==== TABLE OF CONTENTS ====
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//01-introduction.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//02-reduction.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//03-paths.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//04-geometry.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//05-models.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//06-image.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//07-projective.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//arrangement.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//blowdown.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//main.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//path-ladder.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//preamble.tex
A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//references.bib
===========================
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
Let $X$ be a smooth connected projective complex variety, and let
$\widetilde X$ denote its simply connected topological universal cover,
equipped with the lifted complex structure. Shafarevich's
holomorphic-convexity conjecture asks whether $\widetilde X$ is
holomorphically convex. Its particularly direct form for varieties with
large fundamental group asks whether $\widetilde X$ is Stein when it
contains no positive-dimensional compact complex analytic subvariety.
Recall that $X$ has \emph{large fundamental group} if, for every
positive-dimensional closed irreducible subvariety $Z\subset X$, the image
of $\pi_1(Z^\nu)\to\pi_1(X)$ is infinite; here $Z^\nu$ is the
normalization. This is equivalent to the absence of positive-dimensional
compact analytic subvarieties in $\widetilde X$. We prove the following.
\begin{theorem}\label{thm:main}
There exist a smooth connected projective complex fourfold $X$ and a
simple abelian surface $A\subset X$ such that
\begin{enumerate}
\item $X$ has large fundamental group;
\item $\im\bigl(\pi_1(A)\to\pi_1(X)\bigr)\simeq\Z$.
\end{enumerate}
The universal cover $\widetilde X$ contains no positive-dimensional
compact complex analytic subvariety and is not Stein.
\end{theorem}
The obstruction to Steinness is topological and occurs in a closed
complex surface inside $\widetilde X$. Since $\pi_1(A)\simeq\Z^4$,
the kernel of the homomorphism in Theorem~\ref{thm:main} has rank three.
The corresponding cover of $A$ is therefore diffeomorphic to
$\R\times(S^1)^3$ and has nonzero third homology. A Stein surface has
the homotopy type of a CW complex of real dimension at most two.
This contradiction rules out Steinness of the lifted surface and hence
of $\widetilde X$. Simplicity of $A$ ensures that this topological
obstruction introduces no compact complex curve in that covering.
\subsection{Historical context}
Shafarevich's question extends the uniformization viewpoint for curves
to universal covers of projective varieties. Koll\'ar's account
\cite[Introduction, Section~0.3]{Kollar1995} explains both the
holomorphic-convexity conjecture and its formulation for large
fundamental groups. These formulations are related by Remmert
reduction: a holomorphically convex complex manifold admits a proper
holomorphic map to a Stein space with connected compact analytic
fibers \cite[Introduction, p.~1546]{EKPR2012}. If there are no positive-dimensional
compact analytic subvarieties, all those fibers are points and the
reduction is an isomorphism. Theorem~\ref{thm:main} therefore also
disproves the holomorphic-convexity conjecture.
An important development was the construction of almost-holomorphic reduction maps by
Campana and Koll\'ar \cite{Campana1994,Kollar1993,Kollar1995}.
They describe, through a very general point, the normalized subvarieties
whose fundamental groups have finite image in the ambient group.
Their existence provides an algebraic
and geometric description of the part of the variety that such
subvarieties fill. Holomorphic convexity asks in addition for a
proper holomorphic reduction of the universal cover to a Stein space.
The large case makes this additional analytic requirement especially
visible: the expected reduction has no positive-dimensional fibers.
Linearity of the fundamental group supports strong affirmative
results. Katzarkov and Ramachandran proved holomorphic convexity for
projective surfaces whose fundamental group admits a faithful complex
representation with reductive Zariski closure
\cite{KatzarkovRamachandran1998}. Eyssidieux treated the reductive
linear case in arbitrary dimension \cite{Eyssidieux2004}.
Eyssidieux, Katzarkov, Pantev, and
Ramachandran proved the linear Shafarevich conjecture for smooth
projective varieties \cite{EKPR2012}. More recently, Bakker,
Brunebarbe, and Tsimerman developed linear Shafarevich theory for
normal algebraic spaces and quasiprojective varieties
\cite{BBT2024}. The construction here concerns the unrestricted
fundamental group. Its infinite-order argument uses compatible
local covers of an arrangement, so it does not require a faithful
linear representation of that group.
A closely related line of inquiry appears in Bogomolov and Katzarkov's
constructions of projective surfaces and their potential counterexamples
to the Shafarevich conjecture
\cite[Section~4]{BogomolovKatzarkov1998}. Their proposed obstruction
uses infinite chains of compact curves and depends on group-infiniteness
statements left conjectural there.
Here the obstruction is a rank-three lattice cover of a simple abelian
surface. The group-theoretic task is to prove that its cyclic image
survives all global relations; the local path criterion supplies that
infiniteness argument.
Several established tools make the construction possible. Arithmetic
ball quotients supply arbitrarily large regions with exactly prescribed
finite hyperplane configurations. Stover and Toledo's virtual
ramified-cover theorem supplies the required sign covers
\cite{StoverToledo,LlosaPy}, and Zarhin's theorem supplies a
genus-two curve with simple Jacobian \cite{Zarhin}. Standard
hyperbolic geometry \cite{BridsonHaefliger} underlies the local path
criterion. Hamm--L\^e's quasi-projective Lefschetz theorem
\cite{HammLe}, together with a deformation argument, allows us to
retain a prescribed surface while passing
from an orbifold to a smooth projective ambient variety. Each use is
stated with its hypotheses at the relevant point below.
\subsection{The central construction}
The principal task is to embed a simple abelian surface in a smooth
projective variety so that its fundamental group has infinite cyclic
image. Once such an embedding is available, an ample complete
intersection in a product with a sufficiently large abelian variety
gives Theorem~\ref{thm:main}. The complete intersection contains the
prescribed surface, and its other fibers over the added abelian variety
are finite. This separates the two reasons that a subvariety has
infinite fundamental-group image: it either moves in the added abelian
variety or lies in the simple surface. Section~\ref{sec:reduction}
proves this reduction first.
To obtain the cyclic image, we begin with a genus-two curve $C$ whose
Jacobian is simple. Its presentation as a double cover of $\PP^1$
uses four branch points in one small disk and two outside that disk.
We call the corresponding meridians red and blue. A complex hyperbolic
arrangement realizes this marked line as an exceptional fiber. Extra
hyperplanes meeting the red part of the pencil impose relations that
identify all four red meridians. The sphere relation then identifies
the two blue meridians. The image of the genus-two surface group is
therefore cyclic: it lies in the cyclic subgroup generated by the product
of the resulting red and blue involutions.
These relations alone give only an upper bound. The central difficulty
is to prove that the cyclic image is infinite after all global
identifications and fillings. We use covers defined separately on
large local models of the arrangement. We model their monodromies
using the infinite dihedral group generated by two involutions $r,b$;
its even words $(rb)^n$ carry the integer translation label $n$.
These local monodromies need not
define a representation of the full projective fundamental group.
Instead, they agree on the endpoint tests needed to read short paths
in overlapping charts. The graphs of these local covers are uniformly
Gromov hyperbolic. A local path criterion then prevents a loop detected
by one chart from becoming trivial through global relations.
The edges of these graphs are paths with bounded \emph{projected
diameter}, rather than bounded length. Consequently every power of a
fixed loop near the distinguished fiber is a single edge. The local
criterion detects every nonzero power separately and gives the
infinite-order lower bound. This feature is essential to the argument;
neither a global coloring nor a family of finite quotients is needed
for that lower bound.
Two copies of the pencil give a distinguished fiber $C\times C$.
An order-four symmetry exchanges the factors and rotates their normal
directions. Resolving its coarse quotient over this fiber produces a
map from a point blowup of
$\Sym^2 C$, itself the blowup of $\Jac(C)$ at a point. The same local
argument detects the sum of the two translation labels. We then
realize the resulting surface map in an ordinary smooth projective
ambient variety and remove its point blowups by projective
modifications. Throughout these operations we preserve the actual
homomorphism from the surface group.
The local path criterion and the ambient blowdown construction are
formulated separately from the example. The former converts bounded
chart compatibility and hyperbolicity into nontriviality of loops.
The latter realizes a point blowdown of an embedded surface inside a
new smooth projective ambient variety, preserving the ambient
fundamental group and the induced surface homomorphism. These two
statements isolate the group-theoretic and projective-geometric
mechanisms of the construction.
\subsection{Structure of the proof}
Section~\ref{sec:paths} proves the local path criterion independently
of the geometric application. Its hypotheses are bounded-range path
readings, compatibility of endpoint equality, and uniform hyperbolicity
of the reading graphs; no local compactness is required.
Section~\ref{sec:geometry} constructs the arithmetic arrangement,
its sign covers, and the distinguished surface map.
