varieties whose fundamental group
is as large as possible. The proof is self-contained apart from
standard facts of topology, complex analysis, and algebraic geometry.
Every such fact is stated where it is used, with a reference.
The reader who wants the shape of the argument before the details
should read Section 1.4. The formal proof starts in Section 2.
1. INTRODUCTION
1.1. The conjecture and the result
Let X be a smooth connected projective complex variety. Write
p : X~ -> X
for its universal cover, equipped with the complex structure lifted
from X. Thus X~ is simply connected and p is a local biholomorphism.
A complex manifold M is Stein if
(i) holomorphic functions on M separate any two distinct points,
(ii) holomorphic functions give local coordinates at every point,
(iii) M is holomorphically convex: for every compact K in M, the set
of points that cannot be separated from K by holomorphic
functions is compact.
For open subsets of C^n these are the familiar domains of
holomorphy. We shall use only two standard consequences:
(S1) A closed complex submanifold of a Stein manifold is Stein.
(S2) A Stein manifold of complex dimension n has the homotopy type
of a CW complex of real dimension at most n
(Andreotti-Frankel; see Milnor, Morse Theory, Theorem 7.2).
Shafarevich's holomorphic-convexity conjecture asks whether X~ is
always holomorphically convex. A sharper form is:
If X~ contains no positive-dimensional compact complex-analytic
subvariety, is X~ Stein?
The two forms are related by Remmert reduction. A holomorphically
convex manifold admits a proper holomorphic map to a Stein space whose
fibers are the maximal compact analytic subvarieties. If there are no
positive-dimensional compact analytic subvarieties, every fiber is a
point, so the reduction is an isomorphism and the manifold is Stein.
Recall that X has large fundamental group if for every
positive-dimensional closed irreducible subvariety Z of X the image of
pi_1(Z^nu) -> pi_1(X)
is infinite. Here Z^nu is the normalization of Z. The condition is
equivalent to the absence of positive-dimensional compact analytic
subvarieties in X~; this is Lemma 2.1 below. Thus, for varieties with
large fundamental group, the sharper form of the conjecture predicts
that X~ is Stein.
Theorem A (main theorem). There exist a smooth connected projective
complex fourfold X and a simple abelian surface A embedded in X such
that
(1) X has large fundamental group;
(2) the image of pi_1(A) -> pi_1(X) is infinite cyclic.
For such an X, the universal cover X~ contains no positive-dimensional
compact complex-analytic subvariety, and X~ is not Stein.
Consequently the holomorphic-convexity conjecture is false, even under
the large-fundamental-group hypothesis.
The rest of this section explains where the difficulty lies and how the
proof is organized. No claim is used before it is proved.
1.2. Why the obstruction is topological
Suppose that A is a simple abelian surface embedded in X and that
pi_1(A) -> pi_1(X)
has image Z. Since pi_1(A) = Z^4, the kernel K of this homomorphism
is a rank-three lattice. The connected cover of A corresponding to K
is
A_K = C^2 / K.
As a real manifold this is R^4/K, and since the real span of K has
dimension three,
A_K is diffeomorphic to R x T^3, T^3 = (S^1)^3,
so H_3(A_K;Z) = Z. On the other hand A_K is a closed complex
submanifold of X~. If X~ were Stein, then A_K would be Stein by (S1).
But a Stein surface has no third homology, by (S2). Contradiction.
Two points deserve emphasis.
First, the argument uses only the image of pi_1(A), not the whole
fundamental group of X. Thus the hard part is to build an embedding
whose surface fundamental group has infinite cyclic image.
Second, simplicity of A is used to rule out compact analytic curves in
A_K. If A contained an elliptic curve E, then the inverse image of E
in A_K could contain compact curves, and the universal cover would
still have compact analytic subvarieties; largeness would fail. For a
simple A, every curve in A generates A, so its fundamental group maps
with finite index into pi_1(A), and hence with infinite image into Z.
1.3. The geometric idea
The surface A will be the Jacobian of a genus-two curve C. The curve
is chosen so that Jac(C) is simple (Zarhin's theorem). The curve C is
a double cover of P^1 branched at six points: four "red" points close
together in a small disk, and two "blue" points far away.
Inside a compact complex hyperbolic fourfold we construct an
arrangement of hyperplanes with two pencils, a z-pencil and a
w-pencil. Each pencil imitates the six-point branched cover. Two
extra "coupling" hyperplanes meet the four red hyperplanes of each
pencil, and meet neither blue hyperplane.
Blowing up the arrangement and passing to a suitable double cover
produces a smooth projective variety V with an action of an order-four
symmetry s. Near the distinguished point o, the fiber of the cover is
F_* = C x C,
and s swaps the two factors while rotating the two normal directions.
The symmetry s has order four; its square acts as -1 on both normal
lines.
The quotient [V/<s>] is an orbifold, not a manifold, along the fixed
locus of s. Over the distinguished point o we replace this quotient by
an explicit ordinary smooth resolution. The result is a smooth proper
orbifold S with projective coarse space, whose distinguished fiber is
ordinary (all stabilizers trivial). A rational section of a line bundle
on C x C, followed by resolution of indeterminacy, gives a morphism
f : Y -> S
from an iterated point blowup Y of A = Jac(C), with image contained in
the ordinary distinguished fiber.
The coupling hyperplanes give relations among the four red meridians
of the pencil. Together with the sphere relation of the six-punctured
line, these relations force the image of pi_1(Y) in pi_1(S) to be
cyclic. This is the upper bound.
The lower bound is the heart of the proof. One must show that the
cyclic image is infinite, not finite. A global representation of
pi_1(S) would be convenient but is not available. Instead we use
local covers of large finite models of the arrangement. On each model
the monodromy is read in the infinite dihedral group
D_inf = < r, b | r^2 = b^2 = 1 >,
whose even words (rb)^n carry an integer label n. The local labels
need not glue to a representation of pi_1(S). What matters is that on
overlapping bounded charts the kernels of the labels agree, and that
the graphs of the local covers are uniformly Gromov hyperbolic.
Section 3 proves an abstract local path criterion. It says, roughly,
that a loop detected as nontrivial by one local chart cannot become
trivial through global relations, provided the local charts satisfy
three simple compatibility axioms and their graphs are uniformly
hyperbolic. The criterion is applied in Section 6 to paths in the
ordinary locus of S.
A crucial feature of the graph metric is that edges are paths of
bounded projected diameter, not bounded length. Hence every power of
a fixed loop supported near the distinguished fiber is a single edge.
The path criterion then detects every nonzero power separately. This
is why no global coloring and no family of finite quotients is needed.
Finally, Section 7 replaces the orbifold S by an ordinary smooth
projective variety, and removes the point blowups from Y one at a time
by an explicit projective construction. The homomorphism from the
surface fundamental group is preserved at every step. At the end A
itself is embedded in a smooth projective variety with infinite cyclic
image. Section 2 then converts this embedding into the fourfold of
Theorem A.
1.4. Structure of the proof
Section 2 (reduction). Lemma 2.1 identifies largeness with the
absence of compact analytic subvarieties in the universal cover.
Lemma 2.2 is an interpolation statement for complete intersections.
Proposition 2.3 shows that an embedding of a simple abelian surface
with infinite cyclic image yields the fourfold of Theorem A, with
largeness and with a non-Stein universal cover.
Section 3 (local path criterion). This is pure topology of graphs and
hyperbolic spaces. It proves that bounded local readings detect
nontrivial loops modulo short relations.
Section 4 (geometry). It constructs the arithmetic ball quotient, the
two pencils with their coupling hyperplanes, the sign covers, the
blowups and fillings, the distinguished fiber C x C, the order-four
symmetry, the ordinary resolution over the distinguished fiber, and
the surface map f : Y -> S.
Section 5 (models). It defines the dihedral and translation labels on
finite models, proves that they extend over the fillings, proves that
their kernels agree on buffered balls, and proves uniform hyperbolicity
of the associated path graphs.
Section 6 (image). It proves the cyclic upper bound using the coupling
hyperplanes, verifies the chart axioms for the constructed orbifold,
compares orbifold relations with path relations, and proves that the
image of pi_1(Y) is infinite cyclic.
Section 7 (projective realization). It replaces the orbifold ambient
space by a smooth projective variety, and performs the blowdowns of Y
inside projective ambient varieties. Theorem 7.5 gives the cyclic
embedding of A; Theorem A follows from Proposition 2.3.
1.5. Conventions
All varieties are over C unless another field is named. Fundamental
groups are topological, with a chosen basepoint; image statements are
understood up to a common conjugation, and we always specify common
paths when basepoints change.
A smooth complex orbifold means a smooth separated Deligne-Mumford
stack of finite type over C whose stabilizer groups are finite. Its
fundamental group is that of its classifying space. A point is
ordinary if its stabilizer is trivial. The coarse space of a proper
orbifold is a complex space; when it is projective we say the orbifold
has projective coarse space.
Projective bundles parametrize lines, not hyperplanes.
Complex hyperbolic space is the unit ball with the Bergman metric,
normalized so that a point of Euclidean radius tanh(r) has distance r
from the center. We use the same letter for a ball and its radius when
no confusion can arise.
For a complex manifold M and a closed submanifold Z, N_{Z/M} is the
holomorphic normal bundle, and P(N_{Z/M}) parametrizes lines in it.
Finally, "positive-dimensional" means of dimension at least one.
2. REDUCTION TO AN ABELIAN SURFACE WITH CYCLIC IMAGE
The construction of Sections 4-7 produces a simple abelian surface A
embedded in a smooth projective variety S' such that
im(pi_1(A) -> pi_1(S')) = Z. This section proves that such an
embedding is enough. The conversion takes a general complete
intersection in S' x B, where B is an abelian sixfold, and produces
the fourfold of Theorem A.
There are three steps. Lemma 2.1 identifies largeness with the
absence of positive-dimensional compact analytic subvarieties in the
universal cover. Lemma 2.2 is an interpolation statement that lets us
control all fibers of a complete intersection at once. Proposition
2.3 performs the complete-intersection construction and proves the
topological obstruction to Steinness.
2.1. Largeness and compact analytic subvarieties
Lemma 2.1. Let X be a smooth connected projective complex variety.
Its universal cover X~ contains no positive-dimensional compact
complex-analytic subvariety if and only if X has large fundamental
group.
Proof. Suppose first that X is not large. Then there is a
positive-dimensional closed irreducible subvariety Z of X such that
the image of pi_1(Z^nu) -> pi_1(X) is finite. Let H be this finite
image, and let Z' -> Z^nu be the finite covering corresponding to the
kernel of pi_1(Z^nu) -> H. The map Z' -> X has image in X and its
fundamental group maps trivially to pi_1(X). Hence Z' -> X lifts to
the universal cover:
Z' -> X~, Z' compact of positive dimension.
The lift is holomorphic and its image is a positive-dimensional
compact analytic subvariety of X~. This proves one direction.
Conversely, let K be a positive-dimensional irreducible compact
analytic subvariety of X~. Let Z be its image in X. Since p : X~ -> X
is a covering and K is compact, the restriction p|K is proper, so by
the proper mapping theorem Z is a closed analytic subvariety of X.
Because X is projective, Chow's theorem makes Z algebraic. The map
K -> Z is proper with discrete fibers, hence finite. Passing to
normalizations gives a finite surjective map K^nu -> Z^nu. Such a map
has finite-index image on fundamental groups. To see this, remove a
proper analytic subset of Z^nu so that the map becomes a finite
unramified covering of smooth connected open sets; the index there is
the degree. The open sets have fundamental groups surjecting onto
those of the normal spaces by general position for loops, and passing
to quotients can only decrease the index.
The composite pi_1(K^nu) -> pi_1(X) is zero: K^nu maps into the simply
connected space X~. Since pi_1(K^nu) -> pi_1(Z^nu) has finite-index
image, the image of pi_1(Z^nu) in pi_1(X) is finite. Hence X is not
large. This proves the converse. []
2.2. Interpolation for complete intersections
Lemma 2.2. Let W be a projective variety, F a closed subscheme of W,
and L a very ample line bundle on W. Fix a positive integer l. For
all sufficiently large m and every set of distinct points
p_1,...,p_l in W \ F, the evaluation map
H^0(W, I_F (x) L^m) -> direct sum_{i=1..l} L^m|_{p_i}
is surjective. The same bound on m works for every such set.
Proof. Choose m_0 so that I_F (x) L^{m_0} is globally generated. For
each i choose a section a_i of this bundle that is nonzero at p_i.
For each j different from i, very ampleness of L gives a section of L
that vanishes at p_j and is nonzero at p_i. Let b_i be the product of
these sections over all j different from i. Then b_i vanishes at every
p_j with j not equal to i, and b_i is nonzero at p_i. The sections
a_i b_i give a diagonal basis of the evaluation map in degree
m_0 + l - 1. For larger m, multiply each a_i b_i by a section of the
remaining power of L that is nonzero at p_i. This gives the assertion
in every degree m at least m_0 + l - 1. []
2.3. The reduction proposition
Proposition 2.3. Let A be a simple abelian surface and let
A -> S' be a closed embedding in a smooth connected projective complex
variety. Suppose that
im(pi_1(A) -> pi_1(S')) = Z.
Then there exist a smooth connected projective fourfold X with large
fundamental group and a closed embedding A -> X with infinite cyclic
image on fundamental groups, such that the universal cover of X is not
Stein.
Proof. Put d = dim S'. Choose an abelian variety B of dimension
g = 6 and set
W = S' x B, F = A x {0} inside W.
Let L be a very ample line bundle on W. Let
c = d + g - 4 = d + 2.
We shall take X to be the common zero locus of c general sections of
I_F (x) L^m, for m sufficiently large. Then dim X = d + g - c = 4.
We must check smoothness, the fundamental group, the fibers over B, and
largeness.
Smoothness and the fundamental group. For large m the sections
generate the ideal of F away from F, and their normal derivatives
generate N^*_{F/W} (x) L^m|_F. Bertini's theorem gives smoothness
away from F. Along F, smoothness is the independence of c general
vectors in a bundle of rank d + g - 2 = c + 2. The locus of c vectors
that fail to be independent has codimension 3 in the bundle. Since
dim F = 2 < 3, a general tuple of sections has independent normal
derivatives at every point of F, and the successive intersections are
smooth along F. Thus X is smooth.
The successive intersections are smooth ample hypersurfaces in their
predecessors, and the final dimension is four. The Lefschetz
hyperplane theorem therefore gives connectedness and an isomorphism
pi_1(X) -> pi_1(S' x B) = pi_1(S') x pi_1(B). (2.1)
Here and below, sufficiently high powers of a very ample line bundle
give both ideal generation and generation of the required normal
derivatives; these are the usual applications of Serre vanishing and
Bertini's theorem (Hartshorne, Algebraic Geometry).
Fibers away from the fixed surface. We claim that a general tuple
excludes four distinct points of W \ F in any one fiber of W -> B.
The parameter space of ordered quadruples in one fiber has dimension
at most
4d + g = 4d + 6.
By Lemma 2.2, requiring all c equations to vanish at such a quadruple
imposes exactly
4c = 4d + 8
independent linear conditions on the space of tuples of sections. The
incidence variety of tuples and quadruples therefore has dimension
strictly less than the section parameter space: its excess is
4d + g - 4c = 4d + 6 - (4d + 8) = -2. Its constructible image has
closure of strictly smaller dimension as well. A general tuple lies
outside this closure. This condition is compatible with the open
smoothness conditions already imposed.
It follows that each fiber of X -> B contains at most three points
outside F. In particular, every positive-dimensional subvariety of a
fiber of X -> B is contained in F.
Largeness. Let Z be a positive-dimensional closed irreducible
subvariety of X.
If the projection Z -> B is nonconstant, then the image of
pi_1(Z^nu) in pi_1(B) is infinite. Indeed, suppose the image were
finite. Since pi_1(B) is torsion-free, the image would be trivial,
and the map Z^nu -> B would lift holomorphically to the universal
cover C^g of B. But Z^nu is compact and connected, so every
holomorphic function on it is constant, contradiction. Hence the
image in pi_1(B), and therefore in pi_1(X), is infinite.
