% This is a plain tex file:
% 
% branched.tex
%
% Daniel Allcock

%					
%	dja2.tex	
%	Daniel Allcock   (allcock@math.harvard.edu)
%						
%	A set of macros for plain tex
%

% FONTS
%
% calligraphic font
\newfam\calfont
\font\tencal=eusm10 \font\sevencal=eusm7 \font\fivecal=eusm5
\textfont\calfont=\tencal 
\scriptfont\calfont=\sevencal 
\scriptscriptfont\calfont=\fivecal
\def\cal{\fam=\calfont}	
% fraktur
\newfam\frakturfont
\font\tenfrak=eufm10 \font\sevenfrak=eufm7 \font\fivefrak=eufm5
\textfont\frakturfont=\tenfrak 
\scriptfont\frakturfont=\sevenfrak 
\scriptscriptfont\frakturfont=\fivefrak
\def\frak{\fam=\frakturfont}
% blackboard bold (includes many math symbols)
\newfam\bbbfont
\font\tenbbb=msbm10 \font\sevenbbb=msbm7 \font\fivebbb=msbm5	
\textfont\bbbfont=\tenbbb 
\scriptfont\bbbfont=\sevenbbb 
\scriptscriptfont\bbbfont=\fivebbb
\def\bbb{\fam=\bbbfont}

% `IF NOT DEFINED' :
%
% use it like a regular \if
% you can use \else and must use \fi
\long\def\ifndef#1{\expandafter\ifx\csname#1\endcsname\relax}

% MAKING DEFINITIONS THAT \ifndef CAN DETECT
%
\def\DEF#1#2{\expandafter\def\csname #1\endcsname{#2}}
\def\GDEF#1#2{\expandafter\global\expandafter\def\csname #1\endcsname{#2}}
\def\LDEF#1#2{\expandafter\long\expandafter\def\csname #1\endcsname{#2}}

% WARNINGS
%
\def\WARNING#1{\immediate\write16{l.\the\inputlineno: #1}}

% USE \bye TO TELL TeX TO QUIT
\let\ORIGINALend\end
\outer\def\bye{%
  \AnnounceUnclosedEnvironments
  \addressblock
  \par
  \vfill
  \supereject
  \ORIGINALend
}

% ENVIRONMENTS
%
% \defenv{name of environment}{stuff at beginning}{stuff at end}
\long\def\defenv#1#2#3{\LDEF{ENVBEGIN#1}{#2}\LDEF{ENVEND#1}{#3}}
% for example, \begin{titleblock}
\def\begin#1{%
  \begingroup
  \DEF{CURRENVNAME}{#1}%
  \expandafter\def\expandafter\CURRENVLINE\expandafter{\the\inputlineno}%
  \ifndef{ENVBEGIN#1}%
    \WARNING{entering undefined environment `#1'}%
    \DEF{ENVBEGIN#1}{}%
  \fi
  \csname ENVBEGIN#1\endcsname
}
% for example, \end{titleblock}
\def\end#1{%
  \def\TEMPENVNAME{#1}%
  \ifndef{CURRENVNAME}%
    \WARNING{extra end{#1}; ignored}%
  \else
    \checklastenvironment{#1}
  \fi
}
% used in \end{}; separated out to avoid nesting
\def\checklastenvironment#1{%
  \ifx\CURRENVNAME\TEMPENVNAME
  \else
    \WARNING{end{#1} after begin{\CURRENVNAME} on line \CURRENVLINE}%
  \fi
  \ifndef{ENVEND#1}%
    \WARNING{leaving undefined environment `#1'}%
    \DEF{ENVEND#1}{}%
  \fi
  \csname ENVEND#1\endcsname
  \endgroup
  \ignorespaces
}
% called by \bye to report unclosed envirnoments
\def\AnnounceUnclosedEnvironments{%
  \ifndef{CURRENVNAME}%
  \else
    \WARNING{unterminated environment `\CURRENVNAME' from line \CURRENVLINE}%
    \endgroup
    \AnnounceUnclosedEnvironments
  \fi
}