Section~\ref{sec:models} defines the local monodromy labels, proves
their buffered compatibility, and establishes hyperbolicity of the
associated graphs. Section~\ref{sec:image} verifies the path
criterion for the constructed orbifold and proves that the surface
image is infinite cyclic. Section~\ref{sec:projective} carries this
image into an ordinary smooth projective variety, removes the point
blowups, and completes the proof using Section~\ref{sec:reduction}.
\subsection{Conventions}
All varieties are over $\C$ unless a field is specified, and all
fundamental groups are taken in the usual topology. For a smooth
complex orbifold, equivalently a smooth Deligne--Mumford stack in the
constructions below, the fundamental group means that of its
classifying space. An \emph{ordinary} point has trivial stabilizer.
Fundamental-group image statements are understood up to a common
choice of basepoints and connecting paths. Projective bundles
parametrize lines. Complex hyperbolic distance is normalized so that
a point of Euclidean radius $\tanh r$ in the unit ball has distance
$r$ from its center.
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//01-introduction.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//02-reduction.tex ====
\section{Reduction to an abelian surface with cyclic image}
\label{sec:reduction}
The main construction will produce a simple abelian surface inside a
smooth projective variety, with infinite cyclic image on fundamental
groups. We first explain why that is enough. This separates the
geometric obstruction to Steinness from the group-theoretic construction
that occupies the intervening sections.
\begin{proposition}\label{prop:large-reduction}
Let $A$ be a simple abelian surface and let $A\hookrightarrow S'$ be a
closed embedding in a smooth connected projective complex variety.
Suppose that
\[
\im\bigl(\pi_1(A)\longrightarrow\pi_1(S')\bigr)\simeq\Z.
\]
There exist a smooth connected projective fourfold $X$ with large
fundamental group and a closed embedding $A\hookrightarrow X$ with
infinite cyclic image on fundamental groups, such that the universal
cover of $X$ is not Stein.
\end{proposition}
We begin with the equivalence between largeness and the condition on
the universal cover used in Theorem~\ref{thm:main}.
\begin{lemma}\label{red:compact-criterion}
Let $X$ be a smooth connected projective complex variety. Its universal
cover contains no positive-dimensional compact complex-analytic
subvariety if and only if $X$ has large fundamental group.
\end{lemma}
\begin{proof}
Suppose that $Z\subset X$ has positive dimension and that
$\pi_1(Z^\nu)$ has finite image in $\pi_1(X)$. The covering of $Z^\nu$
corresponding to the kernel has finite degree and is compact. Its map
to $X$ lifts holomorphically to $\widetilde X$. The image of the lift
is a positive-dimensional compact analytic subvariety, by the proper
mapping theorem.
Conversely, let $K\subset\widetilde X$ be a positive-dimensional
irreducible compact analytic subvariety. Its image $Z$ in $X$ is
analytic by the same theorem and algebraic by Chow's theorem. The
restriction $K\to Z$ is finite: it is proper and its fibers are discrete,
since the covering map is locally biholomorphic. Thus
$K^\nu\to Z^\nu$ is finite and surjective. Such a map has finite-index
image on fundamental groups. To see the latter assertion, remove a
proper analytic subset of the target so that the map becomes a finite
unramified cover of smooth connected spaces. On these open sets the
index is its covering degree. Their fundamental groups surject onto
those of the normal spaces, by general position for loops; passing to
these quotients can only decrease the index. But the composite
$\pi_1(K^\nu)\to\pi_1(X)$ is zero, because $K^\nu$ maps to
$\widetilde X$. Hence $\pi_1(Z^\nu)$ has finite image in $\pi_1(X)$.
\end{proof}
The next elementary interpolation observation allows us to control all
fibers of a general complete intersection simultaneously.
\begin{lemma}\label{red:interpolation}
Let $W$ be a projective variety, $F\subset W$ a closed subscheme, and
$L$ a very ample line bundle. Fix a positive integer $\ell$. For all
sufficiently large $m$ and every set of distinct points
$p_1,\ldots,p_\ell\in W\setminus F$, the evaluation map
\[
H^0(W,\mathcal I_F\otimes L^m)
\longrightarrow \bigoplus_{i=1}^{\ell}L^m|_{p_i}
\]
is surjective. The same bound on $m$ works for every such set.
\end{lemma}
\begin{proof}
Choose $m_0$ so that $\mathcal I_F\otimes L^{m_0}$ is globally
generated. For each $i$, choose a section $a_i$ nonzero at $p_i$.
For every $j\ne i$, very ampleness gives a section of $L$ vanishing at
$p_j$ and nonzero at $p_i$. Their product $b_i$ vanishes at all the
$p_j$ with $j\ne i$ and is nonzero at $p_i$. The sections $a_ib_i$
therefore give a diagonal basis for evaluation in degree
$m_0+\ell-1$. Multiplying each by a section of the remaining power
of $L$ nonzero at $p_i$ gives the assertion in every higher degree.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:large-reduction}]
Put $d=\dim S'$, choose an abelian variety $B$ of dimension $g=6$, and
write
\[
W=S'\times B,\qquad F=A\times\{0\},\qquad
c=d+g-4=d+2.
\]
Fix a very ample line bundle $L$ on $W$. We shall take $X$ to be the
common zero locus of $c$ general sections of
$\mathcal I_F\otimes L^m$, for a sufficiently large $m$.
There are two requirements: $X$ must be smooth with the expected
fundamental group, and its fibers over $B$ must be finite away from $F$.
\smallskip
\noindent\emph{Smoothness and the fundamental group.}
For large $m$, the sections generate the ideal away from $F$ and their
normal derivatives generate
$N^*_{F/W}\otimes L^m|_F$. 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 rank-deficient matrices
have codimension $3$, larger than $\dim F=2$, so no rank loss occurs
for a general tuple. The same count for each prefix of the tuple
shows that the successive intersections are smooth. They are ample
hypersurfaces in their predecessors, with final dimension four.
The Lefschetz hyperplane theorem consequently gives connectedness and
an isomorphism, induced by inclusion,
\begin{equation}\label{red:fundamental-group}
\pi_1(X)\xrightarrow{\ \sim\ }
\pi_1(S'\times B)=\pi_1(S')\times\pi_1(B).
\end{equation}
Here and below we use sufficiently high powers of a very ample line
bundle to obtain both ideal generation and generation of the indicated
normal derivatives; these are the usual applications of Serre
vanishing and Bertini's theorem \cite{Hartshorne}.
\smallskip
\noindent\emph{Fibers away from the fixed surface.}
We claim that a general such tuple excludes four distinct points of
$W\setminus F$ in any one fiber of $W\to B$. Excluding four points
is enough to exclude every positive-dimensional fiber component
outside $F$; the choice $g=6$ makes the following incidence count
strict. The parameter space of
ordered quadruples in one fiber has dimension at most $4d+g=4d+6$.
By Lemma~\ref{red:interpolation}, 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 thus has dimension strictly less than
the section parameter space. 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 $X\to B$ contains at most three points
outside $F$. In particular, every positive-dimensional subvariety of
a fiber is contained in $F$.
\smallskip
\noindent\emph{Largeness.}
Let $Z\subset X$ be a positive-dimensional closed irreducible
subvariety. If its projection to $B$ is nonconstant, then
$\pi_1(Z^\nu)\to\pi_1(B)$ has infinite image. Otherwise its image,
a finite subgroup of the torsion-free group $\pi_1(B)$, would be
trivial. The map $Z^\nu\to B$ would then lift holomorphically to
$\C^g$. Every holomorphic function on the connected compact normal
space $Z^\nu$ is constant, a contradiction.
If the projection is constant, the fiber property gives $Z\subset F$.
For $Z=F$, its image in $\pi_1(X)$ is infinite cyclic by
\eqref{red:fundamental-group} and the hypothesis. The only other
possibility is an irreducible curve in $A$. Let $D$ be its smooth
normalization. After a translation in $A$, the nonconstant map
$D\to A$ extends to a homomorphism $\Jac(D)\to A$. Its image is a
nonzero abelian subvariety, and therefore all of $A$ by simplicity.
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 $\pi_1(A)\simeq\Z^4$, and its
image under the homomorphism onto the infinite cyclic group remains
infinite. This proves largeness in every case. Lemma~\ref{red:compact-criterion}
then gives the required absence of compact analytic subvarieties in
$\widetilde X$.
\smallskip
\noindent\emph{The obstruction to Steinness.}
Let $K$ be the kernel of $\pi_1(A)\to\pi_1(X)$. Its quotient is
infinite cyclic, so $K\simeq\Z^3$. A connected component of the
inverse image of $F$ in $\widetilde X$ is the connected covering
\[
A_K=\C^2/K.