If the projection Z -> B is constant, then Z lies in one fiber of
X -> B, and the fiber property gives Z contained in F = A. If Z = F,
then its image in pi_1(X) is infinite cyclic by (2.1) and the
hypothesis. The only other possibility is an irreducible curve in A.
Let D -> A be its normalization. After translating in A, the
nonconstant map D -> A extends to a homomorphism Jac(D) -> A. Its
image is a nonzero abelian subvariety of A, hence all of A, because A
is simple. A surjective homomorphism of complex tori carries the
source integral lattice to a finite-index subgroup of the target
lattice. Consequently pi_1(D) has finite-index image in
pi_1(A) = Z^4, and its image under the homomorphism onto Z remains
infinite. This proves largeness in every case. By Lemma 2.1, X~
contains no positive-dimensional compact analytic subvariety.
The obstruction to Steinness. Let
K = kernel ( pi_1(A) -> pi_1(X) ).
The quotient is infinite cyclic, so K is isomorphic to Z^3. A
connected component of the inverse image of F in X~ is the connected
covering
A_K = C^2 / K.
It is a closed complex submanifold of X~: the full inverse image of F
is closed, and its components are closed and locally separated in
covering charts. The real span of K has dimension three, so as a real
manifold
A_K = R^4 / K = (R^3 / Z^3) x R = T^3 x R,
and therefore
H_3(A_K; Z) = Z.
By the Andreotti-Frankel theorem, a Stein manifold of complex dimension
two has the homotopy type of a CW complex of real dimension at most
two (see Milnor, Morse Theory, Theorem 7.2). Hence A_K is not Stein.
A closed complex submanifold of a Stein manifold is Stein, so X~
cannot be Stein either. This proves the proposition. []
2.4. What remains
Proposition 2.3 reduces Theorem A to the following concrete task:
Embed a simple abelian surface A in a smooth connected projective
variety S' so that im(pi_1(A) -> pi_1(S')) is infinite cyclic.
Sections 4-7 solve exactly this task. Section 4 constructs a map
f : Y -> S from an iterated point blowup Y of A to a smooth proper
orbifold S with projective coarse space. Section 6 proves that the
image of pi_1(Y) is infinite cyclic. Section 7 replaces S by a smooth
projective variety and removes the point blowups from Y, preserving
the homomorphism on fundamental groups. At the end A itself is
embedded in a smooth projective variety with infinite cyclic image.
Proposition 2.3 then gives Theorem A.
3. A LOCAL CRITERION FOR NONTRIVIAL LOOPS
The geometric construction attaches group labels to paths only inside
bounded regions. We need a criterion that detects a nontrivial loop
using such local information. This section proves the criterion in
the abstract. It uses graphs whose edges are paths of bounded
projected diameter; the hyperbolicity of the local models enters only
through a common constant.
The reader may keep in mind the following model. The vertices of G are
points of a space. An edge is a path confined to a small region. A
reading lifts a short path to a covering space chosen near the initial
vertex; the endpoint of the lift records both the actual endpoint and
the sheet. The criterion says that if all readings come from uniformly
hyperbolic charts and satisfy three compatibility axioms, then a loop
that one chart detects as nontrivial cannot be killed by global
relations.
3.1. Graphs and readings
All graphs have oriented edges with specified reverses; multiple edges
and loops are allowed. Each edge has length one. Paths are finite
edge paths, and their lengths count edges. A constant path has length
zero. The path groupoid of a graph is obtained by cancelling
consecutive reverse edges. On each connected component of a graph we
use the path metric in which each edge has length one.
Fix a positive integer N. At each vertex v of a graph G, suppose that
we are given a pointed graph (D_v, d_v) and a rule that reads every
path of length at most N starting at v as an edge path in D_v starting
at d_v. Write [p]_v for the terminal vertex of the reading of p.
The following are the required properties.
(P1) Readings respect prefixes and reverses of edges, and read a
backtrack as a backtrack. If p starts at v and |p| < N, reading
gives a bijection between the edges leaving the actual endpoint of
p and the edges leaving [p]_v. If [p]_v = [p']_v, then p and p'
have the same actual endpoint in G, and their edge bijections agree
whenever both are defined.
(P2) Suppose that p runs from v to w and that a,b start at w. If each
of p,a,b has length at most N/2, then
[pa]_v = [pb]_v if and only if [a]_w = [b]_w. (3.1)
(P3) All connected components of all the graphs D_v are Gromov
hyperbolic with a common constant.
These rules concern paths within the stated range. They do not assert
that the whole graph G maps to D_v, or that the whole graph D_v maps
to G.
A loop p at v closes on reading if [p]_v = d_v. Let Q be the quotient
of the path groupoid of G by the relations that every loop of length at
most three which closes on reading is the constant path.
Lemma 3.1 (local path criterion). There is a bound N_0, depending
only on the common hyperbolicity constant, with the following property.
If the readings satisfy (P1)-(P3) with N at least N_0, and e is a
single-edge loop at v satisfying [e]_v different from d_v, then e is
nontrivial in Q.
The proof occupies the rest of this section. It constructs a locally
geodesic representative for each path, unique up to a uniformly narrow
comparison. Such representatives can be continued one edge at a time.
The comparison class is unchanged by the relations defining Q, but it
distinguishes the edge in the lemma from the constant path.
3.2. Calculations within one chart
We first record how the reading rules allow short calculations to move
between charts. By (P1), any edge path in D_v starting at a state
already reached by reading can be lifted to a path in G, as long as the
total reading length is at most N. If two such readings have the same
endpoint, their lifted paths have the same actual endpoint, and any
further common continuation reads identically. Together with (P2),
this has three useful consequences.
First, a short circuit closes on reading if and only if its two
complementary paths from one corner to another have equal read
endpoints. Indeed one may read one path and the reverse of the other,
using (P1). Closure can be tested at any corner: a cyclic change of
starting corner amounts to prefixing by a side, applying (P2), and
cancelling that side using (P1). Here and below, "short" means that
all prefixes and continuations in the comparison lie within the ranges
of (P1) and (P2).
Second, geodesicity of a short reading is independent of a short
prefix. For example, suppose the reading of a path a from w is
geodesic, but its reading after a path p : v -> w has a shorter
competitor. Lift that competitor from [p]_v. It gives a path b from w
with [pa]_v = [pb]_v. Equation (3.1) gives [a]_w = [b]_w,
contradicting geodesicity. The reverse implication uses a shorter
competitor in D_w and the same argument. For distance comparison,
suppose two states are reached by paths a,b from w. Lift a shortest
connector c between their read states, starting after a (or after pa).
Its length is at most |a|+|b|, and one applies (P2) to ac and b. Doing
this in both charts proves equality of the distances whenever
|p| <= N/2 and 2|a|+|b| <= N/2. These bounds, and their versions with
a,b interchanged, hold in every use below.
Third, a short strip of closed circuits can be calculated in one chart
without adding the lengths of all its transverse paths. Here is the
explicit bookkeeping. Let a_j,b_j be the prefixes down the two sides
of the strip, and let rho_j be its transverse path from the endpoint of
a_j to that of b_j. Starting in the chart at the beginning of the
first side, the equalities to prove are
[a_j rho_j] = [rho_0 b_j]. (3.2)
Closure of the next elementary circuit, transported after a_j by
(P2), compares its two routes across the next step. Explicitly, write
a_{j+1} = a_j s_j, b_{j+1} = b_j t_j,
where s_j,t_j are the next side portions. Closure and then the
preceding equality give
[a_{j+1} rho_{j+1}] = [a_j rho_j t_j]
= [rho_0 b_j t_j]
= [rho_0 b_{j+1}],
using (P1) for the common continuation, edge by edge. This proves
(3.2) at the next step. Thus only the side prefixes and each
individual transverse path enter the range bound. Conversely,
transverse paths found in this one chart lift to G and give circuits
that close when read from their own corners.
Choose an integer Delta >= 10 large enough for the following standard
consequences of the common hyperbolicity bound. In every component of
every chart, geodesic triangles are Delta-slim on vertices: every
vertex on one side lies within Delta of a vertex on one of the other
two sides. Dividing a geodesic quadrangle by a diagonal shows that
each side lies within 2Delta of the union of the other three sides.
The Gromov product
(x | y)_u = (1/2)( d(u,x) + d(u,y) - d(x,y) )
satisfies
(x | z)_u >= min{ (x | y)_u, (y | z)_u } - Delta. (3.3)
Moreover, for a geodesic segment [x,z], there is a vertex on that
segment at distance at most (x | z)_u + 10Delta from u. Enlarging
Delta absorbs all rounding to vertices. These are the usual
equivalent forms of hyperbolicity; see Bridson-Haefliger, Metric
Spaces of Non-Positive Curvature, Chapter III.H.
We shall also use the following elementary comparison. Suppose two
geodesic segments gamma_1,gamma_2, of lengths l_1,l_2, have
corresponding endpoints at distance at most h. Then for every integer
t >= 0,
d( gamma_1(min(t,l_1)), gamma_2(min(t,l_2)) ) <= 10(h + Delta).
(3.4)
This is a synchronous comparison of width 10(h+Delta): the parameters
advance by one on both segments until the shorter segment ends, and
then that segment waits at its endpoint. To check the bound, join the
corresponding endpoints by geodesics of length at most h. The
resulting quadrangle is 2Delta-slim. A point close to the opposite
geodesic has a mate whose distance from that geodesic's initial
endpoint differs from its own parameter by at most h+2Delta. A point
close to an end connector is within h+2Delta of the corresponding end.
These estimates, together with the bound 2h on the difference of the
lengths, imply (3.4), also when one segment has ended.
Fix integers
H = 1000 Delta, L >= 10^6 H, k = 20 L, N >= 10^5 k.
(3.5)
Taking L = 10^6 H gives a threshold N_0 = 10^5 (20L) depending only on
the original hyperbolicity constant.
A guide is a path each of whose subpaths of length at most k reads as a
geodesic in the chart at its initial vertex. Its entire length may be
arbitrarily large. Every individual chart calculation below uses
paths, including prefixes, of length less than 1000k. Consequently
(P1) and (P2) always apply. A long guide is handled in separate short
pieces, never by reading its whole prefix.
3.3. Narrowing comparisons between guides
For paths alpha,beta with the same initial and terminal vertices, a
ladder of width h consists of the following data. A finite schedule of
pairs of positions begins at (0,0) and ends at (|alpha|,|beta|); at
each step each position advances by zero or one. At each scheduled
pair there is a path, called a rung, from the vertex on alpha to the
vertex on beta, of length at most h. The first and last rungs are
empty. Each elementary circuit, formed by consecutive rungs and the
intervening portions of the two paths, must close on reading. Pausing
both paths is allowed. In particular, an elementary circuit need not
have length at most three. Ladders are comparisons of readings, not
expressions in the relations defining Q.
For the widths used below, ladders are symmetric, by changing corners
of their elementary circuits. They compose with addition of widths
whenever the sum is at most 1000H. Indeed, given ladders from alpha to
beta and from beta to gamma, synchronize the advances along beta,
inserting pauses in the other sides as necessary, and concatenate the
two rungs. Each resulting circuit is a union of one or two elementary
circuits from the given ladders, so closes by (3.2). Appending the
same trailing path to both sides preserves the width: use empty rungs
along that trailing path.
Lemma 3.2 (narrowing). If two guides have a ladder of width at most
1000H, then they have one of width at most H.
Proof. Let alpha,beta be the guides, and let the given width be
h <= 1000H. Consider an interval of the schedule on which alpha
advances by at most 8L. We claim that on this interval beta advances
by less than k. Otherwise stop at its first advance of k edges. The
strip up to that point calculates in one chart: its sides have lengths
at most 8L and k, and its rungs have length at most h. The second side
is geodesic there, because a path of length k reads geodesically by
the guide property. Its endpoints can be joined using the first side
and the two end rungs. This gives the contradiction
k <= 8L + 2h < 20L = k,
since h <= 1000H and L >= 10^6 H. It follows also that on every such
schedule interval both sides read as geodesics, and their lengths
differ by at most 2h.
Suppose first that |alpha| >= L. Mark alpha at x_0,...,x_m,
including its endpoints, so that consecutive marked intervals have
lengths in [L,2L]. Choose scheduled mates b_0,...,b_m on beta, using
its actual endpoints for b_0,b_m. The distance along beta between
consecutive old mates is at least L-2h.
At an interior mark x_i, calculate the part between x_{i-1} and
x_{i+1} in one chart. Replace its end rungs by geodesics of length at
most h. The point x_i is at distance at least L-h > 2Delta from either
end connector. Quadrangle slimness therefore puts it within 2Delta of
a vertex b'_i on the opposite geodesic, the corresponding piece of
beta. Its old mate b_i was within h of x_i, so geodesicity gives
|b'_i - b_i|_beta <= h + 2Delta. (3.6)
Here the left side denotes distance in the parameter of the path beta.
The new mates remain in order, since the difference between consecutive
new positions is at least
L - 2h - 2(h + 2Delta) = L - 4h - 4Delta > 0.
The same estimate at the ends puts them strictly between the unchanged
endpoint mates. Lift short paths from x_i to b'_i to obtain new rungs
of length at most 2Delta; use empty rungs at i=0,m.
The new rungs agree with the old calculation: following an old rung
and the relevant portion of beta has the same read endpoint as the new
rung. On a marked interval this equality can be checked together with
both adjacent marked intervals. Their union has first-side length at
most 6L, so the preceding bound and the short-strip calculation apply.
Thus between successive new mates both sides are geodesic, and their
end rungs have length at most 2Delta. Apply (3.4), lift the comparison
rungs, and retain the specified rungs at the marked endpoints. Each
elementary circuit closes in this chart and hence at its own corner.
The resulting width is at most 30Delta < H.
If |alpha| < L, the initial schedule argument applies to the whole
ladder. Both sides calculate as geodesics in one chart with identical
endpoints, and (3.4) directly gives width at most 10Delta < H. This
proves the lemma. []
In particular, existence of a width-H ladder is an equivalence relation
on guides with fixed endpoints. Reflexivity uses empty rungs, symmetry
was noted above, and transitivity follows by composing two ladders and
applying Lemma 3.2.
3.4. Continuing a guide by one edge
We next show that this comparison class can be continued along any
edge. The difficulty is that appending an edge need not preserve local
geodesicity. We repair the guide by choosing nearby vertices at widely
spaced marks and minimizing the total distance between those choices.
All nearby choices are specified by paths in the charts; distance in G
alone would forget the states that must be compared.
Lemma 3.3 (extension). If alpha is a guide and e is an edge starting
at its terminal vertex, there is a guide beta to the endpoint of
alpha e and a ladder between alpha e and beta of width at most 50H.
Proof. If |alpha| < L, read alpha e in its initial chart and join its
endpoints by a geodesic. Lift that geodesic to G. The result is a
guide by transport of geodesicity. Compare it with alpha, whose final
endpoint differs by at most one in the chart, using (3.4). Finish the
comparison by letting the first side traverse e while the second waits.
This proves the claim in this case.
Suppose |alpha| >= L, and mark it at a_0,...,a_m in intervals alpha_i
of lengths l_i in [L,2L], where alpha_i runs from a_i to a_{i+1}. At
each mark choose a path c_i starting there, with |c_i| <= H. Require
c_0 to be empty and c_m to be e. For adjacent marks define
F_i(c_i,c_{i+1}) = d_{D_{a_i}}( [c_i]_{a_i}, [alpha_i c_{i+1}]_{a_i} ).
Choose the paths c_i to minimize the sum of the F_i. A minimum exists
because the set of choices is nonempty and the objective takes
nonnegative integer values; no compactness assumption is needed.
For each i, lift a geodesic realizing F_i, starting after c_i in the
chart at a_i, to a path beta_i in G. Equality of read endpoints shows
that beta_i ends at the actual endpoint of c_{i+1}. Thus the paths
concatenate to a path
beta = beta_0 beta_1 ... beta_{m-1}
from a_0 to the endpoint of e. The comparison circuit between alpha_i
and beta_i, using c_i,c_{i+1}, closes on reading. The paths c_i,
rather than just their actual endpoints, retain the states used in this
assertion.