% TITLE BLOCK
%
\defenv{titleblock}{\begintitleblock}{\endtitleblock}
%
\def\begintitleblock{%
  \long\def\title##1{\GDEF{PAPERtitle}{##1}}%
  \long\def\runningtitle##1{\GDEF{PAPERrunningtitle}{##1}}%
  \long\def\author##1{\GDEF{PAPERauthor}{##1}}%
  \long\def\support##1{\GDEF{AUTHORsupport}{##1}}%
  \long\def\runningauthor##1{\GDEF{PAPERrunningauthor}{##1}}%
  \long\def\date##1{\GDEF{PAPERdate}{##1}}%
  \long\def\subject##1{\GDEF{PAPERsubject}{##1}}%
  \long\def\keywords##1{\GDEF{PAPERkeywords}{##1}}%
  \long\def\note##1{\GDEF{PAPERnote}{##1}}%
  \long\def\address##1{\GDEF{AUTHORaddress}{##1}}%
  \long\def\email##1{\GDEF{AUTHORemail}{##1}}%
  \long\def\homepage##1{\GDEF{AUTHORhomepage}{##1}}%
}
% These three macros are here to avoid nested \if's
\def\assignPAPERrunningtitle{%
  \ifndef{PAPERrunningtitle}%
    \GDEF{PAPERrunningtitle}{\PAPERtitle}%
  \fi
}
\def\displayAUTHORsupport{%
  \ifndef{AUTHORsupport}%
  \else
    \footnote{*}{\AUTHORsupport}%	
  \fi
}
\def\assignPAPERrunningauthor{%
  \ifndef{PAPERrunningauthor}%
    \GDEF{PAPERrunningauthor}{\PAPERauthor}%
  \fi
}
%
\def\endtitleblock{%
  \parindent=0pt
  \ifndef{PAPERtitle}%
    \WARNING{untitled manuscript}
  \else
    \par{\bf\PAPERtitle}%
    \assignPAPERrunningtitle	
    \medskip
  \fi
  \ifndef{PAPERauthor}%
    \WARNING{unauthored manuscript}%
  \else
    \par\PAPERauthor
    \displayAUTHORsupport
    \assignPAPERrunningauthor
  \fi
  \ifndef{PAPERdate}%
    \WARNING{undated manuscript}%
  \else
    \par\PAPERdate
  \fi
  \ifndef{PAPERsubject}%
  \else
    \footnote{}{2000 mathematics subject classification: \PAPERsubject}
  \fi
  \ifndef{PAPERkeywords}%
  \else
    \footnote{}{Keywords: \PAPERkeywords}
  \fi
  \vskip-\lastskip
  \medskip
  \ifndef{PAPERnote}%
  \else
    \par\PAPERnote
  \fi
  \vskip-\lastskip
  \bigskip
  \ifndef{PAPERrunningtitle}%
    \GDEF{PAPERrunningtitle}{}%
  \fi
  \ifndef{PAPERrunningauthor}%
    \GDEF{PAPERrunningauthor}{}%
  \fi
%  \global\headline={%
%    \ifodd\pageno
%      \tenrm	
%      \PAPERrunningtitle
%      \hfil
%    \else
%      \tenrm
%      \hfil
%      \PAPERrunningauthor
%    \fi
%  }
}

% ADDRESS BLOCK
%
% These 3 macros are here to avoid nested \if's in \addressblock
\def\displayAUTHORaddress{%
    \ifndef{AUTHORaddress}%
    \else
      \par\AUTHORaddress
    \fi
}
\def\displayAUTHORemail{%
  \ifndef{AUTHORemail}%
  \else
    \par{\it\AUTHORemail}%
  \fi
}
\def\displayAUTHORhomepage{%
  \ifndef{AUTHORhomepage}%
  \else
    \par{\it\AUTHORhomepage}%
  \fi
}
%
\def\addressblock{%
  % we'll only make the address block if there is enough data:
  % an author and  one of: physical address, email and homepage
  \let\letsdoit\relax
  \ifndef{AUTHORaddress}
  \else
    \def\letsdoit{yes}
  \fi
  \ifndef{AUTHORemail}
  \else
    \def\letsdoit{yes}
  \fi
  \ifndef{AUTHORhomepage}
  \else
    \def\letsdoit{yes}
  \fi
  \ifndef{PAPERauthor}%
    \let\letsdoit\relax
  \fi
  \ifndef{letsdoit}
  \else
    \parindent=0pt
    \bigskip\PAPERauthor
    \par\displayAUTHORaddress
    \par\displayAUTHORemail
    \par\displayAUTHORhomepage
  \fi
}

% ABSTRACT
%
\defenv{abstract}{\beginabstract}{}
\long\def\beginabstract{%
  \bigskip
  \noindent
  {\bf Abstract.}%
  \par
  \noindent
  \ignorespaces
}

% SECTIONS
\let\sectionsymbol\S
%
\def\beginsection#1#2\par{%
  \vskip\baselineskip\penalty-250\vskip-\baselineskip % extra line if needed
  \bigskip
  \leftline{\bf\sectionsymbol\refno{#1}. #2}%
  \nobreak\bigskip\noindent
}
\defenv{section}{\beginsection}{}

% THEOREMS, etc
%
% #1 the code to be used in the file
% #2 the word to appear in the text
\def\defproclaim#1#2{%
  \expandafter\def\csname beginproclaim#1\endcsname##1{%
    \medbreak
    \noindent
    {\bf #2~\refno{##1}.\enspace}%
    \it\ignorespaces
  }%
  \defenv{#1}{\csname beginproclaim#1\endcsname}{\endproclaim}%
}
%
\def\endproclaim{%
  \ifdim\lastskip<\medskipamount 
    \removelastskip
    \penalty55\medskip
  \fi
  \noindent
}
%
\defproclaim{lemma}{Lemma}
\defproclaim{theorem}{Theorem}
\defproclaim{proposition}{Proposition}
\defproclaim{corollary}{Corollary}
\defproclaim{conjecture}{Conjecture}

% PROOFS
%
% usage: \begin{proof}{Proof}
%     or \begin{proof}{Proof of theorems foo and bar}
\def\beginproof#1{%
  \ifdim\lastskip<\smallskipamount
    \removelastskip
    \smallskip
  \fi
  {\it #1:\/}%
}
%
\def\endproof{\qedbox\smallskip}
\defenv{proof}{\beginproof}{\endproof}

% THE BOX FOR THE END OF A PROOF
%
% syntax: ``\drawbox{internal width}{internal height}{line
% 	thickness}''; arguments should be dimens.
\def\drawbox#1#2#3{%
  \leavevmode
  \vbox{%
    \hrule height #3%
    \hbox{%
      \vrule width #3 height #2%
      \kern #1%
      \vrule width #3%
    }%
    \hrule height #3%
  }%
}
%
\def\drawqedbox{\drawbox{1.2ex}{1.2ex}{.1ex}}
\def\qedbox{%
  \leavevmode
  \vrule height0pt width0pt depth0pt
  \nobreak\hfill
  \drawqedbox
  \smallskip
}
\let\qed=\qedbox