\]
It is a closed complex submanifold of $\widetilde X$: 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,
\[
A_K\simeq(\R^3/\Z^3)\times\R,
\qquad H_3(A_K,\Z)\simeq\Z.
\]
By the Andreotti--Frankel theorem \cite{AndreottiFrankel1959}, a Stein
manifold of complex dimension two has the homotopy type of a CW complex of real dimension at most
two; see \cite[Theorem~7.2]{MilnorMorse}. Thus $A_K$ is not Stein.
A closed complex submanifold of a Stein manifold is Stein, so
$\widetilde X$ cannot be Stein either.
\end{proof}
Proposition~\ref{prop:large-reduction} leaves one concrete task: embed a
simple abelian surface in a smooth projective variety so that its
fundamental group has infinite cyclic image. We first construct the
corresponding map from a point blowup of the surface to a smooth
orbifold, and then convert it to the required embedding in
Section~\ref{sec:projective}.
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//02-reduction.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//03-paths.tex ====
\section{A local criterion for nontrivial loops}\label{sec:paths}
The geometric construction will attach group labels to paths only within
bounded regions. We therefore need a criterion that detects a nontrivial
loop using such local information. The criterion below uses hyperbolic
graphs as the local models. It does not require a global map to a model,
local finiteness of a graph, or a bound on the information carried by one
edge.
A model to keep in mind is a graph whose vertices are points of a space
and whose edges are paths confined to small regions. A reading lifts
such paths to a covering space chosen near the initial vertex; its
endpoint records both the actual endpoint and the sheet of the cover.
The argument follows the coarse local-to-global viewpoint of hyperbolic
geometry \cite{Gromov1987}; we prove here the version for bounded path
readings in compatible charts.
\subsection{Paths and their 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.
We use the path metric on each connected component of a graph.
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 for reading 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.
\begin{enumerate}
\item[\textnormal{(P1)}] Readings respect prefixes and reverses of edges,
and read a backtrack as a backtrack. If $p$ starts at $v$ and $|p|2\Delta$ from
either end connector. Quadrangle slimness therefore puts it within
$2\Delta$ 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
\begin{equation}\label{path:mate-shift}
|b'_i-b_i|_{\beta}\le h+2\Delta.
\end{equation}
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
\[
L-2h-2(h+2\Delta)=L-4h-4\Delta>0.
\]
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 $2\Delta$; 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 $2\Delta$. Apply
\eqref{path:geodesic-comparison}, 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 $30\Delta2\Delta$.
Also, the distances from $x_0$ increase at every mark by at least
$L-2\Delta$, since
\[
d(x_0,x_{i+1})-d(x_0,x_i)
=d(x_i,x_{i+1})-2(x_0\mid x_{i+1})_{x_i}.
\]
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 \eqref{path:block-products}, every interior $x_i$
has a mate on a geodesic $[x_0,x_q]$ at distance at most $12\Delta$.
Such a mate is farther than $2\Delta$ from either of the geodesic
connectors $[x_0,A]$ and $[x_q,B]$: its distance from those connectors
is at least $L-H-14\Delta$. Quadrangle slimness therefore gives a
vertex $y_i$ on $[A,B]$ with
\begin{equation}\label{path:block-mates}
d(x_i,y_i)\le20\Delta0.
\]
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 \eqref{path:block-mates}. 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~\eqref{path:geodesic-comparison}
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 $50H$.
\end{proof}
\subsection{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~\ref{path:extension}, 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~\ref{path:extension}, composition gives a ladder between them of
width at most
\[
50H+H+50H=101H.
\]
Lemma~\ref{path:narrowing} 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~\ref{path:extension} 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 $\mathcal Q$.
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.
\begin{proof}[Proof of Lemma~\ref{thm:path-lemma}]
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 $\mathcal Q$, 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~\eqref{path:strip-calculation}, with the empty final rung,
would then give $[e]_v=d_v$, a contradiction.
\end{proof}
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//03-paths.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//04-geometry.tex ====
\section{An arithmetic arrangement and a distinguished surface}\label{sec:geometry}
We construct a smooth proper effective orbifold $\cS$ with projective
coarse space and a morphism $f:Y\to\cS$, where $Y$ is obtained from a
simple abelian surface by point blowups. The image of $f$ will lie in
the ordinary locus of $\cS$. The arrangement constructed here provides
both the relations and the local covering spaces used to prove that
$f_*\pi_1(Y)$ is infinite cyclic. The fundamental-group calculation is
made in Theorem~\ref{thm:orbifold-cyclic}.
\subsection{Two pencils and their coupling hyperplanes}\label{geo:prototype}
The two pencils will use the same six marked directions on $\PP^1$.
We first choose these markings so that their double cover has simple
Jacobian, and then choose the arithmetic ball containing the pencils.
Choose six distinct real algebraic numbers $\lambda_1,\ldots,\lambda_6$ with
\begin{equation}\label{geo:slopes}
|\lambda_i|<0.01\quad(1\le i\le4),
\qquad \lambda_5<-4,\quad \lambda_6>4,
\end{equation}
and such that the smooth double cover
\begin{equation}\label{geo:curve}
\pi:C\longrightarrow\PP^1,
\qquad y^2=\prod_{i=1}^6(t-\lambda_i),
\end{equation}
has simple Jacobian. We call the first four slopes \emph{red} and the
last two \emph{blue}. These requirements can be imposed simultaneously,
as follows.
Start with six disjoint real open intervals satisfying
\eqref{geo:slopes}. 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
\[
(6),\qquad(5,1),\qquad(2,1,1,1,1).
\]
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 its splitting
field. Then $F$ is totally real, $F\ne\Q$, and all six slopes belong
to $F$. Zarhin's theorem gives
$\End(\Jac(C))=\Z$ over an algebraic closure, hence $\Jac(C)$ is simple
\cite[Theorem~2.1]{Zarhin}.
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
\begin{equation}\label{geo:hermitian}
h=\operatorname{diag}(1,1,1,1,-a).
\end{equation}
At the distinguished embedding the negative lines form complex
hyperbolic space of dimension four. In the affine chart with last
homogeneous coordinate one, write this ball as
\[
\BB=\{(z,w)\in\C^2\times\C^2:|z|^2+|w|^2a$.
Every intersection involving the two families has a local product
description in the $z$ and $w$ variables. In particular, the phrase
``product'' here concerns local analytic coordinates, rather than a
product decomposition of the hyperbolic metric.
Figure~\ref{fig:arrangement} shows the incidences in one family; the
second family is obtained by using the other coordinate pair.
\input{figures/arrangement}
\subsection{Exact models in congruence quotients}\label{geo:separation}
The arithmetic lattice theorem of Borel--Harish-Chandra
\cite[Corollary~12.4]{BorelHarishChandra1962}, applied to
\eqref{geo:hermitian}, gives a
cocompact arithmetic lattice of simplest type in $\mathrm{PU}(4,1)$.
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
\cite[Theorem~10.11]{BailyBorel1966}. 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 \cite[Section~3.1]{StoverToledo}.
Fix positive normal vectors $n_i\in E^5$ for the members of $\cD$,
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.
\begin{proposition}[Exact local arrangements]\label{prop:arrangement-local}
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 $\mathcal A_0$ be the union of all $\Gamma_0$-translates
of the labeled hyperplanes of $\cD$.
\begin{enumerate}
\item The arrangement $\mathcal A_0$ is locally finite. The hyperplanes
meeting any ball of radius $W$ have distinct labels and are carried to
the corresponding subarrangement of $\cD$ by a holomorphic isometry
induced by a unitary map sending each prescribed translated normal to
its prototype normal.
\item The injectivity radius of $\Gamma_0\backslash\BB$ is greater than
$R_0$.
\item In the ball of radius $R_0$ about $o$, the arrangement is exactly
$\cD$ restricted to that ball. The analogous assertion holds in every
$\Gamma_0$-translate of this ball.
\end{enumerate}
All these conclusions concerning the lifted arrangement remain valid
on any subsequent finite cover of $\Gamma_0\backslash\BB$, provided one
uses the full inverse image of the arrangement.
\end{proposition}
\begin{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
\[
\sinh d(x,H_n)=\frac{|h(n,x)|}{\sqrt{h(n,n)}}.
\]
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
\[
h(\gamma n_i,\gamma'n_j)\equiv h(n_i,n_j)
\]
modulo the level, in this fractional ideal. Choose the level to
separate all the finitely many possible nonzero differences. We obtain
exact equality:
\begin{equation}\label{geo:gram}
h(\gamma n_i,\gamma'n_j)=h(n_i,n_j)
\end{equation}
whenever the corresponding hyperplanes are visible together.
Here 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 \eqref{geo:gram} 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 \eqref{geo:gram}, 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 \eqref{geo:gram}, 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 $\BB$. 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.