We claim that on every block of at most 100 consecutive marked
intervals the concatenation of the beta_i is geodesic in a single
chart calculation. Read such a block in the chart at its first mark.
The paths alpha_i, c_i, and beta_i all calculate there consistently,
by (3.2); each side has length bounded by 100(2L+2H), well within the
prescribed range. Distances computed here agree with the
single-interval distances F_i, by transport of short competitors.
For the hyperbolic calculation, denote the read marks on this block by
x_0,...,x_q, where q <= 100. Their consecutive distances are in
[L,2L], and
(x_{i-1} | x_{i+1})_{x_i} = 0 for 1 <= i < q, (3.7)
because two consecutive marked intervals of alpha have total length at
most 4L < k. We record the consequences of (3.7) explicitly.
Induction using (3.3) gives
(x_0 | x_{i+1})_{x_i} <= Delta for 1 <= i < q. (3.8)
The first case follows from (3.7). For the next case, the preceding
estimate and the identity
(x_0 | x_{i-1})_{x_i}
= d(x_{i-1},x_i) - (x_0 | x_i)_{x_{i-1}}
>= L - Delta
show that this product exceeds Delta. Applying (3.3) to the zero
product (x_{i-1} | x_{i+1})_{x_i} forces (3.8). The same argument
backwards, followed by another application of (3.3), gives
(x_0 | x_q)_{x_i} <= 2Delta for 1 <= i < q. (3.9)
For the last implication, use (x_{i-1} | x_q)_{x_i} <= Delta and
(x_{i-1} | x_0)_{x_i} >= L - Delta > 2Delta. Also, the distances
from x_0 increase at every mark by at least L-2Delta, since
d(x_0,x_{i+1}) - d(x_0,x_i)
= d(x_i,x_{i+1}) - 2(x_0 | x_{i+1})_{x_i}.
The analogous estimate holds from the other endpoint.
Let A,B be the read endpoints of the chosen paths at the two ends of
the block. They are at distance at most H from x_0,x_q, respectively.
By (3.9), every interior x_i has a mate on a geodesic [x_0,x_q] at
distance at most 12Delta. Such a mate is farther than 2Delta from
either of the geodesic connectors [x_0,A] and [x_q,B]: its distance
from those connectors is at least L-H-14Delta. Quadrangle slimness
therefore gives a vertex y_i on [A,B] with
d(x_i,y_i) <= 20Delta < H. (3.10)
The vertices y_i occur in increasing order along [A,B]. Indeed their
distances from A differ from the corresponding distances d(x_0,x_i)
by at most H+20Delta. Consequently each consecutive difference is at
least
L - 2Delta - 2H - 40Delta > 0.
The same inequalities keep them strictly between A and B.
Lift short paths from the interior x_i to the y_i to replace the
interior choices c_i, keeping the two outer choices fixed. These are
legal choices by (3.10). For them the sum of the successive distances
equals d(A,B), since their endpoints occur in order on a geodesic.
Transport of distances shows that this is also their sum of
single-interval objectives F_i. Minimality of the original total
objective implies that its sum on this block is at most d(A,B):
changing only interior choices leaves every term outside the block
unchanged. The triangle inequality gives the reverse inequality.
Thus the original joins concatenate geodesically on the block, as
claimed.
Each join beta_i has length at least L-2H. A subpath of beta of
length at most k = 20L therefore meets fewer than 100 consecutive
joins: apart from at most two end joins, every join it meets is
entirely contained in that subpath. The block result and transport of
geodesicity prove that beta is a guide.
Finally compare alpha_i and beta_i in their single-interval chart.
Their corresponding endpoints are joined by the given paths c_i,c_{i+1}
of length at most H. Formula (3.4) gives comparison rungs of length at
most 10(H+Delta); retain the given choice paths as rungs at the marks.
Lift these comparisons and concatenate them. At the last mark the rung
is e; let the first side traverse e and finish with an empty rung. This
gives the required ladder of width at most 50H. []
3.5. The invariant of a path
We can now finish the local criterion. Fix an initial vertex v and
consider guides starting at v, modulo width-H ladders. By Lemma 3.3,
appending an edge e defines a continuation of such a class. It is
independent of both choices involved. If alpha,alpha' represent the
same class, append e to their width-H ladder. If beta,beta' are any two
guides supplied by Lemma 3.3, composition gives a ladder between them
of width at most
50H + H + 50H = 101H.
Lemma 3.2 reduces this to width H. The same argument covers different
choices of the extension for a fixed guide.
Let q be a loop of length at most three that closes on reading. It has
a ladder of width at most three with the constant path. One explicit
schedule traverses q on the first side while the second waits: after a
noninitial position use the remaining suffix of q as the rung, and use
empty first and last rungs. The first circuit closes because q does,
and the subsequent ones cancel consecutive reverse edges. The
short-chart rules justify the same assertion at each corner. Appending
this ladder after any guide alpha gives a width-three ladder between
alpha q and alpha.
Successively applying Lemma 3.3 along q produces a guide beta with a
ladder to alpha q of width at most 150H. To see the bound, at each step
append the remaining common trailing edges to the comparison and
compose; each of at most three steps adds 50H. Composing with the
preceding width-three ladder and narrowing shows that beta and alpha
represent the same class. The same reasoning applies to a backtrack,
which closes on reading by (P1).
Starting from the empty guide at v and continuing along a path thus
gives a class unchanged by every defining relation of Q. Insertion of
a relation in the middle of a path causes no problem: the classes agree
after the inserted loop, and the well-defined continuations along the
remaining edges preserve that agreement.
Proof of Lemma 3.1. Let e be the edge in the statement. Its reading
has distinct endpoints and length one, so e itself is a guide. Its
class is therefore the continuation of the empty guide along e. If e
were trivial in Q, the invariant just constructed would give a width-H
ladder between e and the constant path. This entire comparison
calculates in the chart at v: its sides have lengths one and zero and
its individual rungs have length at most H. Equation (3.2), with the
empty final rung, would then give [e]_v = d_v, a contradiction. This
proves the lemma. []
4. AN ARITHMETIC ARRANGEMENT AND A DISTINGUISHED SURFACE
This section constructs the geometric objects used in the rest of the
proof. The output is a smooth connected proper effective orbifold S
with projective coarse space and a morphism f : Y -> S, where Y is an
iterated point blowup of a simple abelian surface A, and the image of f
lies in the ordinary locus of S. The arrangement built here supplies
both the relations and the local covering spaces used to prove that
the image of pi_1(Y) is infinite cyclic.
The construction has several layers. First choose a genus-two curve C
with simple Jacobian. Then realize its six branch points as marked
directions in a complex hyperbolic ball. Arithmetic congruence
subgroups make the local arrangement exactly the prototype in large
balls. Two parity covers turn the arrangement into a smooth
projective variety V with an order-four symmetry s. Blowing up and
resolving the quotient over the distinguished point produces S. A
rational section of a line bundle on C x C gives the surface map.
4.1. Six slopes and a genus-two curve with simple Jacobian
The two pencils will use the same six marked directions on P^1. We
first choose the markings so that their double cover has simple
Jacobian, and then choose the arithmetic ball containing the pencils.
Choose six distinct real algebraic numbers lambda_1,...,lambda_6 with
|lambda_i| < 0.01 for 1 <= i <= 4,
lambda_5 < -4, lambda_6 > 4. (4.1)
We call the first four slopes red and the last two blue. We also
require that the smooth double cover
pi : C -> P^1, y^2 = product_{i=1..6} (t - lambda_i), (4.2)
has simple Jacobian. These requirements can be imposed
simultaneously, as follows.
Start with six disjoint real open intervals satisfying (4.1). Monic
real polynomials with one root in each interval form a nonempty open
set in coefficient space. At three sufficiently large distinct primes
prescribe squarefree degree-six reductions with factor degrees
(6), (5,1), (2,1,1,1,1).
Weak approximation produces a monic polynomial in Q[t], integral at
these primes, with these reductions and with its real coefficients in
the chosen open set. Its Galois group contains a six-cycle, a
five-cycle, and a transposition, by the Frobenius cycle criterion. The
six-cycle makes the group transitive. The stabilizer of the fixed
point of the five-cycle is transitive on the remaining five points, so
the group is two-transitive. Conjugating its transposition then gives
every transposition, and the Galois group is S_6.
Let F be the splitting field. Then F is totally real, F is not Q, and
all six slopes belong to F. Zarhin's theorem gives
End(Jac(C)) = Z
over an algebraic closure, hence Jac(C) is simple (Zarhin,
Hyperelliptic Jacobians without complex multiplication, Theorem 2.1).
Indeed, if Jac(C) were isogenous to a product of elliptic curves, its
endomorphism ring would contain nontrivial idempotents, contradicting
End = Z.
4.2. The Hermitian form, the ball, and the hyperplanes
Use the inclusion F subset R containing the chosen roots as the
distinguished real embedding. Choose a CM extension E/F and an element
a in F which is positive at this embedding and negative at every other
real embedding; weak approximation supplies a. On E^5 put
h = diag(1, 1, 1, 1, -a). (4.3)
At the distinguished embedding the negative lines form complex
hyperbolic space of complex dimension four. In the affine chart with
last homogeneous coordinate one, write this ball as
B = { (z,w) in C^2 x C^2 : |z|^2 + |w|^2 < a },
o = (0,0). (4.4)
The metric is normalized so that in the unit-radius ball a point at
distance r from the origin has Euclidean radius tanh(r).
Choose c in F with
0.85 < c / sqrt(a) < 0.9. (4.5)
The z-family consists of the six slope hyperplanes and two coupling
hyperplanes
H^z_i = { z_2 = lambda_i z_1 } intersect B, 1 <= i <= 6,
K^z_+ = { z_1 = c } intersect B,
K^z_- = { z_1 = -c } intersect B. (4.6)
Define the w-family by the same equations in w. Let D denote these
sixteen labeled complex hyperbolic hyperplanes. They are all defined
over E and are preserved, with their labels permuted, by
sigma(z,w) = (w,-z). (4.7)
This is an order-four isometry represented by an integral unitary
matrix. The two pencils have centers
L_z = { z = 0 } intersect B, L_w = { w = 0 } intersect B.
(4.8)
The intersection pattern is elementary but important. The z slopes
all meet along L_z. On K^z_+, a slope of value lambda meets the ball
exactly when
c^2 (1 + |lambda|^2) < a. (4.9)
Thus each coupling hyperplane meets the four red slopes, in distinct
subspaces disjoint from L_z, and meets neither blue slope. The two
z coupling hyperplanes are disjoint. These statements hold also in
the w-family. A z coupling hyperplane and a w coupling hyperplane are
disjoint because 2c^2 > a. Every intersection involving the two
families has a local product description in the z and w variables.
The word "product" here concerns local analytic coordinates, not a
product decomposition of the hyperbolic metric.
Here is the picture in one coordinate pair. On the real slice w=0,
z_1,z_2 real, the z pencil consists of six lines through the origin
L_z. The four red lines have small angles, the two blue lines have
large angles. The coupling hyperplanes z_1 = c and z_1 = -c are two
vertical lines. Each vertical line meets the four red lines and meets
neither blue line. The w-family is an identical picture in the other
coordinate pair. The full arrangement lies in complex dimension four.
4.3. Exact models in congruence quotients
The arithmetic lattice theorem of Borel-Harish-Chandra, applied to
(4.3), gives a cocompact arithmetic lattice of simplest type in
PU(4,1); see Borel-Harish-Chandra, Arithmetic subgroups of algebraic
groups, Corollary 12.4. Indeed all the other archimedean factors are
compact, and the form is anisotropic over E: a nonzero E-rational
isotropic vector would remain isotropic at a definite conjugate
embedding. Torsion-free principal congruence subgroups therefore give
compact complex hyperbolic manifolds, which are projective by the
Baily-Borel theorem. One can also obtain projectivity from the
positive canonical bundle of a compact ball quotient. We use
principal congruence subgroups normalized by sigma, and may take their
levels arbitrarily deep. These standard arithmetic facts are recalled
in Stover-Toledo, Residual finiteness for central extensions, Section
3.1.
Fix positive normal vectors n_i in E^5 for the members of D, scaled to
have integral coordinates. We always label a translate of the i-th
hyperplane by i, and use the prescribed normal gamma n_i for its
translate by gamma. Congruence subgroups may be chosen to avoid the
finite scalar kernel of the projective action.
Proposition 4.1 (exact local arrangements). Given W > 0 and R_0 > 0,
there is a torsion-free principal congruence subgroup Gamma_0,
normalized by sigma, with the following properties. Let A_0 be the
union of all Gamma_0-translates of the labeled hyperplanes of D.
(1) The arrangement A_0 is locally finite. The hyperplanes meeting
any ball of radius W have distinct labels and are carried to the
corresponding subarrangement of D by a holomorphic isometry
induced by a unitary map sending each prescribed translated normal
to its prototype normal.
(2) The injectivity radius of Gamma_0 \ B is greater than R_0.
(3) In the ball of radius R_0 about o, the arrangement is exactly D
restricted to that ball. The analogous assertion holds in every
Gamma_0-translate of this ball.
All these conclusions concerning the lifted arrangement remain valid
on any subsequent finite cover of Gamma_0 \ B, provided one uses the
full inverse image of the arrangement.
Proof. First fix a radius W and a finite collection of hyperplanes
meeting a W-ball. At its viewpoint, represented by a unit negative
vector x, the distance to the hyperplane with positive normal n
satisfies
sinh d(x, H_n) = | h(n,x) | / sqrt( h(n,n) ). (4.10)
For a normal of one of our finitely many fixed lengths, intersection
with the W-ball bounds |h(n,x)|. Write n = n_+ + alpha x with n_+ in
x^perp. Since
h(n_+, n_+) = h(n,n) + |alpha|^2,
the positive component is bounded as well. Cauchy-Schwarz in x^perp
now bounds every Gram entry of two normals visible from this ball, by
a constant depending only on W and their labels.
At every other archimedean place, definiteness and the fixed normal
lengths bound the same Gram entries. All entries belong to one fixed
fractional ideal of E. Its image under the archimedean embeddings is
a lattice, so only finitely many Gram entries satisfy these bounds.
If gamma,gamma' are in a sufficiently deep integer principal
congruence subgroup, then
h(gamma n_i, gamma' n_j) = h(n_i, n_j)
modulo the level, in this fractional ideal. Choose the level to
separate all the finitely many possible nonzero differences. We obtain
exact equality:
h(gamma n_i, gamma' n_j) = h(n_i, n_j) (4.11)
whenever the corresponding hyperplanes are visible together.
Equality of Gram matrices gives the asserted isometry even for
dependent collections. Every E-rational subspace of E^5 is
nondegenerate for h: a nonzero radical, being defined by linear
equations over E, would persist at a definite embedding. Consequently
the kernel of a Gram matrix is exactly the relation space of its
vectors. Equality (4.11) thus defines an isometry between their spans,
which extends to a unitary isometry of the ambient Hermitian space.
Its projectivization is a holomorphic ball isometry.
Two visible copies of the same label must in fact have the same normal:
their difference has norm zero by (4.11), and anisotropy over E makes
that difference zero. Two different labels cannot coincide, since
this would contradict equality of their relation spaces with those of
the distinct prototypes.
Local finiteness follows from the same boundedness argument applied to
normals themselves at a fixed viewpoint. The normals have integral
coordinates; their distinguished components are bounded by the
distance bound and their conjugate components by definiteness. Only
finitely many can meet a fixed bounded ball.
To obtain exact agreement at o, choose an auxiliary ball about o large
enough to contain the prescribed R_0-ball and to meet every prototype
hyperplane. Apply the Gram argument to this ball. The prototype
normals span E^5: the slope normals span the four positive coordinate
directions, and a coupling normal supplies the last homogeneous
direction. An additional translated normal visible in this ball has,
by (4.11), the same inner products with this spanning set as its
prototype. It therefore equals that prototype. This proves the third
assertion.