% REMARKS and EXAMPLES
%
% usage: \begin{remark}{Remark}
%     or \begin{remark}{Remarks on the pineapple conjecture}
% similarly for \begin{example}{...}
\def\skipbeforeremark{%
  \ifdim\lastskip<\smallskipamount
    \removelastskip
    \smallskip
  \fi
}
%
\def\beginremark#1{\skipbeforeremark{\it #1:\/}}
\defenv{remark}{\beginremark}{\smallskip}
\defenv{example}{\beginremark}{\smallskip}

% FIGURES AND TABLES
%
\let\captionindent=\parindent
%
\def\beginfigureortable#1{%
  \def\caption##1{%
    \advance\leftskip by\captionindent
    \advance\leftskip by\captionindent
    \advance\rightskip by\captionindent
    \advance\rightskip by\captionindent
    \vskip 6pt plus 2pt minus 6pt % a shrinkable medskip
    \noindent
    {\bf\Ref{#1}.}
    ##1%
    \par
  }%
}
%
\defenv{figure}{\beginfigureortable}{}
\defenv{table}{\beginfigureortable}{}

% LISTS
%
% for reference to parts of theorems, such as cases, hypotheses,
% or conclusions. Or for reference to elements of other lists.
%
% syntax: \begin{list}{(iv)} if (iv) is the widest thing on your
% list of things. Then \item{abstract-name-for-crossreferences} ...
\def\beginlist#1{%
  \setbox0=\hbox{#1\enspace}%
  \ifnum\parindent<\wd0
    \parindent=\wd0
  \fi
  \let\ORIGINALitem=\item
  \def\item##1{%
    \ORIGINALitem{\refno{##1}}
  }%
}
\def\endlist{\par}
\defenv{list}{\beginlist}{\endlist}
%
% for convenience; use in lists
\def\rom#1{(\romannumeral #1\relax)}
\def\Rom#1{{\uppercase\expandafter{\romannumeral #1}}}

% CROSS-REFERENCES
%
% #1: the code for cross-references in the soure file
% #2: the type of the item (for use at beginning of sentence)
% #3: the type of the item (for use mid-sentence)
% #4: the number or other label of the item
\def\defref#1#2#3#4{%
  \ifndef{#1NUM}%
  \else
    \WARNING{new definition of reference `#1'}%
  \fi
  \redefref{#1}{#2}{#3}{#4}%
}
%
\def\redefref#1#2#3#4{%
  \GDEF{#1NUM}{#4}%
  \GDEF{#1type}{#3}%
  \GDEF{#1TYPE}{#2}%
}
%
% \Ref and \ref are the usual ways to get at cross-reference info.
% \refno, \reftype and \refType can be used to extract the
% pieces of information seperately.
%
\def\undefinedref#1{\WARNING{undefined reference `#1'}}
\def\Ref#1{%
  \ifndef{#1NUM}%
    {\bf ???}%
    \undefinedref{#1}% 
  \else
    \csname #1TYPE\endcsname~\csname #1NUM\endcsname
  \fi
}
%
\def\ref#1{%
  \ifndef{#1NUM}%
    {\bf ???}%
    \undefinedref{#1}% 
  \else
    \csname #1type\endcsname~\csname #1NUM\endcsname
  \fi
}
%
\def\refno#1{%
  \ifndef{#1NUM}%
    {\bf ??}%
    \undefinedref{#1}% 
  \else
    \csname #1NUM\endcsname
  \fi
}
%
\def\reftype#1{%
  \ifndef{#1NUM}%
    {\bf ??}%
    \undefinedref{#1}% 
  \else
    \csname #1type\endcsname
  \fi
}
%
\def\refType#1{%
  \ifndef{#1NUM}%
    {\bf ??}%
    \undefinedref{#1}% 
  \else
    \csname #1TYPE\endcsname
  \fi
}

% REFERENCES TO EQUATIONS
%
\def\eqref#1{{\rm(\refno{#1})}}
\def\eqlabel#1{\eqref{#1}}
\def\equation#1{\eqno\eqlabel{#1}}

% CITATIONS
%
% #1: the code for the citation in the file
% #2: the number or other label to appear in the final text
\def\defcite#1#2{%
  \ifndef{#1CITENUM}%
  \else
    \WARNING{new definition of citation `#1'}%
  \fi
  \GDEF{#1CITENUM}{#2}%
}
%
\def\cite#1{%
  [%
  \ifndef{#1CITENUM}%
    {\bf ??}%
    \WARNING{undefined citation `#1'}%
  \else
    \csname #1CITENUM\endcsname
  \fi
  ]%
}
%
\def\ecite#1#2{%
  [%
  \ifndef{#1CITENUM}%
    {\bf ??}%
    \WARNING{undefined citation `#1'}%
  \else
    \csname #1CITENUM\endcsname
  \fi
  , #2]%
}
%
\def\nocite#1{}

% BIBLIOGRAPHY
% usage: \begin{bibliography}{99}
%
\def\beginbibliography#1{%
  \setbox0=\hbox{[#1]\enspace}%
  \parindent=\wd0
  \vskip\baselineskip\penalty-250\vskip-\baselineskip  % extra line if needed
  \bigskip
  \leftline{\bf Bibliography}%
  \nobreak\bigskip
  \def\bibitem##1{%
    \item{\cite{##1}}%
  }%
}
%
\defenv{bibliography}{\beginbibliography}{\bigskip}