\end{proof}
We next record the consequences for the divisors and centers that we
will blow up. In $M_0=\Gamma_0\backslash\BB$, each label gives a closed
embedded smooth totally geodesic divisor, possibly disconnected. Its
lifts are locally finite and have disjoint sheets by
Proposition~\ref{prop:arrangement-local}. 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.
\subsection{Two global sign covers}\label{geo:signs}
For the later local models we need global parity characters on the
arrangement complement. We obtain these from the following theorem
of Stover--Toledo, in the form stated by Llosa Isenrich--Py
\cite[Theorem~4.2]{LlosaPy}; see also \cite{StoverToledo}.
\begin{theorem}[Virtual cyclic ramified covers]\label{geo:stover-toledo}
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 $d\ge2$ there is a
finite unramified cover $M_1\to M_0$ such that $M_1$ admits a cyclic
cover of degree $d$, branched over the full inverse image of $D$.
\end{theorem}
Apply Theorem~\ref{geo:stover-toledo} 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~\ref{geo:stover-toledo} 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
\begin{equation}\label{geo:sign-equivariance}
\chi_w=\sigma^*\chi_z.
\end{equation}
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 $M=\Gamma\backslash\BB$ 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 $M$.
\subsection{Filling the arrangement}\label{geo:fill}
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 $k\bmod2$.
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 $\widehat M$ denote the resulting smooth projective base. Extend
the two sign covers from the complement by normalization over
$\widehat M$, 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
\begin{equation}\label{geo:V}
V\longrightarrow\widehat M
\end{equation}
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 $\widehat M\to M$
is $\PP^1\times\PP^1$. Each exceptional line has six branch markings
and has even meridian sign. Its inverse image in $V$ is consequently
\begin{equation}\label{geo:Fstar}
F_*=C\times C.
\end{equation}
Put
\[
L=\pi^*\mathcal O_{\PP^1}(-1),
\qquad N=N_{F_*/V}
=\operatorname{pr}_1^*L\oplus\operatorname{pr}_2^*L.
\]
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 thus
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 \eqref{geo:sign-equivariance} 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
\[
(p,q;u,v)\longmapsto(\pi(p),\pi(q);u,v).
\]
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
\begin{equation}\label{geo:line-action}
s(p,q;u,v)=(q,p;v,-u).
\end{equation}
Its fourth power is a deck transformation fixing $F_*$ pointwise,
so $s^4=1$; a nonidentity element of $(\Z/2)^2$ does not fix $C\times C$
pointwise. Its square is the identity on $F_*$ and is multiplication
by $-1$ on both normal lines.
\subsection{An ordinary neighborhood of the distinguished fiber}
\label{geo:resolution}
The quotient stack $[V/\langle s\rangle]$ 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
$\langle s^2\rangle$. The exceptional divisor is
\begin{equation}\label{geo:P}
P=\PP_{F_*}(N),
\end{equation}
where projectivization parametrizes lines. Before taking the quotient,
$s^2$ acts trivially on $P$ and as $t\mapsto-t$ 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 $C\times C$. On the
projective normal line over that diagonal, it acts by
\[
[u:v]\longmapsto[v:-u].
\]
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 $(\pm i)^2=-1$. 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
\[
(x,t_1,t_2,t_3)\longmapsto(x,-t_1,-t_2,-t_3),
\]
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
$V/\langle s\rangle$ 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 $\cS$, use this smooth scheme near that fiber and use
$[V/\langle s\rangle]$ 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 $\cS$ with
coarse space $S_0$. The map $\cS\to S_0$ 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,
$\cS$ is separated and proper over $\C$. In particular the whole distinguished
fiber lies in the ordinary smooth locus of $\cS$.
\subsection{The surface mapping into the ordinary fiber}\label{geo:surface}
Let $A=\Jac(C)$. For a genus-two curve, the Abel map
\[
\Sym^2C\longrightarrow\Pic^2(C)\simeq A
\]
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 $E_C=|K_C|\simeq\PP^1$. For the degree-two quotient
$q:C\times C\to\Sym^2 C$, 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
\[
2E_C^2=(q^*E_C)^2=\Gamma_\iota^2=2-2g(C)=-2.
\]
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 $C\times C$ where both values are defined and nonzero, put
\begin{equation}\label{geo:rational-section}
(p,q)\longmapsto[\tau(p),i\tau(q)]\in P.
\end{equation}
This section is equivariant for swapping the two factors and the
involution on $P$. Indeed \eqref{geo:line-action} gives
\[
[\tau(p),i\tau(q)]\longmapsto[i\tau(q),-\tau(p)]
=[\tau(q),i\tau(p)].
\]
Away from the diagonal it avoids the two fixed curves, and thus gives
a rational map from $\Sym^2C$ 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 $\cS$. We therefore obtain a morphism
\begin{equation}\label{geo:Ymap}
f:Y\longrightarrow\cS
\end{equation}
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.
\begin{proposition}[The geometric construction]\label{prop:orbifold-surface}
After any prescribed finite lower bounds on the arrangement radii and
injectivity radius, the construction above gives a smooth connected
proper effective orbifold $\cS$ with projective coarse space, a simple
abelian surface $A=\Jac(C)$, an iterated point blowup $Y\to A$, and a
morphism $f:Y\to\cS$ with image in its ordinary locus. The distinguished
fiber is obtained from the product $C\times C$, its normal bundle
$\operatorname{pr}_1^*L\oplus\operatorname{pr}_2^*L$, and the action
\eqref{geo:line-action} by the explicit blowups and coarse quotients
above. Away from that fiber the orbifold is $[V/\langle s\rangle]$,
with $V$ given by the two parity covers and the filling rules of
Section~\ref{geo:fill}.
\end{proposition}
All parameters used in the finite arrangement have now been fixed.
The depth of $\Gamma_0$ has not: Proposition~\ref{prop:arrangement-local}
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
\[
\im(f_*:\pi_1(Y)\to\pi_1(\cS))\simeq\Z.
\]
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//04-geometry.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//05-models.tex ====
\section{Local covers and hyperbolic path graphs}\label{sec:models}
The geometric construction gives relations among the loops in the
surface. To show that these relations leave an element of infinite
order, we will read paths in auxiliary covers of finite arrangements.
This section constructs those covers and proves the two properties
needed for Lemma~\ref{thm:path-lemma}: 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
$\cD$. In particular, they are fixed before choosing the arithmetic
level.
\subsection{Dihedral labels on a finite arrangement}
For a subset $T\subseteq\cD$, set
\[
U_T=\BB\setminus\bigcup_{H\in T}H.
\]
There are characters $\epsilon_z,\epsilon_w:\pi_1(U_T)\to\Z/2$,
with $\epsilon_z$ equal to one on a meridian of a $z$ hyperplane and
zero on a meridian of a $w$ hyperplane, and conversely for
$\epsilon_w$. 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 $E_T\to U_T$ 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, as is seen by filling 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
\[
D_\infty=\langle r,b\mid r^2=b^2=1\rangle,
\qquad \epsilon(r)=\epsilon(b)=1.
\]
Its even subgroup is the infinite cyclic translation subgroup
$\langle rb\rangle$. Conjugation in $D_\infty$ preserves this subgroup
and changes its integer coordinate at most by a sign.
We first construct $q_z:\pi_1(U_T)\to D_\infty$. The construction
ignores all $w$ hyperplanes. If $T$ contains fewer than all six $z$
slope hyperplanes, define
\[
q_z(\gamma)=r^{\epsilon_z(\gamma)}.
\]
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 $r^4b^2=1$.
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 $z_1=\pm c$ in the ball,
\[
|\lambda|<0.7.
\]
Over $|\lambda|<1.5$, the pencil local system has a reduction of
structure group to $\langle r\rangle$. In this reduction, multiply
its transition functions by the two-sheet systems defined, for the
coupling hyperplanes that are present, by
\begin{equation}\label{mod:coupling-functions}
(1-z_1/c)^{1/2},\qquad (1+z_1/c)^{1/2}.
\end{equation}
This means adding their winding parities to the exponent of $r$.
Over $|\lambda|>1$ one has $|z_1|0$ 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
\[
\dist(x,L_z)\le B_1(R).
\]
The analogous assertion holds for $w$. The same bounds hold if
the loops must avoid any additional analytic subsets.
\end{lemma}
\begin{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 $z_1=\pm c$
and the boundary estimate is uniform:
\[
|z_2/z_1|\le\frac{\sqrt{a-c^2}}{c}
<\frac{\sqrt{1-0.85^2}}{0.85}<0.62<0.7.
\]
Eventually the whole ball lies over this red disk and misses the
other coupling hyperplane. Its label system consequently reduces
to $\langle r\rangle$, contradicting the assumed nonzero
translation. This proves the bound. Restricting the class of
allowed loops does not weaken it.