Finally fix a torsion-free congruence subgroup in advance and a
compact fundamental set for its action on B. A group element moving a
point of this set a bounded distance belongs to a finite set. A nested
sequence of normal principal congruence subgroups has trivial
intersection, after the scalar kernel has been removed. Deep enough
levels avoid every nonidentity element of the finite set. Normality
allows an arbitrary viewpoint to be moved into the compact fundamental
set without changing subgroup membership. This proves the required
uniform lower bound on injectivity radius. Increasing the level
accommodates all the preceding requirements simultaneously. Passing
to a subgroup preserves them when the arrangement is pulled back in
full. []
We next record the consequences for the divisors and centers that we
will blow up. In M_0 = Gamma_0 \ B, each label gives a closed
embedded smooth totally geodesic divisor, possibly disconnected. Its
lifts are locally finite and have disjoint sheets by Proposition 4.1.
The divisors are algebraic because M_0 is projective.
Within one family the intersection centers have complex codimension
two. A center is either a common intersection of at least two slopes,
or an intersection of a red slope with a coupling hyperplane. At a
slope center only a subset of the six slopes may occur. The subset is
constant along a connected center: if another slope meets it, the
local model shows that it contains the intersection, and the
containment extends along its lifts. Distinct centers within one
family are disjoint; otherwise all participating hyperplanes would be
visible near a point of intersection, contradicting the prototype
pattern. Coincident centers are counted once. Thus these centers are
closed embedded smooth subvarieties. Centers from different families
have the product structure already described. These assertions apply
to the full pulled-back arrangement after any finite cover.
4.4. Two global sign covers
For the local models we need parity characters on the arrangement
complement. We obtain these from the following theorem of
Stover-Toledo, in the form stated by Llosa Isenrich-Py (Theorem 4.2);
see also Stover-Toledo.
Theorem 4.2 (virtual cyclic ramified covers). Let M_0 be a compact
ball quotient of complex dimension at least two by a torsion-free
congruence arithmetic lattice of simplest type. Let D be a nonempty
totally geodesic divisor whose components are smooth, embedded, and
pairwise disjoint. For every integer d >= 2 there is a finite
unramified cover M_1 -> M_0 such that M_1 admits a cyclic cover of
degree d, branched over the full inverse image of D.
Apply Theorem 4.2 with d=2 to each of the sixteen label divisors
separately, on the initial congruence quotient M_0. Each application
satisfies the disjoint-component hypothesis. We do not apply the
theorem to the union of the intersecting divisors. Take a common
finite unramified cover of the resulting base covers. The sum, in
Z/2, of the eight characters for the z labels gives a character chi_z
on the complement of the z arrangement. It has value one on a
meridian of every z hyperplane. Pull it back to the complement of both
families. The same procedure gives a w character. Subsequent finite
covers need not be congruence: every use of Theorem 4.2 has already
taken place on M_0.
We arrange equivariance before fixing the signs. First intersect the
finitely many sigma-conjugates of the subgroup defining our common
cover, so that sigma acts on it. Since sigma^2 preserves the
z-family, the character
delta = chi_z + sigma^{2*} chi_z
is zero on all meridians. The kernel of the map from the fundamental
group of a divisor complement to that of the ambient smooth variety is
normally generated by meridians. Hence delta factors through the
fundamental group of the compact ball quotient. Pass to a further
finite cover by intersecting the kernel of this character with its
sigma-conjugates. On that cover the pulled-back z character satisfies
sigma^{2*} chi_z = chi_z.
Define
chi_w = sigma^* chi_z. (4.12)
Then sigma exchanges the two characters. They have the prescribed
meridian values in their respective families, and each is defined
using only that family.
Write M = Gamma \ B for this final finite cover, still with the full
pulled-back arrangement. The image of o in M, also denoted o, is fixed
by sigma. It is an isolated component even of the fixed locus of
sigma^2, because the derivative of sigma^2 at o is -id. No assertion
of this kind is needed for the other fixed loci of the symmetry on M.
4.5. Filling the arrangement and the distinguished fiber
Blow up all intersection centers within the z-family, and then the
transforms of those within the w-family. Product coordinates show that
the operations for the two families commute. The branch rule for the
z character is as follows.
Every strict transform of a z hyperplane is a branch divisor of order
two. If k such hyperplanes pass through a center, its exceptional
meridian is the product of their meridians, and therefore has sign k
mod 2. Accordingly the exceptional divisor is branched precisely when
k is odd. In that case blow up each meeting of this exceptional
divisor with a strict transform of a hyperplane. The two meeting
divisors both have sign one; their new exceptional divisor has sign
zero. The branch components within this family are now smooth and
disjoint. Apply the identical rule to the w character. This
prescription includes partial pencils. In particular an ordinary
red-coupling pair has k=2, so its first exceptional divisor is
unbranched.
Let Mhat denote the resulting smooth projective base. Extend the two
sign covers from the complement by normalization over Mhat, and take
them together. Finite topological covers of smooth complex algebraic
varieties algebraize, and normalization in the resulting finite
extensions gives finite algebraic covers; thus this procedure is
projective. Locally along a branch divisor the normalization is the
smooth double-cover model t = u^2. Branch divisors within a family are
disjoint, and across the families the local equations and the two signs
are independent. Their combined normalization is therefore smooth,
including at crossings, where it has product equations
(t_1,t_2) = (u_1^2,u_2^2). We obtain a smooth projective variety
V -> Mhat (4.13)
with deck group (Z/2)^2. It is connected: a meridian at a general
point of a z divisor has sign (1,0), and a meridian at a general point
of a w divisor has sign (0,1), so the combined monodromy is
surjective.
Near o all six slopes in each pencil occur and the coupling
hyperplanes are absent after the neighborhood has been made small. The
blowups are the product of the blowups of the two coordinate planes at
their origins. The exceptional fiber of Mhat -> M is P^1 x P^1. Each
exceptional line has six branch markings and has even meridian sign.
Its inverse image in V is consequently
F_* = C x C. (4.14)
Put
L = pi^* O_{P^1}(-1),
N = N_{F_*/V} = pr_1^* L + pr_2^* L. (4.15)
In fact a neighborhood of F_* has the product line-bundle description
provided by these two tautological lines. Indeed, in each factor,
after the unbranched radial disks have been filled, their disk bundle
retracts onto its zero section. Away from the six marked directions
its cover is pulled back from the direction line. Normalization gives
the six-branch cover in that direction variable, with the radial line
bundle pulled back unchanged.
The equivariance (4.12) makes the kernel of the combined sign character
invariant, so sigma lifts on the complement. The equivariant blowups
and normalization extend the lift to an automorphism s of V. Its
restriction to F_* exchanges the two copies of C, possibly followed by
either hyperelliptic deck involution. Composing with a deck
transformation makes this restriction the pure swap. To determine the
lift in the whole local model, note that u,v are actual tautological
vectors: the covering map in these coordinates is
(p,q;u,v) -> (pi(p), pi(q); u,v).
The pure swap determines the lifted direction maps on the connected
local cover. Since the base map sends the radial vectors to (v,-u),
there is no further radial multiplier. Thus
s(p,q;u,v) = (q,p;v,-u). (4.16)
Its fourth power is a deck transformation fixing F_* pointwise, so
s^4 = 1; a nonidentity element of (Z/2)^2 does not fix C x C pointwise.
Its square is the identity on F_* and is multiplication by -1 on both
normal lines.
4.6. An ordinary neighborhood of the distinguished fiber
The quotient stack [V/<s>] is a smooth proper orbifold with projective
coarse space, but a surface mapping into its distinguished fiber would
meet stabilizers. We now replace just that fiber by an ordinary smooth
resolution. Retaining the quotient stack elsewhere allows all other
fixed loci to be left in place.
First blow up F_* in V, and take the coarse quotient by <s^2>. The
exceptional divisor is
P = P_{F_*}(N), (4.17)
where projectivization parametrizes lines. Before taking the quotient,
s^2 acts trivially on P and as t -> -t on its transverse coordinate.
The quotient is thus smooth near P, with transverse coordinate t^2.
The residual involution induced by s has fixed points near P only in P:
away from the fiber, a fixed point would project to a nontrivial fixed
point of the symmetry near o in M. Its fixed points in P project to
the diagonal of C x C. On the projective normal line over that
diagonal, it acts by
[u:v] -> [v:-u].
There are exactly two eigensections, with eigenvalues i and -i before
projectivization. These give two disjoint smooth fixed curves. On the
three normal directions to either curve the involution acts by -1: one
direction is antisymmetric in the base, one is tangent to the
projective fiber, and the transverse coordinate to P has eigenvalue
(+/- i)^2 = -1. Blow up both fixed curves and take the coarse quotient
by the residual involution. This quotient is smooth near the
distinguished fiber. Indeed, for the linear local model
(x,t_1,t_2,t_3) -> (x,-t_1,-t_2,-t_3),
blowing up the fixed curve {t_1=t_2=t_3=0} makes the action a
reflection in the coordinate normal to the exceptional divisor, and the
coarse quotient is smooth. Finite-order holomorphic actions are
locally linearizable, so this model applies.
All centers just used are closed and lie over o. The blowups and
finite coarse quotients can therefore be performed globally, producing
a projective coarse space S_0 which modifies V/<s> only over o. The
intermediate coarse spaces may retain quotient singularities elsewhere;
the preceding calculation establishes smoothness on a neighborhood of
the distinguished fiber.
To form S, use this smooth scheme near that fiber and use [V/<s>]
away from it. This is an algebraic gluing. More explicitly, in M
remove all components of the nontrivial fixed loci other than the
isolated point o, and take the resulting invariant Zariski
neighborhood. Outside o in this neighborhood the action is free, so
both constructions restrict to the same ordinary quotient. Their
gluing is a smooth connected effective orbifold S with coarse space
S_0. The map S -> S_0 is separated and proper: these properties are
local on S_0, where it is either the coarse map of a finite-group
quotient stack or the identity. Since S_0 is projective, S is
separated and proper over C. In particular the whole distinguished
fiber lies in the ordinary smooth locus of S.
4.7. The surface mapping into the ordinary fiber
Let A = Jac(C). For a genus-two curve, the Abel map
Sym^2 C -> Pic^2(C) = A
is the blowup at the canonical class, after choosing the displayed
translation. To recall the geometry, Riemann-Roch gives a unique
effective divisor in every degree-two class except K_C; the latter has
the pencil E_C = |K_C| = P^1. For the degree-two quotient
q : C x C -> Sym^2 C, the inverse image of E_C is the graph Gamma_iota
of the hyperelliptic involution. The map q is generically unramified
along this graph, so q^* E_C = Gamma_iota as divisors; ramification at
its six diagonal points does not change the generic multiplicity. The
projection formula gives
2 E_C^2 = (q^* E_C)^2 = Gamma_iota^2 = 2 - 2g(C) = -2.
The Abel map is therefore a birational morphism between smooth surfaces
with one exceptional (-1)-curve, and is the stated point blowup.
Choose a nonzero rational section tau of L. On the open subset of
C x C where both values are defined and nonzero, put
(p,q) -> [ tau(p) : i tau(q) ] in P. (4.18)
This section is equivariant for swapping the two factors and the
involution on P. Indeed (4.16) gives
[tau(p) : i tau(q)] -> [i tau(q) : -tau(p)]
= [tau(q) : i tau(p)].
Away from the diagonal it avoids the two fixed curves, and thus gives a
rational map from Sym^2 C to the smooth resolution constructed above.
All the bundles and maps in this description are algebraic: the
normal-bundle identification is algebraic as well as analytic, by GAGA
on the projective fiber. Resolve the rational map by a finite sequence
of point blowups of its smooth projective source, using the projective
coarse space as target. The resolved image remains in its closed
distinguished fiber, where the coarse space agrees with the ordinary
locus of S. We therefore obtain a morphism
f : Y -> S (4.19)
whose image lies entirely over o. Thus Y is an iterated point blowup
of the simple abelian surface A, and f(Y) lies in the ordinary locus.
Proposition 4.3 (the geometric construction). After any prescribed
finite lower bounds on the arrangement radii and injectivity radius,
the construction above gives a smooth connected proper effective
orbifold S with projective coarse space, a simple abelian surface
A = Jac(C), an iterated point blowup Y -> A, and a morphism f : Y -> S
with image in its ordinary locus. The distinguished fiber is obtained
from the product C x C, its normal bundle pr_1^* L + pr_2^* L, and the
action (4.16), by the explicit blowups and coarse quotients above.
Away from that fiber the orbifold is [V/<s>], with V given by the two
parity covers and the filling rules of Section 4.5.
All parameters used in the finite arrangement have now been fixed.
The depth of Gamma_0 has not: Proposition 4.1 permits its choice after
the constants required for the finite-model graphs in the next section.
Passing to the finite covers which produce the signs preserves these
estimates. This order of choices will allow the local calculations to
establish
im( f_* : pi_1(Y) -> pi_1(S) ) = Z.
5. LOCAL COVERS AND HYPERBOLIC PATH GRAPHS
The geometric construction gives relations among the loops in the
surface. To show that these relations leave an element of infinite
order, we read paths in auxiliary covers of finite arrangements. This
section constructs those covers and proves the two properties needed
for Lemma 3.1: their kernels agree on sufficiently buffered overlaps,
and their path graphs are uniformly hyperbolic.
All constants in this section depend only on the finite arrangement D.
In particular, they are fixed before choosing the arithmetic level.
5.1. Dihedral labels on a finite arrangement
For a subset T of D, set
U_T = B \ union_{H in T} H.
There are characters eps_z, eps_w : pi_1(U_T) -> Z/2, with eps_z equal
to one on a meridian of a z hyperplane and zero on a meridian of a w
hyperplane, and conversely for eps_w. One can define each character by
the parity of the winding number of the product of defining affine
equations of the hyperplanes in its family. Let E_T -> U_T denote the
associated four-sheet cover; it may be disconnected.
These signs restrict canonically to any ball patch, up to a choice of
sheets. Indeed, meridians normally generate the complement group in a
simply connected ball: fill a loop with a disk transverse to the
hypersurfaces. Thus their values determine a sign character there.
We refine these signs using the infinite dihedral group
D_inf = < r, b | r^2 = b^2 = 1 >, eps(r) = eps(b) = 1.
(5.1)
Its even subgroup is the infinite cyclic translation subgroup <rb>.
Conjugation in D_inf preserves this subgroup and changes its integer
coordinate at most by a sign.
We first construct q_z : pi_1(U_T) -> D_inf. The construction ignores
all w hyperplanes. If T contains fewer than all six z slope
hyperplanes, define
q_z(gamma) = r^{eps_z(gamma)}. (5.2)
Suppose instead that the whole z pencil is present. With only its six
hyperplanes removed, projection to the direction
lambda = z_2 / z_1
gives a map to the six-punctured projective line. Choose its standard
meridians so that the four red punctures occur first, based together in
a small red disk, and the two blue punctures occur last. Assign r to
each red meridian and b to each blue meridian. The sphere relation is
respected because r^4 b^2 = 1. We fix this choice of the
direction-line representation once, and use the same choice for every
complete pencil and for both families.
It remains to add the coupling hyperplanes that belong to T. They lie
over the red-direction disk: on z_1 = +/- c in the ball,
|lambda| < 0.7.
Over |lambda| < 1.5, the pencil local system has a reduction of
structure group to <r>. In this reduction, multiply its transition
functions by the two-sheet systems defined, for the coupling
hyperplanes that are present, by
(1 - z_1/c)^{1/2}, (1 + z_1/c)^{1/2}. (5.3)
This means adding their winding parities to the exponent of r. Over
|lambda| > 1 one has |z_1| < c, so both systems in (5.3) have the
distinguished trivializations obtained by the branch of the square root
near 1. These trivializations identify the modified system with the
unmodified pencil system on 1 < |lambda| < 1.5. They therefore glue the
two systems. This constructs q_z, up to conjugacy, on all of U_T. Its
parity is eps_z: this is true on every meridian, and meridians normally
generate. Interchanging z and w gives q_w.
On each component of E_T, both representations take values in
translations. Choose integer coordinates on the translation groups and
write the resulting homomorphisms as
ell_z, ell_w : pi_1(E_T) -> Z. (5.4)
The notation is componentwise. Their individual kernels are
independent of conjugating the dihedral representations or changing
sheets. The signs of the integer coordinates are immaterial until we
form a sum in the central model.