%--------------------------end-of-dja2.tex--------------------------

% MISC MATH STUFF
\def\({\left(} 
\def\){\right)}
\def\Z{{\bbb Z}} % integers
\def\R{{\bbb R}} % real numbers
\def\C{{\bbb C}} % complex numbers
\let\sset=\subseteq		
\let\tensor=\otimes	
\let\isomorphism=\cong%		
%
% an \llap that you can use inside formulas without
% tex changing to textstyle inside the \hbox of \llap
\def\mathllap#1{%
  \mathchoice
    {\llap{$\displaystyle #1$}}%
    {\llap{$\textstyle #1$}}%
    {\llap{$\scriptstyle #1$}}%
    {\llap{$\scriptscriptstyle #1$}}%
}
%
%the set of all #1 such that #2; makes the braces and the
%middle bar big enough to accomodate everything.
\def\set#1#2{%
  \left\{%
  \,%
  #1%
  \mathllap{\phantom{#2}}%
  \mathrel{}%
  \right|%
  \left.%
  #2%
  \mathllap{\phantom{#1}}%
  \,%
  \right\}%
}


\parskip=0pt
\magnification=1100
\baselineskip=15pt

% computer-maintained bibliographic references
\defcite{allcock:ch4-cubic-moduli}{1}
\defcite{allcock:period-lattice-for-enriques-surfaces}{2}
\defcite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}{3}
\defcite{charney-davis:branched-covers-of-riemannian-manifolds}{4}
\defcite{charney-davis:K-pi-1-problem-for-hyperplane-complements}{5}
\defcite{ghys:sur-les-groupes-hyperboliques}{6}
\defcite{gromov:hyperbolic-groups}{7}
\defcite{gromov-thurston:pinching-constants-for-hyperbolic-manifolds}{8}
\defcite{horikawa:periods-of-enriques-surfaces-I}{9}
\defcite{horikawa:periods-of-enriques-surfaces-II}{10}
\defcite{namikawa:periods-of-enriques-surfaces}{11}
% end of computer-maintained bibliographic references

% computer-maintained cross-reference definitions
\defref{#1}{}{}{0.1}
\defref{intro}{Section}{section}{1}
\defref{sec-br-covers}{Section}{section}{2}
\defref{main-theorem}{Theorem}{theorem}{2.1}
\defref{thm-iterated-br-cover}{Theorem}{theorem}{2.2}
\defref{punctured-balls-are-path-connected}{Lemma}{lemma}{2.3}
\defref{sec-apps}{Section}{section}{3}
\defref{thm-hyperplane-complements-aspherical}{Theorem}{theorem}{3.1}
\defref{thm-neighborhoods-described}{Lemma}{lemma}{3.2}
\defref{thm-weak-homotopy-equiv}{Lemma}{lemma}{3.3}
\defref{cubic-surfaces}{Corollary}{corollary}{3.4}
\defref{enriques-surfaces}{Corollary}{corollary}{3.5}
% end of computer-maintained cross-reference definitions

% stuff for intro
\def\cp{\C P}		  % complex projective space
\def\ch{\C H}	  	  % complex hyperbolic space
\def\moduli{{\cal M}} 	  % moduli space
\def\PGL{{\rm PGL}}	  % projective linear group
\def\U{{\rm U}}	          % unitary group


% stuff for background
\def\x{\kappa}		  % quick greek letter
\def\M{M_{\x}^2}	  % comparison space
\def\sx{\sqrt\x}	  % abbrev

% stuff for simple branched covers
\def\Xhat{{\widehat X}}	  % base space
\def\Yhat{{\widehat Y}}	  % branched cover
\def\D{\Delta}		  % branch locus
\def\Dtil{\tilde{\D}}	  % preimage of branch locus
\def\cat{CAT($\x$)}	  % abbreviation
\def\diam{\mathop{\rm Diam}\nolimits}  % diameter of a set
\def\G{\Gamma}		  % a homotopy
\def\btil{\tilde{\b}}	  % a lifted path
\def\dtil{\tilde{\d}}	  % a lifted path
\def\ytil{\tilde y}	  % a lifted point
\def\geo#1{\overline{#1}} % geodesic joining 2 points
\def\tri#1{\triangle#1}	  % geodesic triangle
\def\Util{\tilde{U}}	  % a ball in the cover
\def\cA{({\bf A})}	  % 7 cases in a proof
\def\cB{({\bf B})}	  
\def\cC{({\bf C})}	  
\def\cD{({\bf D})}	  
\def\cE{({\bf E})}	  
\def\cF{({\bf F})}	  
\def\cG{({\bf G})}	  

% stuff for the proof that triangles with edge in \D satisfy \cat
\def\Tbar{\bar{T}}
\def\Ubar{\bar{U}}
\def\Kbar{\bar{K}}
\def\Bbar{\bar{B}}
\def\abar{\bar{a}}
\def\Abar{\bar{A}}
\def\atil{\tilde{\a}}
\def\xbar{\bar{x}}