\end{proof}
\begin{proposition}[Compatibility of kernels]\label{prop:kernel-compatibility}
For every $R>0$ there are constants $B(R)>R$ and $J_0(R)$, depending
only on $\cD$, with the following properties.
\begin{enumerate}
\item 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.
\item In the full model, if $\dist(x,o)>J_0(R)$, then on such loops
\[
\ker(\ell_z+\ell_w)=\ker\ell_z\cap\ker\ell_w.
\]
\end{enumerate}
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.
\end{proposition}
\begin{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~\ref{mod:buffer-bound}, 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
\[
\{x:\dist(x,L_z)\le B_1(R)\},\qquad
\{x:\dist(x,L_w)\le B_1(R)\}
\]
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~\ref{mod:buffer-bound}. 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.
\end{proof}
\subsection{Graphs defined by projected diameter}
We now turn the local covers into graphs to which the path lemma
applies. Over each connected component of $E_T$, let
$\widehat E_T$ be the connected regular cover defined by
$\ker\ell_z\cap\ker\ell_w$. 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 $\BB$ 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 $\widehat s$. 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 $\mathcal G_T(u)$, with
$\mathcal G_{\mathrm{cen}}(u)$ 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.
\begin{proposition}[Hyperbolicity of the models]\label{prop:model-hyperbolicity}
There are $u>0$ and $\delta\ge0$, depending only on $\cD$, such that
every connected component of every $\mathcal G_T(u)$ and of
$\mathcal G_{\mathrm{cen}}(u)$ is $\delta$-hyperbolic.
The same $\delta$ can be used if an arbitrary locally finite union
of proper complex analytic subsets of $\BB$ is additionally
omitted, with invariant omissions in the central case. On the
vertices retained after such an omission, graph distances are
unchanged.
\end{proposition}
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.
\begin{lemma}\label{mod: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.
\end{lemma}
\begin{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.
\end{proof}
\subsection{The arrangement at the sphere at infinity}
Rescale the ball to unit radius and write
\[
x=\tanh(r)\xi,\qquad |\xi|=1.
\]
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~\ref{prop:arrangement-local}. More explicitly, write
an affine intersection as $a_0+V_0$ with $a_0\perp V_0$. 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 $r\ge r_0$ 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
\begin{equation}\label{mod:collar-shift}
O(1-\tanh r)=O(e^{-2r})
\quad\hbox{in Euclidean distance.}
\end{equation}
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
\[
d_K(\xi,\eta)=|1-\langle\xi,\eta\rangle|^{1/2}.
\]
For completeness, it satisfies the triangle inequality. If
$a=d_K(\xi,\zeta)$ and $b=d_K(\zeta,\eta)$, then
$|\xi-\zeta|\le\sqrt2a$ and
$|\zeta-\eta|\le\sqrt2b$. Expanding the Hermitian inner product
around $\zeta$ gives
\[
|1-\langle\xi,\eta\rangle|
\le a^2+b^2+|\xi-\zeta|\,|\eta-\zeta|
\le(a+b)^2.
\]
Positivity and symmetry are immediate. In particular,
\eqref{mod:collar-shift} is an $O(e^{-r})$ displacement in $d_K$.
The ball distance in our normalization satisfies
\begin{equation}\label{mod:ball-distance}
\cosh d(x,y)=
\frac{|1-\langle x,y\rangle|}
{\sqrt{1-|x|^2}\sqrt{1-|y|^2}}.
\end{equation}
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
\[
1-\langle x,y\rangle
=1-\rho\tau+\rho\tau(1-\langle\xi,\eta\rangle).
\]
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 \eqref{mod:ball-distance} 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
\begin{equation}\label{mod:ideal-metric}
d_I(\xi',\eta')=
\min\left\{1,\inf_\alpha\diam_{d_K}(\pi\alpha)\right\},
\end{equation}
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.
\subsection{A cone over an arbitrary bounded metric space}
We isolate the metric calculation, a discrete form of the familiar
hyperbolic-cone construction; compare \cite[Section~7]{BonkSchramm2000}.
We give the proof for arbitrary bounded metric spaces, as no
compactness hypothesis is available here. For a metric space $(I,d_I)$
with $d_I\le1$, let $\mathcal C(I)$ have vertices $(\xi',t)$,
$\xi'\in I$, $t\in\Z_{\ge0}$. Add vertical unit edges between
successive heights at the same point, and horizontal unit edges
at height $t$ whenever $d_I(\xi',\eta')\le e^{-t}$.
In particular all vertices at height zero are pairwise adjacent.
\begin{lemma}\label{mod:cone}
The graph $\mathcal C(I)$ is hyperbolic with an absolute constant,
independent of the bounded metric space $I$. More precisely, put
\[
j=\min\{t,t',-\log d_I(\xi',\eta')\},
\qquad -\log0=+\infty.
\]
Its distance $D$ satisfies
\begin{equation}\label{mod:cone-distance}
t+t'-2j\le
D\bigl((\xi',t),(\eta',t')\bigr)
\le t+t'-2j+3.
\end{equation}
\end{lemma}
\begin{proof}
If $\xi'=\eta'$, both the formula without its additive error and
the claim are immediate from vertical distance. Otherwise,
descend to height $\lfloor j\rfloor$, take one horizontal edge,
and ascend. This gives the upper bound.
For the lower bound, let any connecting path have minimum height
$m$ and $n\ge1$ horizontal edges. The number of its vertical
edges is at least $t+t'-2m$. By the triangle inequality in $I$,
\[
d_I(\xi',\eta')\le n e^{-m}.
\]
Since also $m\le t,t'$, this implies $m\le j+\log n$.
The path length is therefore at least
\[
t+t'-2j+n-2\log n\ge t+t'-2j.
\]
The last inequality holds for every integer $n\ge1$ (indeed its
minimum over positive real $n$ is $2-2\log2>0$).
To deduce hyperbolicity, fix a height-zero basepoint $p$.
Its distance to $(\xi',t)$ differs from $t$ by at most one.
It follows from \eqref{mod:cone-distance} 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
\[
-\log d_I(\xi',\eta')
\ge\min\{-\log d_I(\xi',\zeta'),
-\log d_I(\zeta',\eta')\}-\log2.
\]
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; see \cite[Chapter~III.H]{BridsonHaefliger}.
\end{proof}
\subsection{Comparison with the model graphs}
We can now complete the proof of
Proposition~\ref{prop:model-hyperbolicity}. 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 \eqref{mod:ideal-metric} to be unattained. Place that
path at radius $r_0+t$ with the collar. The angular estimates and
\eqref{mod:ball-distance} 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~\ref{mod:bounded-core} 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
\eqref{mod:ball-distance} 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
\[
d_I(\xi',\eta')\le C_u e^{-r}.
\]
Its rounded heights differ by a bounded amount, so
\eqref{mod:cone-distance} 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
\cite[Chapter~III.H]{BridsonHaefliger} prove hyperbolicity of the
model graph.
For the central model, the collar and the comparison maps on the
end are equivariant under $\widehat s$. The lifted action preserves
$d_I$, since its projection is unitary and hence preserves $d_K$.
For the finite group $H=\langle\widehat s\rangle$, the formula
\[
\overline d_I(H\xi',H\eta')
=\min_{h\in H}d_I(\xi',h\eta')
\]
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~\ref{mod:cone} 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~\ref{thm:path-lemma}.
Only after these choices will we choose the arithmetic level and
the geometric sizes of the charts.
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//05-models.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//06-image.tex ====
\section{An infinite cyclic image}\label{sec:image}
We now compute the image of the surface constructed in
Proposition~\ref{prop:orbifold-surface}. 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 the preceding section exist in charts, and need not define a
homomorphism on the fundamental group of the whole orbifold.
\begin{theorem}\label{thm:orbifold-cyclic}
For a sufficiently deep choice of the arithmetic level in the
construction of Proposition~\ref{prop:orbifold-surface}, the map
$f:Y\longrightarrow\cS$ satisfies
\[
\im\bigl(f_*:\pi_1(Y)\longrightarrow\pi_1(\cS)\bigr)\cong\Z.
\]
Here $Y$ is an iterated point blowup of the simple abelian surface
$\Jac(C)$, and its image lies in the ordinary locus of $\cS$.
\end{theorem}
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.
\subsection{Surface loops and the cyclic upper bound}
Recall the distinguished fiber $F_*=C\times C$ in $V$, the order-four
lift $s$, and the rational section used to construct $Y$ in
Section~\ref{geo:surface}. Put
\[
\mathcal K=[V/\langle s\rangle].
\]
There is a natural homomorphism
\begin{equation}\label{app:comparison-map}
\pi_1(\mathcal K)\longrightarrow\pi_1(\cS).