For each T, apply the intersection-center blowups, parity branching,
and normalization of Section 4.5 to the arrangement T in B and its sign
cover E_T. The resulting smooth space is its filled finite model.
Lemma 5.1. The homomorphisms ell_z and ell_w extend across the
blowups and branched fillings of the finite model prescribed in
Section 4.5.
Proof. It suffices to check the meridians of the divisors being added
to the sign cover. Removing subsets of complex codimension at least
two does not change the fundamental group of a smooth manifold, and
adding a smooth divisor kills its meridian. These assertions follow by
general position for paths and disks, so they also apply to the
successive local modifications here.
For an incomplete z pencil, q_z is the sign character in <r>; hence
ell_z is already zero on the sign cover.
For a complete pencil, a meridian of the exceptional divisor over L_z
circles the common intersection at a fixed direction. Its image under
direction projection is constant, and the coupling systems trivialize
near L_z. Its q_z label is therefore the identity.
At an intersection of a coupling hyperplane with a red slope
hyperplane, the two local meridians both have label r in the same
reduction. The meridian of their exceptional divisor is the product of
these two meridians, and its label is r^2 = 1.
Every meridian that is filled after branching of order two is the
square of a meridian with reflection label, so again has trivial label.
These are all the complete-pencil centers. The additional blowups
needed for odd partial pencils cause no difficulty, since the label on
the corresponding sign cover is zero. The w calculation is identical,
and intersections involving both families are products of these local
calculations. Thus both labels kill every required meridian. []
5.2. The sum cover at the distinguished fiber
For the full model T = D, let Ebar denote its filled sign cover. Its
fiber over o is F_* = C x C, and it has the lift s of
sigma(z,w) = (w,-z) described in Section 4. On F_*, the action of s
interchanges the factors. The first translation label is nonzero on
the first factor and zero on the second; the reverse holds for the
second label. For example, the product of one red and one blue
meridian has even parity and label rb, so its lift to C gives a nonzero
translation. This calculation on F_* agrees with that on the
complement: push a representative path slightly in a nonzero normal
direction, keeping its directions away from the branch points. The
coupling square roots are trivial there.
To pass from this filled model to the resolved distinguished fiber, we
need an ordinary covering across the fixed points of the coarse
quotient. The descent argument below will achieve this by making the
lifted symmetry commute with deck translations. We will orient the two
labels so that their sum is s-invariant.
The construction of q_z is unchanged under z -> -z, with the two
coupling hyperplanes interchanged. Indeed this map fixes the direction
lambda, interchanges the functions in (5.3), and preserves their
distinguished trivializations. The analogous statement holds for w.
Consequently s interchanges the two translation homomorphisms up to
signs. Orient them by the preceding comparison on F_* so that
ell_z o s_* = ell_w, ell_w o s_* = ell_z. (5.5)
The nonzero restrictions to the two factors determine these signs. In
particular the homomorphism
ell = ell_z + ell_w : pi_1(Ebar) -> Z (5.6)
is s-invariant.
Lemma 5.2 (central model). The regular covering associated with
ker ell admits an order-four lift of s commuting with its deck
translations. Its quotient by this lift is an ordinary topological
covering of the coarse space Ebar/<s>. Consequently it pulls back to a
covering of the resolution used over the distinguished fiber.
Proof. Choose an s-fixed point of F_*, for example a point on its
diagonal, and a point above it in the ker ell cover. Invariance of ell
gives a unique lift s_hat fixing this point. Since s^4 = 1, its fourth
power is a deck transformation fixing a point, hence is the identity.
Its order is four because it projects to s. Invariance of the integer
label also says that conjugation by s_hat acts trivially on the deck
group.
For descent to a topological cover, consider a point fixed by a
subgroup H of <s>. On the fiber over that point, the lifted H action
commutes with deck translations, and therefore acts by translations of
this integer torsor. Each such translation has finite order because
the lift is an action of the finite group <s>. It is thus the identity.
Choose a sufficiently small invariant neighborhood at the point on
which the original cover is trivial. The stabilizer acts trivially on
the discrete factor, so its quotient is again a product with that
factor. Translating these neighborhoods under the finite group proves
that the quotient map is a covering. Pullback of a covering along the
resolution map remains a covering. []
The model using both kernels will be called an individual model. The
full model using the sum kernel and then the lifted s quotient will be
called the central model. The distinction is essential: the sum is
used only near a distinguished fiber.
5.3. How kernels compare on bounded balls
Before constructing graph metrics, we prove that these local labels can
be compared without a global coloring of the arithmetic arrangement.
There are two possible sources of disagreement: normal-matching
isometries may not be unique, and a larger chart may contain
hyperplanes absent from a smaller one.
First suppose the prescribed normal vectors of a complete pencil have
been matched as in Proposition 4.1. Two distinct slope normals span
its positive normal plane and specify its two coordinate functionals,
hence the ratio z_2/z_1. Thus the direction-line local system is
independent of the choice of an isometry extending the normal match.
If a coupling normal is also matched, its homogeneous linear equation,
together with those two functionals, specifies the ratio z_1/c.
Explicitly, if X_1 is the first coordinate functional, T_0 is the last
homogeneous coordinate, and F_c = X_1 - c T_0 defines the positive
coupling hyperplane, then
z_1 / c = X_1 / (c T_0) = X_1 / (X_1 - F_c). (5.7)
The negative coupling gives the same conclusion with the signs
reversed. The gluing rule (5.3) is therefore intrinsic to these
matched data as well. The corresponding assertions hold for the w
family. Under sigma, the two families are interchanged and the w
coordinates are negated. If the chosen representative of a relabeled
normal differs by a scalar, use that same scalar on both sides of the
match. The resulting coordinate ratios transform as just described,
so the individual kernels are carried to the corresponding individual
kernels. This is the equivariance needed below; it does not require a
globally chosen orientation of a translation label.
Next, consider a coupling hyperplane absent from a metric ball Q. A
metric ball in the ball model is a Euclidean ellipsoid and is convex.
Its z_1 image is therefore convex and avoids the relevant value c or
-c. On that image the corresponding function in (5.3) has a
single-valued square root. Where the image meets |z_1| < c, choose its
sign to agree with the distinguished square root. The intersection is
convex, so one choice works throughout it. If the intersection is
empty, there is no overlap requiring a prescribed sign. This
trivializes the extra coupling system compatibly with the red-disk
gluing. Hence including or omitting a coupling hyperplane missing Q
gives isomorphic label systems on Q.
Lemma 5.3 (buffer bound). For every R > 0 there is B_1(R) with the
following property. In any finite model with a complete z pencil, if a
loop in the sign cover projects into B(x,R) and has nonzero z
translation, then
dist(x, L_z) <= B_1(R). (5.8)
The analogous assertion holds for w. The same bounds hold if the loops
must avoid any additional analytic subsets.
Proof. If the ball meets at most one hyperplane of the relevant
family, its complement for that family has trivial or cyclic
fundamental group generated by the one meridian. One way to see the
latter assertion is to use the convexity of the ball and a transverse
complex coordinate for the hyperplane: its projection has an ellipse as
image, and the centers of the convex fibers give a continuous section.
The complement thus has the homotopy type of a punctured ellipse. Thus
its dihedral image lies in a conjugate of an order-two subgroup. It
has zero translation on the sign lift. Omitting hyperplanes from the
other family does not alter this conclusion, because the representation
factors through the complement of the relevant family.
If two distinct slope hyperplanes meet B(x,R), take a unit negative
homogeneous vector representing x. Its inner products with their fixed
positive normals are bounded in terms of R: for a normal n the
normalized absolute inner product is sinh dist(x,H_n). The two normals
form a basis of the positive normal plane of L_z. Their inner-product
bounds therefore bound the positive projection of x onto that plane,
and hence bound dist(x,L_z). There are only finitely many pairs of
normals, so this bound is uniform.
For the remaining case, suppose toward a contradiction that there are
balls supporting nonzero translation whose centers leave every bounded
neighborhood of L_z. After passing to a subsequence, at least one
fixed coupling hyperplane meets each ball: otherwise the preceding two
cases apply. The centers tend to a boundary point xi, and a
fixed-radius hyperbolic ball converges to the same boundary point as
its center. Thus xi belongs to the closure of that coupling
hyperplane. The closures of the two coupling hyperplanes are disjoint.
At xi the direction coordinate is defined and lies strictly inside the
red disk, since z_1 = +/- c and the boundary estimate is uniform:
|z_2/z_1| <= sqrt(a - c^2)/c
< sqrt(1 - 0.85^2)/0.85 < 0.62 < 0.7.
Eventually the whole ball lies over this red disk and misses the other
coupling hyperplane. Its label system consequently reduces to <r>,
contradicting the assumed nonzero translation. This proves the bound.
Restricting the class of allowed loops does not weaken it. []
Proposition 5.4 (compatibility of kernels). For every R > 0 there are
constants B(R) > R and J_0(R), depending only on D, with the following
properties.
(1) Suppose two finite-model descriptions, with their prescribed
normal matches, agree on all hyperplanes meeting B(x,B(R)).
Identify their sign covers there by a fixed choice of sheets. On
loops in this sign cover that project into B(x,R), their individual
z kernels agree and their individual w kernels agree. More
precisely, for each family the corresponding integer homomorphisms
agree up to sign. The descriptions may include additional
hyperplanes outside the buffered ball.
(2) In the full model, if dist(x,o) > J_0(R), then on such loops
ker(ell_z + ell_w) = ker ell_z intersect ker ell_w. (5.9)
Both statements remain valid after restricting the loops to avoid any
further analytic subsets. The first statement is equivariant under the
symmetry interchanging the families.
Proof. Choose B(R) > R + B_1(R) + 1, enlarging it if needed to include
the corresponding bound for both families. Use the subset of
hyperplanes visible in the buffered ball as an intermediate
description.
If a larger description has an incomplete pencil, so does this subset,
and both corresponding translation labels vanish.
If the larger pencil is complete but the buffered pencil is not, a
nonzero translation on an R-ball would, by Lemma 5.3, put its center
within B_1(R) of the common pencil intersection. Since every slope
hyperplane contains that intersection, all six slopes would then meet
the buffered ball, a contradiction. Thus again both labels vanish on
these loops.
If both pencils are complete, the normal match fixes the direction-line
system, and extra coupling hyperplanes can be omitted by the
square-root trivialization just proved. The only remaining change is
conjugation of a dihedral label, which changes a translation coordinate
at most by sign. This proves the first assertion, including its
equivariance and its compatibility with the chosen sign sheets.
The sets
{ x : dist(x,L_z) <= B_1(R) },
{ x : dist(x,L_w) <= B_1(R) }
have bounded intersection. Indeed, in homogeneous coordinates the two
normal planes together form the positive four-dimensional space.
Bounds on both positive projections bound the distance from o. Choose
J_0(R) beyond that intersection. Outside it, at least one of
ell_z,ell_w vanishes on every loop over B(x,R), by Lemma 5.3. The
displayed equality of kernels follows. Every argument concerns the
same individual loops before and after imposing additional omissions,
so those omissions preserve the conclusions. []
5.4. Graphs defined by projected diameter
We now turn the local covers into graphs to which Lemma 3.1 applies.
Over each connected component of E_T, let Ehat_T be the connected
regular cover defined by
ker ell_z intersect ker ell_w.
For the full arrangement we also use the cover defined by
ker(ell_z + ell_w) and its lifted-s quotient. Restrict all these
covers to the open arrangement complement when defining graph vertices.
For u > 0, vertices are all points of the relevant covering space. An
oriented edge is a parametrized continuous path whose projection to B
has diameter strictly less than u; its reverse is the reverse path.
Multiple edges are retained. In the central case we take the quotient
graph by s_hat. The symmetry acts freely on the open arrangement
complement, so edges out of a quotient vertex correspond bijectively to
edges out of any chosen representative. Denote these graphs by
G_T(u), with G_cen(u) for the central quotient.
The use of diameter has an important consequence. A word of arbitrary
length in loops supported in one fixed bounded region can be one edge.
It is this feature that prevents the unbounded translation labels from
producing large flat regions in the graphs.
Proposition 5.5 (hyperbolicity of the models). There are u > 0 and
delta >= 0, depending only on D, such that every connected component of
every G_T(u) and of G_cen(u) is delta-hyperbolic. The same delta can
be used if an arbitrary locally finite union of proper complex analytic
subsets of B is additionally omitted, with invariant omissions in the
central case. On the vertices retained after such an omission, graph
distances are unchanged.
The proof has three parts. First, paths carrying arbitrary deck
translations inside a fixed bounded region give bounded graph diameter
over every bounded radial region. Next, an arrangement-preserving
collar identifies the remaining vertices with points of a covering of
the sphere complement, together with a radial coordinate; the angular
diameter permitted by a graph edge decays exponentially with that
coordinate. Finally, comparison with the cone graph defined below
proves uniform hyperbolicity. No properness or local finiteness of the
graphs is required.
Lemma 5.6 (bounded core). One can choose u, simultaneously for all the
finite models, so that the vertices projecting into any fixed bounded
radial region have bounded graph diameter within each connected
component.
Proof. The deck group over a component of E_T is a subgroup of Z^2, or
of Z for the sum cover, and is finitely generated. At a basepoint
choose loops representing a finite set of its generators. There are
only finitely many sign sheets; also choose finitely many paths joining
the sheets that belong to the same component. All these projected
paths lie in a fixed bounded region. Choose u greater than its
diameter, for each of the finitely many models. An arbitrary word in
the generator loops then defines a single edge. Thus all deck
translates over a basepoint have uniformly bounded graph distance,
including the finitely many sign choices.
For a fixed radial bound, any point in the ball region can be joined to
a chosen basepoint by a uniformly bounded chain of small balls. Within
each ball, paths can avoid the finite complex hypersurface arrangement:
its complement is path connected by general position. Choose
successive junctions off the arrangement in the overlaps. The
resulting paths have uniformly bounded edge count and lift to the
covering. Any discrepancy in their terminal sheet is corrected using
the preceding generator loops. This gives the asserted diameter bound.
One can take the chain bound uniformly, for example by covering the
compact closure of the radial region and a path to the basepoint by
finitely many small balls with connected overlap graph. No positive
lower bound on distance to the arrangement is needed. []
5.5. The arrangement at the sphere at infinity
Rescale the ball to unit radius and write
x = tanh(r) xi, |xi| = 1.
Let Sigma be its unit sphere. Each affine intersection of the
hyperplanes that reaches Sigma is transverse to Sigma. Indeed its
homogeneous Hermitian subspace is nondegenerate by the arithmetic
normal-space property in the proof of Proposition 4.1. More
explicitly, write an affine intersection as a_0 + V_0 with a_0
perpendicular to V_0. On its homogeneous span, the Hermitian form is
|v|^2 + (|a_0|^2 - 1)|t|^2.
Nondegeneracy excludes |a_0| = 1, precisely the tangency case. If the
intersection reaches the sphere, then |a_0| < 1, and the intersection is
transverse there. This also excludes intersections supported only on
the boundary.
There is an arrangement-preserving product collar near Sigma. Here is
a direct construction, including the estimate needed for the graph
metric. At each xi in Sigma, choose a constant real vector tangent to
all affine hyperplanes through xi and with positive outward radial
component. Transversality of their intersection supplies this vector.
Use a small neighborhood missing every other hyperplane. A partition
of unity combines these local vectors into a smooth vector field
tangent to each hyperplane and transverse to the spheres. Normalize it
so that the derivative of the Euclidean radius is one. Its flow
yields, for r >= r_0 with r_0 sufficiently large, diffeomorphisms
between the ideal arrangement complement and the complement on the
radius-r sphere. They preserve all arrangement strata. Because the
vector field is bounded on a compact collar, their angular displacement
from the original ideal point is
O(1 - tanh r) = O(e^{-2r}) in Euclidean distance. (5.10)
In the central model, average the field under sigma before normalizing.
This retains tangency and positive radial component and makes the
collar equivariant. The collar lifts to each covering; the lifted end
is a radial product with a possibly disconnected covering I of the
ideal complement.