% stuff for iterated covers
\def\S{{\cal S}}  % union of vector subspaces
\def\H{{\cal H}}  % union of submanifolds
\def\Htil{{\widetilde{\cal H}}}  % preimage of union of submanifolds
\def\s{\sigma}    % generator of \pi_1
\def\Mhat{{\hat M}} % base manifold
\def\Nhat{{\hat N}} % branched cover
\def\pihat{{\hat\pi}} % branched covering map
\def\xtil{{\tilde x}} % point in Ntil above x
\def\ytil{{\tilde y}} % point in Ntil above y
\def\ztil{{\tilde z}} % point in Ntil above z
\def\ctil{{\tilde\gamma}} % path in Ntil
\def\Vtil{{\widetilde V}} % a neighborhood of \xtil
\def\Btil{{\widetilde B}} % a closed neighborhood of \xtil
\def\Atil{{\widetilde A}} % a branched cover of a standard disk
\def\Gammatil{{\widetilde\Gamma}} % lifted homotopy

% stuff for applications
\def\w{\omega} % a cube root of 1
\def\E{{\cal E}} % eisenstein numbers
\def\I{{\cal I}} % the hyperplanes passing through a given point
\def\DD{{\cal D}} % symmetric space for O(2,10)

% misc stuff
\def\a{\alpha}
\def\b{\beta}
\def\c{\gamma}
\def\d{\delta}
\def\e{\varepsilon}
\def\image{\mathop{\rm image}\nolimits}

%-----------------Beginning-of-Document---------------------------

\begin{titleblock}
  \title   {Asphericity of moduli spaces via curvature} 
%  \title   {Metric curvature of infinite branched covers}
  \author  {Daniel Allcock}
  \support {Supported in part by an NSF	Postdoctoral Fellowship}
  \date    {26 March 2001}
% previous versions
%
% nov 13, 2000
% october 15, 2000
% July 24, 2000
% 25 May 1999
% 14 May 1999
%  2 May 1999
% 14 Feb 1999
%  9 Oct 1998
  \address {Department of Mathematics, Harvard University,
		  Cambridge, MA 02138}
  \email   {allcock@math.harvard.edu}
  \homepage{http://www.math.harvard.edu/$\sim$allcock}
  \subject {53C23 (14J28, 57N65)}
  \keywords{branched cover, ramified cover, Alexandrov space,
		  cubic surface, Enriques surface}
%  \note    {}
\end{titleblock}

\begin{abstract}
We show that 
under suitable conditions a branched cover satisfies the same upper
curvature bounds as its base space. First we do this when the
base space is a metric space satisfying
Alexandrov's curvature condition \cat\ and the branch locus is 
complete and convex. Then we treat  branched covers of
a Riemannian manifold over suitable mutually
orthogonal submanifolds. In neither setting do we require that the
branching be locally finite. We apply our results to hyperplane
complements in several Hermitian symmetric spaces of nonpositive
sectional curvature in order to prove that two moduli spaces arising
in algebraic geometry are aspherical. These are the moduli spaces of
the smooth cubic surfaces in $\cp^3$ and of the 
smooth complex Enriques surfaces.
\end{abstract}

\begin{section}{intro}{Introduction}

It is well-known that taking branched covers
usually introduces negative curvature. One
can see this phenomenon in elementary examples using Riemann
surfaces, and the idea also plays a role in the construction
\cite{gromov-thurston:pinching-constants-for-hyperbolic-manifolds}
of exotic manifolds with negative sectional curvature. In this
paper we work in the setting of Alexandrov's comparison geometry;
for background see
\cite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}.
In this setting we will establish 
the persistence of upper curvature bounds in
branched covers. 
A simple way to build a cover $\Yhat$ of a space $\Xhat$
branched over $\D\sset\Xhat$ is to take any covering space $Y$
of $\Xhat-\D$ and define $\Yhat=Y\cup\D$. We call $\Yhat$ a
simple branched cover of $\Xhat$ over $\D$. Our first result,
\ref{main-theorem}, states that if $\Xhat$
satisfies Alexandrov's
\cat\ condition and $\D$ is complete and satisfies a convexity 
condition then the natural
metric on $\Yhat$ also satisfies \cat.
%We include examples showing that an important
%completeness hypothesis cannot be dropped; our examples also
%disprove several claims in the literature.

The question which motivated this investigation is whether the
moduli space of smooth cubic surfaces in $\cp^3$ is aspherical
(i.e., has contractible universal cover). It is, and
our argument also establishes the analogous result for the
moduli space of smooth complex Enriques surfaces. To prove these
claims, we use the fact that each of these moduli spaces is
known to be covered by a Hermitian symmetric space with
nonpositive sectional curvature, minus an arrangement of complex
hyperplanes. In each case the hyperplanes have the property that
any two of them are orthogonal wherever they meet. In
\ref{sec-apps} we show that such a hyperplane complement
is aspherical. The result
(\ref{thm-hyperplane-complements-aspherical}) is more general
because the symmetric space structure is not needed.
Theorem~5.3 of
\cite{charney-davis:branched-covers-of-riemannian-manifolds}
morally contains this result and also suggests a substantial
generalization of it. However, there are some difficulties with the proof. 

If $\Mhat$ is the symmetric space and $\H$ is the union of the
hyperplanes, then the idea is to apply standard nonpositive
curvature techniques like the Cartan-Hadamard theorem to the
universal cover $N$ of $M=\Mhat-\H$. The problem is that $N$ is
not metrically complete. One can pass to its metric completion
$\Nhat$, but this introduces problems of its own. First there is
the issue of how $N$ and $\Nhat$ are related. We resolve this by
a simple trick that shows that the inclusion $N\to\Nhat$ is a
homotopy equivalence. The second problem is that $\Nhat$ is not
a manifold and not even locally compact, so that one cannot use
the techniques of Riemannian geometry. But it is still a metric
space and it turns out to satisfy Alexandrov's CAT(0) condition
locally. It is then easy to show that $N$ and $\Nhat$
are contractible.