\end{equation}
Indeed, the substack $[F_*/\langle s\rangle]$ has complex codimension
two in the smooth stack $\mathcal K$. Removing it does not change the
orbifold fundamental group, and its complement is the open substack on
which the construction of $\cS$ leaves $\mathcal K$ unchanged.
Choose a general point $q_0\in C$. The map
\[
C\longrightarrow\Sym^2 C\longrightarrow\Jac(C),
\qquad p\longmapsto p+q_0,
\]
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 $C\times\{q_0\}$ 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.
\begin{lemma}\label{app:surface-comparison}
With common choices of basing, the images in $\pi_1(\cS)$ of the
preceding generators of $\pi_1(Y)$ equal the images, under
\eqref{app:comparison-map}, of the corresponding loops in
$C\times\{q_0\}\subset V$. The loops can also be represented in the
sign-covered arrangement complement, arbitrarily close to $F_*$, with
both normal coordinates nonzero.
\end{lemma}
\begin{proof}
Lift the bouquet by the rational section to the exceptional divisor of
$\operatorname{Bl}_{F_*}V$. Over this bouquet the section has the form
\[
[\tau(p),\sqrt{-1}\,\tau(q_0)],
\]
with both entries nonzero. It avoids the fixed curves involved in the
last resolution step, because the bouquet avoids the diagonal of
$C\times C$.
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
\[
\varepsilon\bigl(\tau(p),\sqrt{-1}\,\tau(q_0)\bigr).
\]
One sufficiently small $\varepsilon>0$ 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 $\cS$, 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.
\end{proof}
We next compute an upper bound before taking the quotient by $s$.
Write $H=(\Z/2)^2$ for the sign deck group. The map
\[
V\longrightarrow[V/H]
\]
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
\begin{equation}\label{app:pencil-group}
Q=\langle h_1,\ldots,h_6\mid
h_i^2=1,\ h_1h_2h_3h_4h_5h_6=1\rangle.
\end{equation}
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
$\pi_1(C\times\{q_0\})$ to $\pi_1(V)$, followed by the injection just
described.
\begin{lemma}\label{app:cyclic-upper}
The image of $\pi_1(C\times\{q_0\})$ in $\pi_1(V)$ is cyclic.
Consequently the image of $\pi_1(Y)$ in $\pi_1(\cS)$ is cyclic.
\end{lemma}
\begin{proof}
We show that the four red meridians in \eqref{app:pencil-group} 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
\[
t=z_1\ne0\quad\hbox{fixed and small},\qquad
\lambda=z_2/z_1
\]
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
\[
\{t=c\}\ \cup\ \bigcup_{i=1}^4\{\lambda=\lambda_i\}.
\]
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 $kh_i$: in the two transverse coordinates,
the exceptional meridian circles both coordinates once. The filling
rules of Section~\ref{geo:fill} impose no branching along this
exceptional divisor, since two branch components met there. Thus its
meridian is killed. Therefore
\[
kh_i=1\qquad(1\le i\le4).
\]
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 \eqref{app:pencil-group} now gives
$h_5h_6=1$, and the two blue images are one involution $b$. The image
of $Q$ is thus a quotient of
\[
\langle r,b\mid r^2=b^2=1\rangle.
\]
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
$\pi_1(V)\to\pi_1([V/H])$ gives the assertion in $\pi_1(V)$.
Lemma~\ref{app:surface-comparison} and the surjectivity from the chosen
$C$ loops onto $\pi_1(Y)$ give the final assertion.
\end{proof}
This argument allows the cyclic image to be finite. We will exclude
that possibility by applying Lemma~\ref{thm:path-lemma} to paths in a
dense open subset of $\cS$.
\subsection{Paths in the ordinary open set}
Let $O\subset\cS$ 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 $\cS$ 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
$\BB$ of the lifted arrangement and the lifted omitted fixed loci,
pull back the sign data from $V$, and call the resulting space
$O^\#$. Then
\[
O^\#\longrightarrow O
\]
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~\ref{prop:model-hyperbolicity}. Define a graph $G$ whose
vertices are the points of $O$. An edge is a parametrized continuous
path in $O$ whose lifted projection to $\BB$ 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.
\subsection{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~\ref{thm:path-lemma}. Set
\begin{equation}\label{app:radius}
R=(2N+100)(u+1).
\end{equation}
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 $\BB$ 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 $o$:
\begin{enumerate}
\item 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.
\item For $j=1,2,3$, the displacement by $\sigma^j$ exceeds $10R$.
\end{enumerate}
The first requirement follows from
Proposition~\ref{prop:kernel-compatibility}. 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~\ref{prop:kernel-compatibility}, enlarged so that
$B(R)>R$, and choose
\begin{equation}\label{app:chart-radius}
K>10\bigl(J+R+B(R)\bigr).
\end{equation}
All these constants depend only on the finite models. Now take the
initial arithmetic level deep enough that
Proposition~\ref{prop:arrangement-local} 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
Lemma~\ref{app:cyclic-upper}. 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.
\begin{description}
\item[Central chart.] If $\dist(x,\Gamma o)5$. 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 $\mathcal P\to\mathcal T$ induces an isomorphism on fundamental
groups, since its fibers are simply connected. Thus the natural maps
give
\begin{equation}\label{proj:ordinary-group}
\pi_1(U)\simeq\pi_1(\mathcal P)\simeq\pi_1(\mathcal T).
\end{equation}
It remains to cut down inside $U$ while retaining the embedded surface.
Put $N=\dim P$. In a sufficiently high projective embedding of $P$,
take $N-5$ general hyperplane sections through $Y$. Their linear systems
are free off $Y$ and generate the normal differentials along $Y$.
The locus where $r$ general normal differentials fail to be independent
has expected codimension
\begin{equation}\label{proj:rank-loss}
(N-2)-r+1
\end{equation}
on the surface. For $r=N-5$ this is $4>2$, so the final intersection
is smooth along $Y$. Bertini gives smoothness elsewhere in $U$.
Since $P\setminus U$ has codimension greater than five and is disjoint
from $Y$, a general constrained intersection avoids it. We obtain a
smooth projective fivefold $S\subset U$ containing $Y$.
To verify the assertion about $\pi_1$, first take unconstrained general
hyperplane sections in this same embedding. The quasi-projective
Lefschetz theorem of Hamm and L\^e
\cite[Theorem~1.1.3(ii)]{HammLe} says that a smooth open variety of
complex dimension $n$ is obtained, up to homotopy, from a general
hyperplane section by attaching cells of dimensions at least $n$.
At every stage here the dimension before cutting is at least six.
The successive inclusions therefore preserve connectedness and induce
isomorphisms on fundamental groups. The final intersection avoids
$P\setminus U$, by the same dimension count.
Now consider the full product of the hyperplane parameter spaces,
without a containment requirement. The tuples whose intersections
have dimension five, are smooth, and avoid $P\setminus U$ form a
nonempty Zariski-open set $\Omega$. It contains both a general
unconstrained tuple and the constrained tuple just constructed, and it
is path connected. The universal complete intersection over $\Omega$
is a smooth proper family with a map to $U$. Along a path in $\Omega$,
Ehresmann's theorem identifies its fibers by diffeomorphisms and gives
a homotopy of their inclusions into $U$. Thus the isomorphism on
fundamental groups for an unconstrained fiber also holds for $S$.
Composing $S\hookrightarrow U\subset\mathcal P$ with the bundle
projection gives $h$, with $h\circ i=f$, and
\eqref{proj:ordinary-group} proves the lemma.
\end{proof}
Applied to $Y\to\cS$, this gives a smooth connected projective ambient
variety $S$ with an embedding of $Y$ and
\begin{equation}\label{proj:blownup-image}
\im\bigl(\pi_1(Y)\longrightarrow\pi_1(S)\bigr)\simeq\Z.
\end{equation}
The remaining issue is that Proposition~\ref{prop:large-reduction}
requires $A$ itself. We address one point blowup at a time.
\subsection{Realizing a surface blowdown in a new ambient variety}
\begin{lemma}\label{lem:ambient-blowdown}
Let $Y_1\hookrightarrow S$ be an embedding of a smooth projective
surface in a smooth connected projective variety, and let
$b:Y_1\to Y_0$ be the blowdown of a $(-1)$-curve to a smooth projective
surface. There exist a smooth connected projective variety $S_0$, an
embedding $Y_0\hookrightarrow S_0$, and an isomorphism
$\theta:\pi_1(S)\xrightarrow{\sim}\pi_1(S_0)$ such that
\[
\theta\circ(\pi_1(Y_1)\to\pi_1(S))
= (\pi_1(Y_0)\to\pi_1(S_0))\circ b_*.
\]
The equality is understood with compatible basepoints, or up to
conjugation.