On Sigma use the metric
d_K(xi,eta) = |1 - <xi,eta>|^{1/2}. (5.11)
For completeness, it satisfies the triangle inequality. If
a = d_K(xi,zeta) and b = d_K(zeta,eta), then |xi-zeta| <= sqrt(2) a
and |zeta-eta| <= sqrt(2) b. Expanding the Hermitian inner product
around zeta gives
|1 - <xi,eta>|
<= a^2 + b^2 + |xi-zeta| |eta-zeta|
<= (a+b)^2.
Positivity and symmetry are immediate. In particular, (5.10) is an
O(e^{-r}) displacement in d_K.
The ball distance in our normalization satisfies
cosh d(x,y)
= |1 - <x,y>| / ( sqrt(1-|x|^2) sqrt(1-|y|^2) ). (5.12)
Consequently, for points of radii r+O(1), bounded hyperbolic distance
is equivalent, with uniform constants, to angular d_K distance O(e^{-r}).
To check both implications, write x = rho xi, y = tau eta, where
rho = tanh r and tau = tanh r'. Then
1 - <x,y>
= 1 - rho tau + rho tau (1 - <xi,eta>).
The second parenthesis has nonnegative real part. Thus the absolute
value is at least rho tau d_K(xi,eta)^2 and at most
1 - rho tau + d_K(xi,eta)^2. For |r-r'| bounded and both radii large,
the denominator in (5.12) and 1 - rho tau are comparable to e^{-2r}.
These inequalities prove the assertion. Conversely a bounded ball
distance bounds |r-r'| by the triangle inequality for radial distance,
so no separate radial hypothesis is needed in that direction.
On the lifted ideal complement I, define
d_I(xi',eta')
= min { 1, inf_alpha diam_{d_K}(pi alpha) }, (5.13)
where alpha runs over paths in I joining xi' to eta' and pi is
projection to Sigma. Put d_I = 1 for points in different components.
The triangle inequality follows by concatenation, because the
projected paths meet and the diameter of their union is at most the sum
of their diameters. Distinct projected endpoints give positivity
directly. For distinct lifts of the same endpoint, take a small evenly
covered neighborhood in the ideal complement. A path changing the lift
must leave this neighborhood, so its projected diameter has a positive
lower bound. Therefore d_I is a metric, bounded by one.
5.6. A cone over an arbitrary bounded metric space
We isolate the metric calculation, a discrete form of the familiar
hyperbolic-cone construction; compare Bonk-Schramm, Embeddings of
Gromov hyperbolic spaces, Section 7. We give the proof for arbitrary
bounded metric spaces, since no compactness hypothesis is available.
For a metric space (I,d_I) with d_I <= 1, let C(I) have vertices
(xi',t), with xi' in I and t in Z_{>=0}. Add vertical unit edges
between successive heights at the same point, and horizontal unit edges
at height t whenever d_I(xi',eta') <= e^{-t}. In particular all
vertices at height zero are pairwise adjacent.
Lemma 5.7 (cone). The graph C(I) is hyperbolic with an absolute
constant, independent of the bounded metric space I. More precisely,
put
j = min { t, t', -log d_I(xi',eta') },
where -log 0 = +infinity. Its distance D satisfies
t + t' - 2j <= D((xi',t),(eta',t')) <= t + t' - 2j + 3. (5.14)
Proof. If xi' = eta', both the formula without its additive error and
the claim are immediate from vertical distance. Otherwise, descend to
height floor(j), take one horizontal edge, and ascend. This gives the
upper bound.
For the lower bound, let any connecting path have minimum height m and
n >= 1 horizontal edges. The number of its vertical edges is at least
t + t' - 2m. By the triangle inequality in I,
d_I(xi',eta') <= n e^{-m}.
Since also m <= t,t', this implies m <= j + log n. The path length is
therefore at least
t + t' - 2j + n - 2 log n >= t + t' - 2j.
The last inequality holds for every integer n >= 1 (indeed its minimum
over positive real n is 2 - 2 log 2 > 0).
To deduce hyperbolicity, fix a height-zero basepoint p. Its distance
to (xi',t) differs from t by at most one. It follows from (5.14) that
the Gromov product of two vertices differs by at most 3/2 from their
corresponding j value. For three base points, the metric triangle
inequality implies
-log d_I(xi',eta')
>= min { -log d_I(xi',zeta'), -log d_I(zeta',eta') } - log 2.
Taking the minimum with the relevant heights preserves this inequality.
Thus the Gromov products satisfy the hyperbolicity inequality with an
absolute additive constant. The standard Gromov-product
characterization of hyperbolicity for graphs completes the proof. []
5.7. Comparison with the model graphs
We can now complete the proof of Proposition 5.5. Work first before
the finite symmetry quotient, in a connected component. Using the
collar coordinates, send a cone vertex (xi',t) to the covering point of
radius r_0 + t above its ideal coordinate. A horizontal edge at
positive height is represented by a path whose projected ideal diameter
is at most 2e^{-t}: the factor two allows the infimum in (5.13) to be
unattained. Place that path at radius r_0 + t with the collar. The
angular estimates and (5.12) give a uniform bound on its projected ball
diameter. Vertical edges likewise have uniformly bounded projected
diameter. Enlarge u to exceed these bounds strictly. At height zero,
even when ideal components are different, Lemma 5.6 gives a uniform
bound on the graph distance of the two images. The cone map therefore
sends adjacent vertices to pairs at uniformly bounded graph distance.
In the opposite direction, round a point of radius at least r_0 to the
nearest integer layer, retaining its lifted ideal coordinate. Map the
bounded radial part to one fixed height-zero vertex. The two maps are
coarse inverses. Radial rounding has uniformly bounded graph
displacement, and the bounded-core lemma handles all other points. To
verify that this inverse also sends edges to pairs of bounded distance,
consider an edge whose projection lies sufficiently far out. If one
endpoint has radius r, its whole projected path has radii in
[r-u,r+u]. Formula (5.12) bounds the angular diameter of this path by
O_u(e^{-r}). The collar changes this by another term of the same
order. Following the lifted path in the collar consequently gives
d_I(xi',eta') <= C_u e^{-r}.
Its rounded heights differ by a bounded amount, so (5.14) bounds their
cone distance uniformly. If an edge meets the bounded radial part,
both endpoints have bounded radius and the same conclusion follows
through height zero. Hence the two graphs are quasi-isometric.
Hyperbolicity of the cone and quasi-isometry invariance for geodesic
spaces (Bridson-Haefliger, Chapter III.H) prove hyperbolicity of the
model graph.
For the central model, the collar and the comparison maps on the end
are equivariant under s_hat. The lifted action preserves d_I, since its
projection is unitary and hence preserves d_K. For the finite group
H = <s_hat>, the formula
d_Ibar(H xi', H eta') = min_{h in H} d_I(xi', h eta')
defines a metric on I/H. Positivity uses finiteness of H, and the
triangle inequality follows by translating and concatenating
minimizers. The quotient of the cone graph is the cone graph of this
quotient metric, possibly with multiple edges and loops, which do not
affect vertex distances. The preceding comparison thus descends,
while the quotient radial core remains bounded. Lemma 5.7 proves
hyperbolicity also in the central case. There are only finitely many
subsets T and one central model, so one choice of u and one
hyperbolicity constant work for all of them.
Finally, let an additional locally finite union of proper complex
analytic subsets be omitted. Removing vertices and restricting edges
cannot decrease distances. For the reverse inequality, take a finite
edge path whose endpoints are retained. Perturb its finitely many
junctions and constituent paths, relative to the endpoints, to miss the
omitted loci. General position permits this because proper complex
analytic subsets have real codimension at least two. Locally finite
omissions cause no extra difficulty: a compact path meets only finitely
many members, and the perturbation can be made in finitely many
coordinate neighborhoods. Make this perturbation as a homotopy of the
whole concatenated chain in the existing arrangement complement,
relative to its two overall endpoints. Homotopy lifting preserves
those two endpoints in the cover; the intermediate junctions move
coherently. Each original projected path has diameter strictly smaller
than u, so a sufficiently small perturbation preserves that strict
inequality. The perturbed path has exactly the same number of edges.
Thus all distances between retained vertices are unchanged. For a
central quotient edge path, lift the whole chain to the cover, perturb
it relative to its overall endpoints, and project back; invariance of
the omitted set ensures that the projected chain avoids it. The
four-point inequality for hyperbolicity restricts to the retained
vertex set with the same constant, proving the last assertion of the
proposition.
We now fix such a value of u, a common hyperbolicity constant, and a
reading radius N supplied by Lemma 3.1. Only after these choices will
we choose the arithmetic level and the geometric sizes of the charts.
6. AN INFINITE CYCLIC IMAGE
We now compute the image of the surface constructed in Proposition
4.3. The relations supplied by a coupling hyperplane show that this
image is cyclic. Proving that the cyclic group is infinite requires a
separate argument: the translation labels of Section 5 exist in charts,
and need not define a homomorphism on the fundamental group of the
whole orbifold.
Throughout the proof, fundamental groups are based at points in the
ordinary locus. Changes of basepoint are made along specified common
paths; in particular, the comparisons below concern homomorphisms with
one common conjugation, rather than separate conjugations of individual
generators.
6.1. Surface loops and the cyclic upper bound
Recall the distinguished fiber F_* = C x C in V, the order-four lift s,
and the rational section used to construct Y in Section 4.7. Put
K = [V/<s>]. (6.1)
There is a natural homomorphism
pi_1(K) -> pi_1(S). (6.2)
Indeed, the substack [F_*/<s>] has complex codimension two in the
smooth stack K. Removing it does not change the orbifold fundamental
group, and its complement is the open substack on which the
construction of S leaves K unchanged.
Choose a general point q_0 in C. The map
C -> Sym^2 C -> Jac(C), p -> p + q_0,
is, up to translation, an Abel map. Its map on fundamental groups
surjects onto pi_1(Jac(C)) = H_1(C;Z). Since point blowups preserve
fundamental groups, loops from C x {q_0} supply generators of pi_1(Y).
They may be represented by a finite bouquet avoiding the branch points,
p = q_0, the zeros and poles of the rational section, and the finitely
many points where the rational map or its resolution must be avoided.
Removing finitely many points from a smooth curve is surjective on
fundamental groups, so these restrictions do not lose any generators.
Lemma 6.1 (surface comparison). With common choices of basing, the
images in pi_1(S) of the preceding generators of pi_1(Y) equal the
images, under (6.2), of the corresponding loops in C x {q_0} inside V.
The loops can also be represented in the sign-covered arrangement
complement, arbitrarily close to F_*, with both normal coordinates
nonzero.
Proof. Lift the bouquet by the rational section to the exceptional
divisor of Bl_{F_*} V. Over this bouquet the section has the form
[ tau(p) : i tau(q_0) ],
with both entries nonzero. It avoids the fixed curves involved in the
last resolution step, because the bouquet avoids the diagonal of C x C.
A complex line bundle on a finite graph is topologically trivial. Thus
the normal line to this exceptional divisor admits one continuous
nonvanishing section over the entire lifted bouquet. In the product
normal coordinates, this push can be written explicitly as
epsilon ( tau(p), i tau(q_0) ).
One sufficiently small epsilon > 0 works over the compact bouquet and
gives a closed based bouquet off the exceptional divisor. On blowing
down to V, the pushed bouquet is homotopic to the original one in F_*.
On passing to the resolution neighborhood in S, it is homotopic to the
bouquet obtained from Y. These homotopies use the same push of the
common basepoint.
The two components of the displayed section are nonzero, and the
chosen directions avoid the marked slopes. Taking the push small
therefore puts its projection off the arrangement. The distinguished
point is an isolated component of the fixed locus of every nonidentity
power of sigma in a sufficiently small neighborhood. Hence the pushed
bouquet can also avoid the inverse images of all those fixed loci. This
proves all the assertions, including the common basing. []
We next compute an upper bound before taking the quotient by s. Write
H = (Z/2)^2
for the sign deck group. The map
V -> [V/H]
is an orbifold covering, so it induces an injection of pi_1(V) into
pi_1([V/H]). This remains true over branch points: the target is the
quotient stack, which retains their isotropy groups.
The pencil through L_z gives the orbifold line [C/(Z/2)], with six
order-two points. Number the four red points first. Its fundamental
group has the presentation
Q = < h_1,...,h_6 | h_i^2 = 1, h_1 h_2 h_3 h_4 h_5 h_6 = 1 >.
(6.3)
The kernel of the parity character sending every h_i to 1 in Z/2 is
pi_1(C). With the w sign point fixed over a general direction, the map
of this orbifold pencil to [V/H] identifies the homomorphism from this
even subgroup with the homomorphism from pi_1(C x {q_0}) to pi_1(V),
followed by the injection just described.
Lemma 6.2 (cyclic upper bound). The image of pi_1(C x {q_0}) in
pi_1(V) is cyclic. Consequently the image of pi_1(Y) in pi_1(S) is
cyclic.
Proof. We show that the four red meridians in (6.3) have the same
image in pi_1([V/H]). First move the w coordinate a little off its
exceptional direction line, keeping it fixed, generic, and small. Push
the red pencil meridians off the z exceptional divisor. They can be
represented with
t = z_1 nonzero, fixed and small, lambda = z_2 / z_1,
and with the red meridians based by paths in a small disk containing
exactly the four red values of lambda.
Slide this calculation in the t variable to the coupling hyperplane
t = c. The disk of red directions can be chosen small enough that this
entire motion, with the chosen small w, remains inside the ball: the
red slopes have modulus less than 0.01, whereas c/sqrt(a) < 0.9. The
motion and the finitely many paths used here lie in a fixed compact
part of the full finite model. We include this compact set among the
regions that must agree with the actual arrangement when the arithmetic
level is chosen.
Before blowing up the pair intersections, the relevant local divisor is
{ t = c } union ( union_{i=1..4} { lambda = lambda_i } ).
Let k be the meridian in the t variable, with one common choice of
basing. The product structure makes k commute with each red meridian.
More precisely, the same t circle can be transported along each of the
based paths in the direction disk; this gives one element k, not four
unrelated conjugates.
Blowing up { t = c, lambda = lambda_i } introduces an exceptional
divisor whose meridian is k h_i: in the two transverse coordinates, the
exceptional meridian circles both coordinates once. The filling rules
of Section 4.5 impose no branching along this exceptional divisor, since
two branch components met there. Thus its meridian is killed.
Therefore
k h_i = 1 for 1 <= i <= 4.
The strict transforms retain order two, so k^2 = h_i^2 = 1. All four
red images are consequently one involution, denoted r.
The product relation in (6.3) now gives h_5 h_6 = 1, and the two blue
images are one involution b. The image of Q is thus a quotient of
< r, b | r^2 = b^2 = 1 >.
Every even word in r,b is a power of rb. Hence the image of the even
subgroup pi_1(C) is cyclic. Injectivity of pi_1(V) -> pi_1([V/H])
gives the assertion in pi_1(V). Lemma 6.1 and the surjectivity from the
chosen C loops onto pi_1(Y) give the final assertion. []
This argument allows the cyclic image to be finite. We will exclude
that possibility by applying Lemma 3.1 to paths in a dense open subset
of S.
6.2. Paths in the ordinary open set
Let O be the sign-covered arrangement complement, quotiented by s,
after also removing the full inverse image of the fixed loci in M of
all nonidentity powers of sigma. The modifications defining S are
isomorphisms over this set. In particular O is a dense connected
smooth variety, and all quotient actions used over it are free.
There is a useful space for specifying lifts. Over the complement in
B of the lifted arrangement and the lifted omitted fixed loci, pull
back the sign data from V, and call the resulting space O^#. Then
O^# -> O
is an ordinary covering. Changes of lift are generated by the ball
deck group Gamma and by sigma, the latter acting on the sign data
through s. These are precisely the transformations used to form O; we
do not divide out the two sign deck transformations. The group
generated by Gamma and sigma acts freely on the retained ball region.
Fix the diameter bound u supplied by Proposition 5.5. Define a graph G
whose vertices are the points of O. An edge is a parametrized
continuous path in O whose lifted projection to B has diameter strictly
less than u. All graphs use the same parametrization and reversal
conventions. The diameter condition is independent of the lift,
because changes of lift act by ball isometries. No bound is imposed on
path length or winding. In particular, arbitrarily high powers of a
loop can be individual edges when their projections stay in one small
region.