In order to understand this curvature bound for $\Nhat$, 
the reader should imagine a closed ball $B$ in 
$\C^n$, equipped with
some Riemannian metric, minus the coordinate
hyperplanes. The metric completion of the universal cover of the
hyperplane complement can be obtained by first taking a simple
branched cover of $B$ over one hyperplane, then taking a
simple branched cover of this branched cover over (the preimage
of) the second hyperplane, and so on. If the hyperplanes are
mutually orthogonal and totally geodesic then \ref{main-theorem}
may be used inductively to bound the curvature of the iterated
branched cover.  Observe that the base space of each branched cover 
fails to be locally compact except in the first step. 
This means that the inductive argument
requires a theorem treating branched covers of
spaces more general than manifolds.
We also note that the condition of mutual orthogonality in branched
covers has appeared before, for example for the
modified Deligne complexes of
\cite{charney-davis:K-pi-1-problem-for-hyperplane-complements},
which are certain metric polyhedral complexes of piecewise
constant curvature. In fact, in this polyhedral setting our results are
already well-established. We need to go beyond the piecewise
constant curvature case for the applications to algebraic
geometry.

For the reader's convenience we recall some definitions from
\cite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}. $D_\x$ is the diameter of the simply-connected complete
surface of constant sectional curvature $\x$. A metric space $X$
is $D_\x$-geodesic if any two points at distance~$<D_\x$ are
joined by a geodesic. A subset $\D$ of $X$ is called
$D_\x$-convex if any two points of $\D$ at distance~$<D_\x$ are
joined by a geodesic of $X$, and every geodesic joining them lies in
$\D$. We say that $X$ is \cat\ if it is $D_\x$-geodesic and any two
points on any geodesic triangle in $X$ with perimeter~$<2D_\x$
satisfy Alexandrov's inequality. We write $\ell(\c)$ for the
length of a path $\c$.

I would like to thank Jim Carlson and Domingo Toledo for their
interest in this work, and for the collaboration
\cite{allcock:ch4-cubic-moduli} that suggested these problems. I
would also like to thank Richard Borcherds, Misha Kapovich and Bruce
Kleiner for helpful conversations, Brian Bowditch for pointing out an
error in an early version, and the referee for suggesting many
improvements.  This paper was distributed in preprint form under the
title ``Metric curvature of infinite branched covers''.

\end{section}

\begin{section}{sec-br-covers}{Branched covers}

We begin by showing that a branched cover satisfies the same upper bounds on
curvature as its base space. 
We treat what
we call simple branched covers. The idea is that
one removes a closed subset $\D$ from a length space $\Xhat$,
takes a cover of what is left, and then attaches a copy of $\D$
in the obvious way. More precisely, if $Y$ is any covering of
$X=\Xhat-\D$, then each component of $Y$ carries a unique length
metric under which projection to $X$ is a local isometry. We
take $\Yhat=Y\cup\D$ and write $\pi:\Yhat\to\Xhat$ for the
obvious projection map. If $x,z\in\Yhat$ then we define
$$ 
d(x,z)=
\inf\Bigl(
\set{d(\pi x,\pi y)+d(\pi y,\pi z)}{y\in\D}
\;\cup\;
\set{\ell(\c)}{\hbox{$\c$ is a path in $Y$ joining
$x$ and $z$}}
\Bigr)\;.
$$
One checks that $d$ is a length metric, and we call $\Yhat$ a
simple branched covering of $\Xhat$ over $\D$.

\begin{theorem}{main-theorem}  
If $\Xhat$ is \cat\ and $\D$ is complete and $D_\x$-convex, then
$\Yhat$ is also \cat.
\end{theorem}

This theorem was stated by Gromov \ecite{gromov:hyperbolic-groups}{section~4.4}, without the
completeness hypothesis.  Without this, $\Yhat$ may fail
to be $D_\x$-geodesic. It may even happen that $\Yhat$ contains
points having no geodesic neighborhoods. On the other hand,
every geodesic triangle in $\Yhat$ of perimeter~$<2D_\x$ still
satisfies \cat.  The same concerns arise for Reshetnyak's gluing
lemma (see for example
\ecite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}{p.~347}),
to which this result is very similar.

% \begin{example}{Example}
% The
% completeness condition in \rom2 cannot be dropped, because of the
% following example. 
% Take $\Xhat$ to be the set of points $(x,y)\in\R^2$ with
% $x\geq0$ and $y>0$, together with the point $(1,0)$. Let $\D$ be
% the positive $y$-axis. Then $\Xhat$ is a convex subset of
% $\R^2$, hence CAT(0), and $\D$ is a closed convex
% subset of $\Xhat$. The set $X=\Xhat-\D$ is contractible, so any
% cover of it is a union of disjoint copies of it. Taking $Y$ to
% be the cover with 2 sheets, $\Yhat$ is isometric to the
% upper half plane in $\R^2$ together with the points
% $(\pm1,0)$. There is no geodesic joining these two points, so
% $\Yhat$ is not a geodesic space. This provides
% a counterexample to several assertions in the literature, such
% as \ecite{gromov:hyperbolic-groups}{4.3--4.4},
% \ecite{januszkiewicz:hyperbolizations}{Lemma~1.1} and 
% \ecite{davis91:hyperbolization_of_polyhedra}{Lemma~2.4}.
% \end{example}