\end{lemma}
The proof first places $Y_1$ in a projective threefold and adjusts the
normal bundle of its exceptional curve to
$\mathcal O(-1)\oplus\mathcal O(-1)$. Blowing up that curve gives an
exceptional $\PP^1\times\PP^1$. Contracting the other ruling then
performs $b$ on the surface. This is the classical local two-ruling
construction; compare \cite{Atiyah1958}. The essential global point
here is to construct a polarization that makes this second contraction
projective.
\begin{proof}
Write $E\subset Y_1$ for the exceptional curve. Replace $S$ by its
product with a projective space and embed $Y_1$ by the graph of
\[
Y_1\xrightarrow{b}Y_0\hookrightarrow\PP^r.
\]
The new ambient fundamental group is naturally $\pi_1(S)$. Let $Q$
be the pullback of $\mathcal O_{\PP^r}(1)$; it is nef, and
$Q|_{Y_1}=b^*L_0$ for a very ample line bundle $L_0$ on $Y_0$.
Further projective-space factors allow us to make the ambient dimension
as large as convenient.
\smallskip
\noindent\emph{A threefold containing the unchanged surface.}
Cut to a smooth projective fourfold $W$ containing $Y_1$, using high
ample degrees. For each successive cut the rank-loss count
\eqref{proj:rank-loss}, with final dimension four, is at least three;
thus it exceeds $\dim Y_1=2$. Together with Bertini, this gives smooth
successive cuts, and the Lefschetz theorem preserves the fundamental
group throughout.
For a final threefold, the rank-loss codimension is now two, equal to
the dimension of the surface. We therefore allow isolated singularities
in the last cut and resolve them while leaving $Y_1$ unchanged.
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
$N^*_{Y_1/W}\otimes\mathcal O_W(Z)|_{Y_1}$. 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
\begin{equation}\label{proj:node}
xu+yv=0,\qquad Y_1=(x=y=0).
\end{equation}
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 \eqref{proj:node}. 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
\[
x=tv,\quad y=-tu
\qquad\text{and}\qquad
v=sx,\quad u=-sy,
\]
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 $Y_1\subset T$.
These operations preserve the required fundamental group. For the
last cut, deform $Z$ to a smooth ample hypersurface in $W$. Away from
disjoint small neighborhoods of its nodes the family is locally
trivial. A three-dimensional ordinary double point has simply
connected link and simply connected Milnor fiber, and the small
resolution neighborhood retracts to $\PP^1$. For completeness, after
writing the node as $\sum_{j=0}^3 z_j^2=0$, its link is the space of
orthonormal two-frames in $\R^4$, hence the unit tangent bundle of
$S^3\simeq\mathrm{SU}(2)$, which is $S^3\times S^2$. Its smoothing
retracts to $S^3$; writing $z=x+iy$ identifies it with the tangent
bundle of $S^3$ before truncation. These are also the elementary
quadratic case of the Milnor-fiber theorem \cite{MilnorSingular}.
Van Kampen therefore gives the same fundamental group for the nodal
hypersurface, its smoothing, and its small resolution, compatibly with
the maps to $W$. Lefschetz for the smoothing yields
$\pi_1(T)\simeq\pi_1(W)\simeq\pi_1(S)$, with the prescribed map
from $\pi_1(Y_1)$.
\smallskip
\noindent\emph{Adjusting the normal bundle.}
The hypersurface degree can be chosen so large that
\[
N_{Y_1/T}|_E\simeq\mathcal O_E(-m),\qquad m\ge1.
\]
In fact there are no nodes on $E$, and the normal exact sequence there
subtracts $\deg\mathcal O_W(Z)|_E$ from the fixed degree of
$\det N_{Y_1/W}|_E$. The sequence
\[
0\longrightarrow\mathcal O_E(-1)
\longrightarrow N_{E/T}
\longrightarrow\mathcal O_E(-m)\longrightarrow0
\]
splits because
$H^1(E,\mathcal O_E(m-1))=0$. 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
$N_{Y_1/T}\otimes\mathcal O_{Y_1}(-E)$, 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
\[
N_{E/T}\simeq\mathcal O_E(-1)\oplus\mathcal O_E(-1),
\]
with the first summand the tangent-normal direction inside $Y_1$.
At each step the same extension vanishing justifies this splitting.
Blow up $E$ once more, obtaining $\widehat T$, and let $\widehat Y$
be the strict transform of $Y_1$. The exceptional divisor is
\begin{equation}\label{proj:exceptional-quadric}
G=\PP^1_{\mathrm{curve}}\times\PP^1_{\mathrm{normal}},\qquad
N_{G/\widehat T}=\mathcal O_G(-1,-1),\qquad
\widehat Y\cap G=\PP^1_{\mathrm{curve}}\times\{q\}.
\end{equation}
The intersection is transverse. Moreover
$\mathcal O_{\widehat T}(\widehat Y)|_G=\mathcal O_G(0,1)$, and
$N_{\widehat Y/\widehat T}|_E=\mathcal O_E$.
We continue to denote by $Q$ the pullback of the earlier nef bundle;
it is trivial on $G$ and restricts on $\widehat Y\simeq Y_1$ to
$b^*L_0$. Figure~\ref{fig:blowdown} shows the two rulings and the
curve on the surface that the second contraction must remove.
\input{figures/blowdown}
\smallskip
\noindent\emph{A polarization for the other contraction.}
We seek an ample line bundle $H$ restricting to
$\mathcal O_G(\alpha,\beta)$, with $\beta>\alpha>0$.
Since $\mathcal O_{\widehat T}(G)|_G=\mathcal O_G(-1,-1)$,
adding $\alpha G$ will then give the restriction
$\mathcal O_G(0,\beta-\alpha)$: it is trivial on the curves to be
contracted and positive on the retained factor.
Choose an ample integral line bundle $H_0$ on $\widehat T$, with
$H_0|_G=\mathcal O_G(\alpha,\beta_0)$, where $\alpha,\beta_0>0$.
Adding $k\widehat Y$ increases the second degree by $k$ and leaves the
first unchanged. Choose $k\ge0$ with $\beta:=\beta_0+k>\alpha$.
On $\widehat Y$, the line bundle $H_0+k\widehat Y$ still has positive
degree on $E$, because $N_{\widehat Y/\widehat T}|_E$ is trivial.
It is therefore relatively ample for $b:\widehat Y\to Y_0$, whose
only positive-dimensional fiber is $E$. Adding a sufficiently large
multiple $lQ$ makes its restriction to $\widehat Y$ ample.
We claim that
\[
H=H_0+k\widehat Y+lQ
\]
is ample on all of $\widehat T$. 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,
$k\ge0$, 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
\[
\bigl((A_0+kD_0)^r-A_0^r\bigr)\cdot V
= kD_0\cdot V\cdot
\sum_{j=0}^{r-1}(A_0+kD_0)^j A_0^{r-1-j}\ge0.
\]
Every term on the right is a mixed intersection of ample restrictions
on the effective cycle $D_0\cap V$. For $V\subset D_0$, positivity
follows directly from the restriction hypothesis. Nakai--Moishezon
now applies. Taking $D_0=\widehat Y$ proves the claim. We have obtained
\begin{equation}\label{proj:ample-degrees}
H|_G=\mathcal O_G(\alpha,\beta),\qquad \beta>\alpha>0.
\end{equation}
The relative ampleness, Serre vanishing, and numerical ampleness facts
used here are standard projective criteria; see \cite{Hartshorne}.
Set $D=H+\alpha G$. We shall 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
\[
L_{n\alpha}=nH,
\qquad L_j|_G=\mathcal O_G\bigl(j,n(\beta-\alpha)+j\bigr).
\]
Serre vanishing gives $H^1(\widehat T,L_{n\alpha})=0$.