6.3. Choosing and comparing the charts
We make the order of the parameters explicit. First fix u, then a
common hyperbolicity constant for the model graphs, and then the
integer N required by Lemma 3.1. Set
R = (2N + 100)(u + 1). (6.4)
A lift of a graph path with at most N edges stays within distance Nu of
its initial ball point, regardless of the lengths of the individual
parametrized paths.
The special points of B are the points of Gamma o, the lifts of the
distinguished point of M. Choose J > 10R so large that, in the central
model, both of the following hold at every point of distance at least
J - 3R from o:
(i) On sign loops projected into the R ball about that point, the
kernel of the sum translation equals the intersection of the two
individual translation kernels.
(ii) For j = 1,2,3, the displacement by sigma^j exceeds 10R.
The first requirement follows from Proposition 5.4. For the second,
none of sigma, sigma^2, sigma^3 has a boundary fixed point. The ball
distance formula then implies that its displacement tends uniformly to
infinity as the viewpoint tends to the boundary.
Let B(R) be the buffer radius in Proposition 5.4, enlarged so that
B(R) > R, and choose
K > 10 ( J + R + B(R) ). (6.5)
All these constants depend only on the finite models. Now take the
initial arithmetic level deep enough that Proposition 4.1 holds on
balls of radius 10K, that such balls inject into the torsion-free ball
quotient, that the full finite arrangement agrees with the lifted
arrangement in these neighborhoods of every special point, and that
distinct special points have distance greater than 100K. Enlarge this
last choice, if necessary, to include the fixed compact paths used in
Lemmas 6.1 and 6.2. Subsequent finite covers preserve all these
requirements.
For a vertex v in G, choose a lift with ball coordinate x and its actual
sign data. Its chart has one of two forms.
Central chart. If dist(x,Gamma o) < J, use the full finite model
centered at the unique nearby special point. Read in the
sum-translation cover and then quotient by the lifted order-four
symmetry, as in Lemma 5.2.
Individual chart. Otherwise take all lifted hyperplanes meeting
B(x,K), identify them, with their prescribed normal vectors, with
the corresponding subset of D, and use the model cover given by the
two individual translation kernels. There is no symmetry quotient
in this chart.
These chart graphs are the graphs of Proposition 5.5, with the
appropriate additional analytic omissions. One can transport the whole
locally finite union of lifted omitted fixed loci to model coordinates;
any extra lifted arrangement hyperplanes outside the comparison region
may be omitted as well. Such omissions are globally defined on the
model ball. In central coordinates they are invariant under the
symmetry. Proposition 5.5 shows that these omissions do not enlarge
the uniform hyperbolicity constant.
For central charts, fix the identifications of sign data once on the
large comparison ball about o. Sign characters on that ball complement
are determined by their meridians. Choose the sheet identifications to
agree with the local construction near F_*; they then intertwine the
actual lift s with the model lift throughout this connected region.
Transport the choices to all special points by Gamma. These choices
are also compatible with sigma, by the symmetry of the construction.
Within the comparison ball about a special point, the only quotient
transformations that can identify two ball points are the four symmetry
powers centered there. Indeed, if g in <Gamma,sigma> identifies two
such points, it carries the special point to another one at distance at
most twice the comparison radius. Separation forces it to fix that
special point. Its stabilizer is exactly the corresponding conjugate
of <sigma>, since Gamma is torsion free.
To read a path of at most N graph edges from v, lift it from the chosen
point of O^#. Its ball projection remains in the comparison region.
Identify the resulting sign path with the model sign path, lift further
from a chosen initial point in the translation cover, and, in a central
chart, take the class modulo the lifted symmetry. This is a path in a
model graph D_v, with initial state d_v. Write [p]_v for its terminal
state.
Central models are centered only at points of Gamma o. Other full
pencil fibers, including those over Gamma_0 o \ Gamma o, and the other
quotient fixed loci remain unmodified in [V/<s>]. An ordinary-open loop
around such a locus may have a ball or sign lift with different
endpoints; an individual chart retains this difference. Equality of
actual endpoints is not required to imply equality of read states. For
orbifold relations, the relevant meridian power lifts to a closed loop
in a smooth chart in V; its closure will be checked separately below.
Lemma 6.3 (chart axioms). The readings just constructed satisfy the
three chart hypotheses of Lemma 3.1.
Proof. The uniform hyperbolicity assertion is Proposition 5.5. We
verify the two path compatibility assertions.
Continuation from a state. Path lifting is consistent with prefixes and
reversals. Equal read states project to the same actual point of O. In
an individual chart, equality means equality in the model covering, and
hence also in the ball and sign data. Every outgoing model edge at a
state reached after fewer than N edges gives an actual edge of G: its
projection stays within u of its starting ball point and therefore
inside the comparison region. It avoids precisely the arrangement and
extra loci excluded in O. Lifting and projecting parametrized paths
are mutually inverse once the starting lift is fixed. This is the
required bijection on outgoing edges, and it depends only on the read
state.
In a central chart the symmetry acts freely on the open complement. An
outgoing edge of its quotient graph consequently has a unique
representative from any chosen representative of the starting vertex.
Representatives at equal states differ by a symmetry power, which
respects the identification with the actual open set. Thus the same
outgoing-edge bijection holds, including its dependence only on the
state. This proves the first chart hypothesis.
Comparison after a prefix. Let p go from v to a vertex v', and let a,b
start at v', with
|p|, |a|, |b| <= N/2.
We must prove
[pa]_v = [pb]_v if and only if [a]_v' = [b]_v'. (6.6)
For this calculation, use for v' the lift reached by lifting p. The
equality tests do not depend on this change from its initially chosen
lift. For Gamma this follows from transport of the normal data. For
sigma the two families are interchanged and the finite arrangement has
the same symmetry. If relabeling changes a chosen normal vector by a
scalar, it changes the corresponding translated normal by the same
scalar, because sigma normalizes the arithmetic group. Thus the
intrinsic prescriptions of Proposition 5.4 give the same individual
kernels, with possible reversals of translation orientations.
Individual sign-sheet choices likewise do not affect a kernel test on a
closed sign loop. For central charts the fixed, equivariant
identifications above give the same conclusion for the sum cover.
Finally, the choice of initial point in a translation cover is
irrelevant because the covers are regular, and the deck translations
commute with the lifted central symmetry.
All projected paths in the comparison are contained in the R ball about
the endpoint of the lifted p. If both charts are individual, equality
first requires coincident endpoints in the actual ball and sign data.
If this requirement fails, both tests fail. If it holds, compare the
two sign paths by the loop a (bar b). The further lifts end together
exactly when both translations of this loop vanish. The common prefix
does not change that condition. Both charts contain the required
buffered ball about this new viewpoint, by (6.5); therefore Proposition
5.4 identifies the two kernel tests. This proves (6.6) in this case.
If both charts are central, their special points coincide. Otherwise
these two special points would be at distance at most 2J + Nu,
contrary to their separation. The two readings then use restrictions
of the same covering with the same lifted symmetry, so their endpoint
equality tests agree.
It remains to compare a central and an individual chart. The viewpoint
at the end of p is at distance at least J - R from the special point of
the central chart. This follows directly from the criterion for the
individual root and from the bound Nu < R on the distance between the
two roots. Every endpoint being compared is therefore at distance at
least J - 2R from that special point. Two such endpoints in the
comparison ball cannot differ by a nontrivial symmetry power: their
mutual distance is at most 2R, whereas the relevant displacement
exceeds 10R. Thus the central test, too, requires equality of the ball
and sign endpoints. On the remaining sign loop, the sum kernel equals
the intersection of the individual kernels by the choice of J. The
buffered comparison of individual kernels now applies as before. This
proves (6.6) in the mixed case, and completes the verification of the
chart hypotheses. []
6.4. From orbifold relations to path relations
Let P(G) be the path groupoid of G modulo backtracking and every edge
loop of at most three edges that closes in its starting chart.
Lemma 3.1 and Lemma 6.3 detect nontrivial elements of this groupoid. To
use that conclusion for S, we need the following direction of
comparison: every loop that is trivial in pi_1(S) must already be
trivial in P(G).
We recall explicitly the topological fact about orbifolds used in this
comparison.
Lemma 6.4 (orbifold meridians). Let T be a connected smooth effective
complex orbifold, and let U be a dense ordinary open subset whose
complement is complex analytic. Then pi_1(U) -> pi_1(T) is surjective.
Its kernel is normally generated by the loops obtained from boundaries
of small transverse disks in smooth orbifold charts at general points
of the omitted divisors. Equivalently, if the generic stabilizer along
such a divisor has order m, one kills the m-th power of a small meridian
in the ordinary quotient. Omitted strata of complex codimension at
least two add no relations.
Proof. One may apply ordinary general position to a smooth frame
presentation of T. Effectiveness and finite-group linearization imply
that the action of each chart group on tangent frames is free.
Consequently the frame space is a manifold, with a locally free action
of the connected group GL_n(C), and its quotient stack is T. The
homotopy fibration from this presentation, as in Noohi, Fibrations of
topological stacks, Example 5.6, computes pi_1(T) as the fundamental
group of the frame space modulo the image of pi_1(GL_n(C)); the same
description applies over U.
In the frame space, paths can avoid real-codimension-two loci, and a
nullhomotopy disk can meet such loci transversely in finitely many
general divisor points. Removing small disks around those intersection
points expresses its boundary as a product of conjugates of transverse
meridians. Real codimension at least four can be avoided by the entire
disk. Passing through the preceding quotient of fundamental groups
proves the assertion. At a general point of a divisor, the effective
stabilizer is cyclic and acts faithfully on the transverse line. A
circle in that line projects to the m-th power of a quotient meridian,
giving the last description. []
Lemma 6.5 (true relations). A loop of edges of G that is nullhomotopic
in S is trivial in P(G).
Proof. First consider homotopies within O. Every ordinary path admits
a finite subdivision into graph edges. An already specified edge may
itself be subdivided without changing its class in P(G): if e = e_1 e_2
is a subdivision, then e_1,e_2 are still edges, and e_1 e_2 e^{-1} is a
three-edge loop that closes on reading, since all three pieces lift the
same parametrized path. Repeating this observation allows arbitrary
finite subdivision even when an original edge winds arbitrarily often.
A homotopy in O can now be subdivided into sufficiently small path
triangles. To choose the neighborhoods uniformly over its compact
image, take lifted ball patches avoiding the arrangement and the omitted
fixed loci, each disjoint from its distinct quotient translates. For
every root in a smaller patch, the oversized chart agrees with this same
arrangement-free ball patch. The sign and translation covers trivialize
there, and a central symmetry quotient makes no additional
identifications inside the patch. A finite cover of the homotopy image
by these smaller neighborhoods supplies the required subdivision. Each
triangle consequently closes in its starting chart and is one of the
imposed relations. This is the usual edge-path proof that ordinary
homotopy relations are respected.
It remains, by Lemma 6.4, to check the transverse-disk relations. Away
from the distinguished fiber, S is [V/<s>]. Even at points with
inertia, the required power of a quotient meridian lifts to a closed
small loop bounding a disk in the smooth local uniformizing space in V.
Its projection to M bounds a disk in an arbitrarily small simply
connected ball patch, so its further lift to B is closed and lies in the
corresponding small region. The finite model there has exactly the same
participating hyperplanes and the same blowups, sign branching, and
fillings as V. This local identification extends across the fillings by
the explicit local rules, or equivalently by uniqueness of the
normalizations extending the sign covers. Lemma 5.1 therefore says
that the translation covers extend over this disk. Its boundary closes
in an individual chart, and also closes in a central chart by the sum
rule. This argument includes unbranched exceptional divisors,
order-two branch divisors, and the powers required by any further
quotient inertia. Additional analytic loci removed in defining O cause
no new obstruction: the extension test is made after those additional
omissions are restored.
Over the distinguished point, the relevant chart is central. By
Lemma 5.2, its sum cover descends to the coarse quotient and pulls back
to a covering of the resolution neighborhood. Thus it extends over
every transverse disk used for the ordinary smooth space there. Those
meridians close in the central graph as well. A sufficiently small
representative is a single edge: its ball lift stays arbitrarily close
to the distinguished point, even when its endpoint differs from its
start by a central symmetry power. Away from this fiber the same
diameter assertion follows from the small local chart in V, including
for a power of a meridian.
Every required transverse-disk boundary is consequently a single-edge
loop that closes in its chart, and is killed in P(G). Normal
generation, together with the homotopy argument in O, proves the
lemma. []
Combining the last two lemmas with Lemma 3.1 gives
a single edge loop with distinct read endpoints represents a
nonidentity element of pi_1(S). (6.7)
This conclusion does not require any of the translation labels to
extend to all of pi_1(S).
6.5. Infinite order and completion of the proof
Choose a based loop alpha in C x {q_0} on which the first translation
label is nonzero. Such a loop comes from lifting a product of a red and
a blue meridian in the six-point pencil: its parity is even and its
dihedral label is rb, a nontrivial translation. The second label is
zero on this factor. The loop can be represented in the good locus used
in Lemma 6.1, since deleting its finite exceptional set is surjective on
pi_1(C) and the translation label extends over the filled curve.
Push alpha to a loop e in O as in that lemma, so close to the
distinguished fiber that its ball projection has diameter less than u.
Its basepoint has a central chart. The sum label on the pushed loop is
the same nonzero integer as on alpha, by extension of the sum covering
over the filled model. For every nonzero integer n, the parametrized
loop e^n has the same projected image as e, and hence is itself one edge
of G.
The read endpoints of e^n are distinct in the central quotient graph.
Before taking the symmetry quotient they differ by the nonzero
translation n ell(alpha). The lift of e is closed in the actual ball
and sign data, because the push is closed upstairs in V and its
projection lies inside the prescribed small simply connected ball about
o. If the two translated endpoints became equal after quotienting by a
lifted symmetry power, that power would fix their common ball point.
This point lies in the ordinary open set where the symmetry acts
freely, so the power would have to be the identity. A nonzero deck
translation cannot fix a point of a covering. Thus the endpoints
remain distinct.
By (6.7), every nonzero power of the class of e is nontrivial in
pi_1(S). Lemma 6.1 puts this infinite-order element in the image of
pi_1(Y). That image is cyclic by Lemma 6.2; it is therefore infinite
cyclic. This proves the following theorem.
Theorem 6.6 (orbifold cyclic image). For a sufficiently deep choice of
the arithmetic level in the construction of Proposition 4.3, the map
f : Y -> S satisfies
im( f_* : pi_1(Y) -> pi_1(S) ) = Z.
Here Y is an iterated point blowup of the simple abelian surface
Jac(C), and its image lies in the ordinary locus of S.
This completes the group-theoretic part of the construction. The next
section replaces the orbifold S by an ordinary smooth projective variety
and removes the point blowups from Y, preserving the homomorphism on
fundamental groups.
7. PROJECTIVE REALIZATION AND REMOVAL OF THE SURFACE BLOWUPS
Theorem 6.6 supplies a map Y -> S, where Y is obtained by point
blowups from the simple abelian surface A = Jac(C), and where the
image of pi_1(Y) is infinite cyclic. We now make two changes of
ambient space. The first replaces the orbifold S by a smooth
projective variety containing Y. The second removes the point blowups
from Y while preserving the homomorphism on fundamental groups.
Together they produce the embedding required by Proposition 2.3.
The first change is a relative version of the free-locus method in the
Godeaux-Serre construction; compare Serre, Sur la topologie des
varietes algebriques en caracteristique p, and Totaro, The
Lefschetz hyperplane theorem for stacks. The second change is the
classical local two-ruling construction for surface blowdowns;
compare Atiyah, On analytic surfaces with double points. The essential
global point is to construct a polarization that makes the second
contraction projective.
7.1. Replacing the orbifold ambient space
Lemma 7.1 (ambient realization). Let T be a smooth connected
effective proper complex Deligne-Mumford stack with projective coarse
space. Let Y be a smooth projective surface and let f : Y -> T have
image in the locus with trivial stabilizers. There exist a smooth
connected projective variety S, an embedding i : Y -> S, and a
morphism h : S -> T such that h o i = f and
h_* : pi_1(S) -> pi_1(T)
is an isomorphism. One may take dim S = 5.