% \begin{example}{Example}
% Although the space $\Yhat$ of the previous example is not
% geodesic, it still has curvature $\leq0$. The following example shows
% that even this may fail if $\D$ is not complete. We take $\Xhat$
% to be the set of points $(x,y,z)$ of $\R^3$ whose first nonzero
% coordinate is positive, together with the origin. That is,
% $\Xhat$ is the union of an open half-space together with an open
% half-plane in its boundary, together with a ray in {\it its}
% boundary. We take $\D$ to be the set of points of $\Xhat$ with
% vanishing $x$-coordinate, which is the union of the open half-plane
% and the ray. Then  $\Xhat$ is a convex
% subset of $\R^3$ and $\D$ is closed and convex in
% $\Xhat$. As before, any cover of $X=\Xhat-\D$ is a union of copies
% of $X$, and we take $Y$ to be the cover with 2 sheets. Then
% $\Yhat$ is isometric to the subset of $\R^3$ given by
% $$ 
% \Yhat=\D\cup\set{(x,y,z)\in\R^3}{x\neq0}, 
% $$
% equipped with the path metric induced by the Euclidean
% metric. It is easy to see that for each $n\geq1$ the points
% $(\pm1/n,-1/n,-1/n)$ are joined by no geodesic of
% $\Yhat$. Since every neighborhood of $0$ contains such a pair of
% points, $0$ has no geodesic neighborhood.
% \end{example}

\begin{proof}{Proof}
We have obtained a proof in full generality, but here we make
the additional assumptions that $\x\leq0$ and $\Xhat$ is
complete.  This is sufficient for our applications.  The idea is
to realize $\Yhat$ as a Gromov-Hausdorff limit of spaces which
are obviously \cat.  For $\e>0$ let $\D_\e$ be the closed
$\e$-neighborhood of $\D$, and let $\Yhat_\e$ be obtained by
gluing together a copy of $\D_\e$ and a copy of $Y$, so that all
preimages in $Y$ of any given point of $\D_\e-\D$ are
identified.  This space has a natural path metric---in fact it
is a simple branched cover of $\Xhat$ over $\D_\e$.  It is
obvious that $\Yhat$ is a Gromov-Hausdorff limit of the
$\Yhat_\e$.  Since $\Yhat$ is complete it suffices by
\ecite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}{Cor.~3.10}
to show that each $\Yhat_\e$ is \cat.  Since $\Yhat_\e$ is complete
and simply connected, it suffices to show that $\Yhat_\e$ is locally
\cat.  If $x\in\Yhat_\e$ lies at distance${}\neq\e$ from $\D$ then $x$
admits a \cat\ neighborhood because it has a neighborhood isometric to
an open subset of $\Xhat$.  Now suppose $d(x,\D)=\e$.  We let $U$ be
the closed $\e/2$-ball about the image of $x$ in $\Xhat$.  Then $x$
admits a neighborhood which is the union of some number of copies of
$U$, glued together along $U\cap\D_\e$.  This neighborhood of $x$ is
\cat\ by Reshetnyak's lemma, since $U\cap\D_\e$ is complete and is
convex in $U$.
\end{proof}

Next we define precisely what we mean by a branched
cover which is locally an iterated branched cover of a manifold
over a family of mutually orthogonal totally geodesic submanifolds. Then
we show that such a branched cover satisfies the
same upper bounds on local curvature as the base manifold. We
prove this only in the case of nonpositive curvature, and
indicate what else is needed in the general case.

We say that a finite set $\{S_1,\ldots,S_n\}$ of codimension-2 subspaces of an
even-dimensional real vector space $A$ is normal if it is
equivalent to some $n$ of the $m$ coordinate hyperplanes in
$\C^m$ under an $\R$-linear isomorphism $A\to\C^m$. If $A$ is
odd-dimensional then we call $\{S_1,\ldots,S_n\}$ normal if it is equivalent
to the products with $\R$ of some $n$ of the $m$ coordinate hyperplanes
of $\C^m$, in
$\C^m\times\R$. We write $\S$ for $\cup_iS_i$.
Now suppose $\H_0$ is a family of immersed submanifolds of a
Riemannian manifold $\Mhat$ with union $\H$. We say that $\H_0$ is normal at
$x\in\Mhat$ if there is a set $\{S_1,\ldots,S_n\}$ of mutually orthogonal
subspaces of $T_x\Mhat$ that are normal in the sense above and
have the following property. We require that there be an open
ball $U$ about $0$ in $T_x\Mhat$ which the exponential map
carries diffeomorphically onto its image $V$, such that
$V\cap\H=\exp_x(U\cap\S)$, and such that each $\exp_x(S_i\cap U)$ is a convex
subset of $V$. 
We say that $\H_0$ is normal if it is normal at
each $x\in\Mhat$. In this case, each element of $\H_0$ is
totally geodesic and intersections of elements of $\H_0$ are orthogonal.

If $x\in\Mhat$ then
$\pi_1(V-\H)\isomorphism\pi_1(U-\S)\isomorphism\pi_1(T_x\Mhat-\S)\isomorphism\Z^n$.
The first two isomorphisms are obvious and the
last follows from the fact that $T_x\Mhat-\S$ is a product of $n$ punctured
planes and a Euclidean space.
We choose generators $\s_1,\ldots,\s_n$ for
$\pi_1(T_x\Mhat-\S)$ by taking a representative for $\s_i$ to be
a simple circular loop that links $S_i$ but none of the other
$S_j$. We say that a connected covering space of $T_x\Mhat-\S$
is standard if the subgroup of $\Z^n$ to which it
corresponds is generated by $\s_1^{d_1},\ldots,\s_n^{d_n}$ for some
$d_1,\ldots,d_n\in\Z$. We apply the same
terminology to the corresponding cover of $V-\H$.
In particular, the universal cover is
standard. An arbitrary covering space of $V-\H$ is called
standard if each of its components is. 