For $0\le j=Latex,
every node/.style={font=\small},
redline/.style={arrred,line width=.8pt},
blueline/.style={arrblue,dashed,line width=.8pt},
redpoint/.style={circle,fill=arrred,draw=arrred,inner sep=1.8pt},
bluepoint/.style={rectangle,fill=arrblue,draw=arrblue,inner sep=1.8pt}]
\node at (2.55,4.55) {The six marked directions};
\draw[->,black!65] (0,2.6)--(5.1,2.6);
\node[right] at (5.1,2.6) {$\lambda$};
\draw[black!45,rounded corners=9pt] (1.35,2.12) rectangle (3.75,3.08);
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\node[redpoint] at (\x,2.6) {};
\node[bluepoint] at (.35,2.6) {};
\node[bluepoint] at (4.75,2.6) {};
\node[align=center,arrred] at (2.55,3.6)
{four red points\\$|\lambda_i|<0.01$};
\node[align=center,arrblue] at (2.55,1.45)
{two blue points: $|\lambda_i|>4$};
\node[align=center] at (2.55,.45)
{$C\longrightarrow\mathbb P^1_{\lambda}$\\double cover branched at these points};
\begin{scope}[shift={(9.15,2.25)}]
\node at (0,2.3) {One pencil and its couplings};
\draw[black!40,line width=.6pt] (0,0) circle (2.05);
\foreach \angle in {-13,-4,4,13}
\draw[redline] (\angle:2.05)--({\angle+180}:2.05);
\foreach \angle in {78,102}
\draw[blueline] (\angle:2.05)--({\angle+180}:2.05);
\draw[line width=1pt] (-1.78,-1.017)--(-1.78,1.017);
\draw[line width=1pt] (1.78,-1.017)--(1.78,1.017);
\foreach \angle in {-13,-4,4,13}{
\fill[arrred] (1.78,{1.78*tan(\angle)}) circle (1.9pt);
\fill[arrred] (-1.78,{-1.78*tan(\angle)}) circle (1.9pt);
}
\fill (0,0) circle (2pt);
\node[fill=white,inner sep=1.5pt,below=8pt] at (0,0)
{$L_z$};
\node[align=center,below=4pt,fill=white,inner sep=1.5pt] at (-1.78,-1.017)
{$z_1=-c$};
\node[align=center,below=4pt,fill=white,inner sep=1.5pt] at (1.78,-1.017)
{$z_1=c$};
\node[arrblue,fill=white,inner sep=1pt] at (.86,1.64) {blue};
\node[arrred,fill=white,inner sep=1pt] at (.92,.57) {red};
\node at (0,-2.55) {$z_2=\lambda_i z_1$};
\end{scope}
\end{tikzpicture}
\endgroup
\caption{The finite arrangement in one coordinate pair. Left: a
schematic placement of the six branch points on the real direction
axis, grouped into four red circles and two blue squares. Right:
the slice $w=0$, $z_1,z_2\in\mathbb R$ of the $z$ pencil in the
ball; solid red slopes meet both coupling
hyperplanes, whereas dashed blue slopes meet neither. The red
angular separation is exaggerated to show these incidences.
The center represents the slice of $L_z$. The $w$ family is an
identical construction in the other coordinate pair; the full
arrangement lies in complex dimension four.}
\label{fig:arrangement}
\end{figure}
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//arrangement.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//blowdown.tex ====
\begin{figure}[htbp]
\centering
\begingroup
\definecolor{blowaccent}{RGB}{22,91,119}
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,
every node/.style={font=\small},
marked/.style={blowaccent,line width=1.5pt}]
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\coordinate (se) at (7.8,.65);
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\draw[black!23] ($(sw)!\t!(nw)$)--($(se)!\t!(ne)$);
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}
\draw[black!55] (sw)--(se)--(ne)--(nw)--cycle;
\coordinate (el) at ($(sw)!.5!(nw)$);
\coordinate (er) at ($(se)!.5!(ne)$);
\draw[marked] (el)--(er);
\node[above=6pt] at (6.65,3.25)
{$G=\mathbb P^1_{\mathrm{curve}}\times\mathbb P^1_{\mathrm{normal}}$};
\node[fill=white,inner sep=2pt,text=blowaccent] at (6.15,1.95)
{$\widehat Y\cap G=E\times\{q\}$};
\node[below=4pt] at (5.7,.65) {curve direction};
\draw[marked] (0,.7)--(2.05,.7);
\node[below=5pt] at (1.025,.7) {$E\subset Y_1\subset T$};
\draw[->,line width=.75pt] (3.45,1.65)--(1.65,1.05);
\node[align=center] at (1.6,2.35)
{original blowdown\\$G\longrightarrow\mathbb P^1_{\mathrm{curve}}$};
\draw[black!65,line width=.8pt] (11.1,.75)--(11.7,2.65);
\fill[blowaccent] (11.4,1.7) circle (2.8pt);
\node[right=5pt,align=left] at (11.4,1.7) {$q$\\in $Y_0$};
\draw[->,line width=.75pt] (8.7,1.9)--(10.7,1.9);
\node[align=center] at (10.35,3.65)
{other blowdown\\$G\longrightarrow\mathbb P^1_{\mathrm{normal}}$};
\node[below=6pt,align=center] at (11.1,.6)
{image curve\\in $T'$};
\end{tikzpicture}
\endgroup
\caption{The two rulings of the last exceptional divisor in
$\widehat T$. The original blowdown contracts the normal-direction
fibers and leaves the curve $E$ in $Y_1$ unchanged. The other
blowdown, whose target threefold $T'$ is constructed below,
contracts the horizontal ruling, including
$\widehat Y\cap G=E\times\{q\}$, so the surface undergoes its
point blowdown $Y_1\to Y_0$. The parallelogram records only the
product structure of $G$, whose normal bundle is
$\mathcal O_G(-1,-1)$.}
\label{fig:blowdown}
\end{figure}
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//blowdown.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//main.tex ====
\newcommand{\DoNotLoadEpstopdf}{}
\documentclass[11pt,reqno]{amsart}
\input{preamble}
\title[Large fundamental group and non-Stein universal cover]{A projective fourfold with large fundamental group\protect\\
and non-Stein universal cover}
\author{OpenAI}
\date{October 5, 2026}
\subjclass[2020]{32Q30, 14F35, 20F67, 14K12}
\keywords{Shafarevich conjecture, universal cover, large fundamental group,
complex hyperbolic arrangement, Stein manifold}
\begin{document}
\begin{abstract}
We construct a smooth projective complex fourfold with large fundamental group
whose universal cover is not Stein. The universal cover contains no
positive-dimensional compact complex-analytic subvariety, so this disproves
Shafarevich's holomorphic-convexity conjecture even under the
large-fundamental-group hypothesis.
\end{abstract}
\maketitle
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-reduction}
\input{sections/03-paths}
\input{sections/04-geometry}
\input{sections/05-models}
\input{sections/06-image}
\input{sections/07-projective}
\bibliographystyle{amsplain}
\bibliography{references}
\end{document}
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//main.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//path-ladder.tex ====
\begin{figure}[htbp]
\centering
\begingroup
\definecolor{ladderaccent}{RGB}{22,91,119}
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,
every node/.style={font=\small},
guide/.style={line width=.9pt},
oldrung/.style={densely dashed,line width=.65pt,black!65},
newrung/.style={line width=1.1pt,ladderaccent}]
\coordinate (am) at (0,2.05);
\coordinate (ai) at (5.4,2.05);
\coordinate (ap) at (10.8,2.05);
\coordinate (bm) at (0,.2);
\coordinate (bo) at (6.7,.2);
\coordinate (bn) at (5.4,.2);
\coordinate (bp) at (10.8,.2);
\draw[guide] (am)--(ap);
\draw[guide] (bm)--(bp);
\draw[oldrung] (am)--node[left] {$\le h$} (bm);
\draw[oldrung] (ap)--node[right] {$\le h$} (bp);
\draw[oldrung] (ai)--node[pos=.50,right,fill=white,inner sep=1.5pt]
{old: $\le h$} (bo);
\draw[newrung] (ai)--node[pos=.55,left,fill=white,inner sep=1.5pt]
{new: $\le 2\Delta$} (bn);
\foreach \p in {am,ai,ap,bm,bo,bp}
\fill (\p) circle (1.7pt);
\draw[newrung,fill=white] (bn) circle (2.2pt);
\node[above=4pt] at (am) {$x_{i-1}$};
\node[above=4pt] at (ai) {$x_i$};
\node[above=4pt] at (ap) {$x_{i+1}$};
\node[below=5pt] at (bm) {$b_{i-1}$};
\node[below=5pt] at (bn) {$b_i'$};
\node[below=5pt] at (bo) {$b_i$};
\node[below=5pt] at (bp) {$b_{i+1}$};
\node[left=18pt] at (am) {$\alpha$};
\node[left=18pt] at (bm) {$\beta$};
\draw[decorate,decoration={brace,amplitude=4pt}]
(0,2.76)--node[above=5pt] {length in $[L,2L]$} (5.4,2.76);
\draw[decorate,decoration={brace,amplitude=4pt}]
(5.4,2.76)--node[above=5pt] {length in $[L,2L]$} (10.8,2.76);
\end{tikzpicture}
\endgroup
\caption{Narrowing a ladder in a single chart. Both displayed guide
pieces read as geodesics. The interior mark $x_i$ is far from the
end rungs, so quadrangle slimness supplies a new mate $b_i'$ on
the opposite side, within $2\Delta$ of $x_i$. The old scheduled
mate $b_i$ is at distance at most $h+2\Delta$ from $b_i'$ along
that side. The drawing is
schematic: the long side intervals and the two rung bounds use
different visual scales.}
\label{fig:path-ladder}
\end{figure}
==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//path-ladder.tex ====
==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//preamble.tex ====
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==== BEGIN A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//references.bib ====
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==== END A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover//references.bib ====