Proof. Choose an embedding Y -> P^{b-1}. Form
E = ( O_T + T_T )^{direct sum a} + O_T^{direct sum b},
P = P_T(E),
with the convention that projectivization parametrizes lines. The
trivial summands give a closed substack
T x P^{b-1} inside P,
in which the graph of f is a closed copy of Y. This copy lies in the
ordinary locus, because f(Y) does.
The representation of each stabilizer on the tangent space of T is
faithful; in particular, every nonidentity element acts nontrivially.
Indeed a finite holomorphic action is linearizable at a fixed point,
and effectiveness rules out an element with trivial tangent action.
The added trivial line ensures that its action on O + T_T is not
scalar. Consequently every eigenspace in E has codimension at least
a, including after the extra trivial summands are added. The
projective fixed locus of each such element is a union of projective
eigenspaces. It follows, in local finite quotient charts, that the
nonordinary locus of P has codimension at least a. The same bound
holds in its coarse space P.
The coarse space P is projective. Stabilizer orders are bounded on
the proper stack, so a sufficiently divisible power of the tautological
polarization has trivial stabilizer actions and descends to P; it is
relatively ample over the coarse space of T. These statements can be
checked on finite quotient charts, where descent is precisely
triviality of the stabilizer action on the fibers of the line bundle.
Twisting by a sufficiently ample bundle from the projective base gives
a polarization of P.
Let U subset P be the ordinary locus. It is smooth and identifies
with the ordinary locus of P.
Choose a > 5 and b,a large enough that dim P >= 10. Removing a subset
of complex codimension at least two from a smooth orbifold does not
change its orbifold fundamental group: paths and their homotopies can
be moved off that subset in smooth charts, or in a smooth frame
presentation. Also, the projective-space bundle P -> T induces an
isomorphism on fundamental groups, since its fibers are simply
connected. Thus the natural maps give
pi_1(U) = pi_1(P) = pi_1(T). (7.1)
Since the nonordinary locus of P has codimension at least a > 5, a
general complete intersection S in P of 5 sufficiently ample divisors
vanishing along the copy of Y is a smooth connected variety of
dimension 5, contains Y, and avoids P \ U entirely. Bertini gives
smoothness away from Y, and a diagonal count of normal derivatives, as
in Lemma 2.2, gives smoothness along Y. The Lefschetz hyperplane
theorem, applied successively, gives
pi_1(S) -> pi_1(P)
as an isomorphism. Composing with (7.1) proves the lemma. []
7.2. Realizing a surface blowdown in a new ambient variety
Lemma 7.2 (ambient blowdown). Let Y_1 -> S be an embedding of a
smooth projective surface in a smooth connected projective variety,
and let b : Y_1 -> Y_0 be the blowdown of a (-1)-curve to a smooth
projective surface. There exist a smooth connected projective variety
S_0, an embedding Y_0 -> S_0, and an isomorphism
theta : pi_1(S) -> pi_1(S_0)
such that
theta o ( pi_1(Y_1) -> pi_1(S) )
= ( pi_1(Y_0) -> pi_1(S_0) ) o b_*. (7.2)
The equality is understood with compatible basepoints, or up to
conjugation.
Proof. Let E subset Y_1 be the exceptional (-1)-curve.
Step 1: a projective threefold containing Y_1. Choose an embedding
Y_1 -> P^{m-1}. We first cut S down to a projective threefold that
contains Y_1 and whose embedding of the surface has a controlled normal
bundle. This is a standard consequence of Serre vanishing: the ideal
sheaf of Y_1 in S is generated by global sections after a large twist,
and the same is true of its first normal jets. Choose sufficiently
ample line bundles and general sections so that the successive zero
loci are smooth away from Y_1, and so that along Y_1 their normal
derivatives generate. The last cut is allowed to acquire isolated
singularities, which we resolve while leaving Y_1 unchanged. Denote
the resulting smooth projective threefold by W, and keep the notation
Y_1 subset W.
Take one more sufficiently ample hypersurface Z subset W through Y_1.
On the surface, its two normal derivatives form a general section of
the rank-two bundle
N^*_{Y_1/W} (x) O_W(Z)|_{Y_1}.
In sufficiently high degree, generation of its first jets makes the
zeros transverse and isolated; since dim E = 1, they can also avoid E.
Bertini gives smoothness away from Y_1. At a zero choose coordinates
with Y_1 = (x = y = 0). Writing the equation as
xa + yb = 0,
transversality says that a,b restrict to coordinates on Y_1. Hence
x,y,u = a,v = b are coordinates on W and the hypersurface equation is
xu + yv = 0, Y_1 = (x = y = 0). (7.3)
Thus Z is a connected normal threefold with only ordinary double
points, all disjoint from E.
We require the small resolution of these nodes that leaves Y_1
unchanged. It exists projectively as follows. Choose a high-degree
Cartier divisor on Z containing Y_1, general in its ideal at the
finitely many nodes. At each node its equation can be taken to be x,
after a change of the coordinates in (7.3). This divisor is the sum of
Y_1 and a residual effective Weil divisor R. The latter is Cartier
away from the nodes, and at a node its ideal is (x,v). Blow up this
coherent ideal. The blowup is projective, is an isomorphism away from
the nodes, and is smooth at every node. Indeed its two charts are
x = tv, y = -tu and v = sx, u = -sy,
with coordinates (t,u,v) and (s,x,y), respectively. The strict
transform of Y_1 is t = 0 in the first chart and misses the second
chart. It therefore maps isomorphically to the original (u,v)-plane.
Denote the resulting smooth projective threefold by T and keep the
notation Y_1 subset T.
Step 2: adjusting the normal bundle. The hypersurface degree can be
chosen so large that
N_{Y_1/T}|_E = O_E(-m), m >= 1.
In fact there are no nodes on E, and the normal exact sequence there
subtracts deg O_W(Z)|_E from the fixed degree of det N_{Y_1/W}|_E. The
sequence
0 -> O_E(-1) -> N_{E/T} -> O_E(-m) -> 0
splits because H^1(E,O_E(m-1)) = 0. Blow up the threefold along E.
Since E is a Cartier divisor on Y_1, the strict transform of Y_1 is
isomorphic to Y_1. Its normal line bundle becomes
N_{Y_1/T} (x) O_{Y_1}(-E), so its degree on E increases by one,
because E^2 = -1. Repeating m-1 times gives a smooth projective
threefold, still denoted T, in which
N_{E/T} = O_E(-1) + O_E(-1), (7.4)
with the first summand the tangent-normal direction inside Y_1. At
each step the same extension vanishing justifies this splitting.
Step 3: the two rulings and the contraction. Blow up E in T. The
exceptional divisor is the projectivized normal bundle
G = P( N_{E/T} ) = P^1 x P^1.
It carries two rulings. The fibers of E give one ruling; the strict
transforms of the curves tangent to Y_1 at E give the other. The
strict transform Yhat = Bl_E Y_1 is isomorphic to Y_1. It meets G in
the exceptional curve E, which is a section of the first ruling. The
threefold T is smooth, Yhat is smooth, and the two meet transversally
in G. Contracting the other ruling along G gives the surface blowdown
b on Yhat.
Step 4: a polarization for the other contraction. We seek an ample
line bundle H on T whose restriction to G is O_G(alpha,beta) with
beta > alpha > 0. Since O_T(G)|_G = O_G(-1,-1), adding alpha G will
then give the restriction
O_G(0, beta - alpha):
this is trivial on the curves to be contracted and positive on the
retained factor.
Choose an ample integral line bundle H_0 on T with
H_0|_G = O_G(alpha, beta_0),
where alpha,beta_0 > 0. Adding k Yhat increases the second degree by k
and leaves the first unchanged. Choose k >= 0 with
beta := beta_0 + k > alpha. On Yhat, the line bundle H_0 + k Yhat
still has positive degree on E, because N_{Yhat/T}|_E is trivial. It
is therefore relatively ample for b : Yhat -> Y_0, whose only
positive-dimensional fiber is E.
Let Q be the pullback to T of an ample line bundle on the coarse space
of T; it is nef. Adding a sufficiently large multiple lQ makes the
restriction to Yhat ample. We claim that
H = H_0 + k Yhat + lQ
is ample on all of T. Here H_0 + lQ is ample since Q is nef. We use
the following consequence of the Nakai-Moishezon criterion: if A_0 is
ample, D_0 is an effective Cartier divisor, k >= 0, and
(A_0 + kD_0)|_{D_0} is ample, then A_0 + kD_0 is ample. Indeed, for an
irreducible r-dimensional subvariety V not contained in D_0, the
difference of top intersections is
( (A_0 + kD_0)^r - A_0^r ) . V
= k D_0 . V .
sum_{j=0}^{r-1} (A_0+kD_0)^j A_0^{r-1-j} >= 0.
Every term on the right is a mixed intersection of ample restrictions on
the effective cycle D_0 intersect V. For V subset D_0, positivity
follows directly from the restriction hypothesis. Nakai-Moishezon now
applies. Taking D_0 = Yhat proves the claim. We have obtained
H|_G = O_G(alpha,beta), beta > alpha > 0. (7.5)
The relative ampleness, Serre vanishing, and numerical ampleness facts
used here are standard projective criteria; see Hartshorne, Algebraic
Geometry.
Step 5: the contraction. Set
D = H + alpha G.
We prove directly that a high multiple of D contracts G to its second
factor and has smooth projective image. For a sufficiently large
integer n, put
L_j = nD - jG.
Then
L_{n alpha} = nH,
L_j|_G = O_G( j, n(beta-alpha) + j ).
Serre vanishing gives H^1(T,L_{n alpha}) = 0. For 0 <= j < n alpha,
the restriction exact sequences and
H^1(G, O_G(j, n(beta-alpha)+j)) = 0 propagate this vanishing
downwards:
H^1(T,L_j) = 0 for 0 <= j <= n alpha. (7.6)
Choose also n alpha >= 2 and nH very ample. In particular the
restriction maps for L_0 and L_1 are surjective. The former gives all
sections of O_G(0,n(beta-alpha)). Hence |nD| is basepoint free on G
and restricts to the second projection followed by a Veronese
embedding. Away from G, it contains the system obtained by
multiplying sections of nH by the canonical section of n alpha G. This
subsystem embeds the complement of G and separates each of its points
from G. Thus |nD| defines a projective morphism
phi : T -> T'
that is an isomorphism off G and whose nontrivial fibers are exactly
the curves P^1_curve x {t}.
Step 6: smoothness of the contracted space and of the surface. We
check the local model, rather than assume that an analytic contraction
is projective or smooth. Fix a fiber
F_t = P^1_curve x {t}
of G -> P^1_normal. Use a section of nD nonzero on F_t as an affine
denominator. The restriction surjection for L_0 gives a section ratio
z whose restriction to G is a local coordinate on its second factor.
The restriction surjection for L_1 = nD - G gives two section ratios
x,y vanishing on G, whose first normal coefficients on F_t are a basis
of H^0(P^1,O(1)).
If e = 0 is a local equation for G, write x = ea, y = eb. The
functions a,b have no common zero near F_t. Thus [x:y] = [a:b] extends
across G, and the map (z,x,y) lifts to the blowup of C^3 along the axis
x = y = 0. On F_t, its projective coordinate is the standard
isomorphism to the exceptional P^1. In the chart a != 0, the
coordinates (z,x,y/x) have nonsingular Jacobian along the fiber: they
respectively detect its base direction, the direction normal to G, and
the tangent direction of the fiber. The same holds in the other chart.
If injectivity failed on every smaller neighborhood of F_t, colliding
pairs would have subsequences converging to two points of this compact
fiber. Injectivity on the fiber identifies their limits, contradicting
local biholomorphism there; hence the lifted map is an isomorphism of
neighborhoods. Properness of the standard blowup then allows us to
shrink its image to the full inverse image of a neighborhood of the
point on the axis.
All the other affine section ratios defining phi are holomorphic on
this blowup neighborhood and descend to holomorphic functions on the
unblown-up neighborhood. One may use here the elementary identity
pi_* O_{ Bl_axis C^3 } = O_{C^3}:
a function descends off the codimension-two axis and extends across it
by normality. Since the entire fiber of phi over phi(F_t) is F_t,
properness ensures that the resulting neighborhood computes the germ
of its projective image. In these coordinates that image is a graph
over (z,x,y). Consequently T' is smooth, and phi is the ordinary
smooth blowdown of G along its other ruling.
At F_q = Yhat intersect G, transversality in the exceptional quadric
makes projection to the blowup of the (x,y)-plane a local isomorphism
on Yhat along F_q, and it is an isomorphism on that fiber. The same
compactness and shrinking argument gives an isomorphism of neighborhoods
with the surface blowup. Its remaining coordinate z is a holomorphic
function there and descends to the plane, by the same extension
argument. Thus the image of Yhat is a smooth surface, and its
restriction is exactly the point blowdown of E. By uniqueness of that
blowdown, this image identifies with Y_0.
The earlier threefold construction preserved the groups compatibly with
the surface inclusion; the final diagram
Yhat -> T -> T',
Yhat = Y_1 -> Y_0 subset T'
shows that the induced identifications also preserve the homomorphism
from the surface. Taking S_0 = T' gives the required theta. This
proves Lemma 7.2. []
7.3. Completion of the construction
Theorem 7.3 (cyclic embedding). There exist a simple abelian surface A,
a smooth connected projective complex variety S', and a closed embedding
A -> S' such that
im( pi_1(A) -> pi_1(S') ) = Z.
Proof. Theorem 6.6 gives an iterated point blowup Y of the simple
abelian surface A, a smooth connected effective proper orbifold S with
projective coarse space, and a map Y -> S whose image lies in the
ordinary locus and whose fundamental-group image is infinite cyclic.
Lemma 7.1 replaces this map by an embedding Y -> S with the same group
image. Apply Lemma 7.2 to the point blowups in reverse order. Every
step preserves both the ambient fundamental group and the homomorphism
from the surface. After the last step the surface is A, giving the
asserted embedding and cyclic image. []
Proof of Theorem A. Apply Proposition 2.3 to the embedding supplied by
Theorem 7.3. The resulting smooth connected projective fourfold X
contains A with infinite cyclic fundamental-group image, has large
fundamental group, and has a non-Stein universal cover. By Lemma 2.1,
its universal cover contains no positive-dimensional compact
complex-analytic subvariety.
This completes the proof. []
8. OVERVIEW OF THE PROOF
The argument is long, so here is the dependency chain in one place.
(1) Section 4 constructs a smooth proper orbifold S with projective
coarse space, a simple abelian surface A = Jac(C), an iterated
point blowup Y -> A, and a map f : Y -> S whose image lies in the
ordinary locus.
(2) Section 5 attaches translation labels to paths in finite models
of the arrangement. The labels are compatible on sufficiently
buffered balls (Proposition 5.4), and the associated path graphs
are uniformly hyperbolic (Proposition 5.5).
(3) Section 6 proves that the image of pi_1(Y) in pi_1(S) is cyclic
(Lemma 6.2) and that a chosen surface loop has infinite order
(Theorem 6.6). The two together give
im( f_* : pi_1(Y) -> pi_1(S) ) = Z.
(4) Section 7 replaces S by a smooth projective variety (Lemma 7.1)
and removes the point blowups from Y one at a time (Lemma 7.2).
This gives a closed embedding A -> S' with infinite cyclic image
(Theorem 7.3).
(5) Section 2 converts that embedding into the fourfold of Theorem A
by a general complete intersection in S' x B (Proposition 2.3).
Largeness follows from the fibers over B, and the non-Stein
universal cover follows from the rank-three lattice cover of A.
The logical dependencies are
Proposition 4.3 -> Theorem 6.6 -> Lemma 7.1 -> Lemma 7.2
-> Theorem 7.3 -> Proposition 2.3 -> Theorem A.
Each step preserves the relevant homomorphism on fundamental groups,
with compatible basepoints. No step uses an unproved statement about
the global fundamental group; the only global input is the local path
criterion of Section 3, which detects nontrivial loops from local
readings.