We write $M$ for $\Mhat-\H$. If $\pi:N\to M$ is a covering space
then we say that $N$ is a standard cover of $M$ if for each
$x\in\Mhat$ with $V$ as above, $\pi:\pi^{-1}(V-\H)\to V-\H$ is a
standard covering in the sense above. In this case, we take
$\Nhat$ to be a certain subset of the metric completion of $N$,
namely those points which map into $\Mhat$ under the extension
of $\pi$. In particular, if $\Mhat$ is complete then $\Nhat$ is
the completion of $N$. We denote the natural extension
$\Nhat\to\Mhat$ of $\pi$ again by $\pi$, and call $\Nhat$ a
standard branched covering of $\Mhat$ over $\H_0$.  The simplest
example of a standard branched cover is $\pi:\C^n\to\C^n$,
carrying $(z_1,\ldots,z_n)$ to
$(z_1^{d_1},\ldots,z_n^{d_n})$. We have just extended this by
making the definition local and allowing infinite branching.

\begin{theorem}{thm-iterated-br-cover}
If a Riemannian manifold $\Mhat$
has sectional curvature bounded above by $\x\leq0$ and
$\pi:\Nhat\to\Mhat$ is a standard branched cover over a
normal family $\H_0$ of immersed submanifolds of $\Mhat$, then
$\Nhat$ is locally \cat.
\end{theorem}

\begin{lemma}{punctured-balls-are-path-connected}
Let $X$ be a length space with metric completion $\Xhat$. Then
every open ball in $\Xhat$ meets $X$ in a path-connected set.
\end{lemma}

\begin{proof}{Proof}
Suppose given an open ball $U$ about $x\in\Xhat$, and $y,z\in U\cap
X$. Choose $x'\in X$ near $x$ and join $y$ and $z$ to $x'$ by paths
in $X$ that are short enough that they are forced to lie in
$U$. 
\end{proof}

\begin{proof}{Proof of~\ref{thm-iterated-br-cover}}
We will write $\Htil$ for $\pi^{-1}(\H)$. Suppose
$\xtil\in\Nhat$ lies over $x\in\Mhat$ and let $S_1,\ldots,S_n$,
$U$ and $V$ be
as in the definition of the normality of $\H_0$ at $x$.  
Let $r$ be the common radius of
$U$ and $V$. 
We write  $T_i$
for $\exp_x(U\cap S_i)\sset V$. It is clear that geodesics from
$\xtil$ to nearby points are lifts of radial geodesics
 from $x$. By choosing $r$ small
enough we may suppose that $\pi^{-1}(V)$ is the disjoint union
of the $r$-balls about the points of $\pi^{-1}(x)$. We also
choose $r$ small enough so that $V$ and all smaller balls
centered at $x$ are convex. We write $\Vtil$ for the open
$r$-ball about $\xtil$; \ref{punctured-balls-are-path-connected} assures us that $\Vtil-\Htil$ is
a connected covering space of $V-\H$. Taking generators
$\sigma_1,\ldots,\sigma_n$ for $\pi_1(V-\H)$ as above, the
standardness of the cover assures us that the covering
$\Vtil-\Htil\to V-\H$ corresponds to the subgroup generated by
$\sigma_1^{d_1},\ldots,\sigma_n^{d_n}$ for some
$d_1,\ldots,d_n$. We take $B$ (resp. $\Btil$) to be the closed
$r'$-ball about $x$ (resp. $\xtil$), where we will choose $r'<r$
later. To show that $x$ admits a \cat\ neighborhood, it suffices
to show that $\Btil$ is
\cat\ under the metric induced by lengths of
paths in $\Btil$. We will prove this by realizing $\Btil$ as an iterated
simple branched cover of $B$.

For each $k=0,\ldots,n$, let
$G_k$ be the subgroup of $G=\pi_1(B-\H)$ generated by
$\s_1^{d_1},\ldots,\s_k^{d_k},$ $\s_{k+1},\ldots,\s_n$. We let $B_k$ be
the metric completion of the cover of $B-\H$ associated to
$G_k$, equipped with the natural path
metric. Then $B_k$ is the standard branched cover of $B$,
branched over the $T_i\cap B$, with branching indices
$d_1,\ldots,d_k,1,\ldots,1$. In particular, $B_0=B$ and
$B_n=\Btil$.
We write $p_k$ for the natural projection $B_k\to B$
obtained by extending the covering map to a map of metric completions. 
Because
$G_{k+1}\sset G_k$, there is a covering map 
$B_{k+1}-p_{k+1}^{-1}(\H)\to 
B_{k}-p_{k}^{-1}(\H)$ whose completion
$q_{k+1}:B_{k+1}\to B_k$ satisfies $p_k\circ
q_{k+1}=p_{k+1}$. 
For each
$k=0,\ldots,n-1$ we let $\D_k=p_k^{-1}(T_{k+1})$. 
It is easy to see that
$q_{k+1}$ is a simple branched covering with branch locus
$\D_k\sset B_k$. 

In order to use \ref{main-theorem}
inductively, we will need to know that $\D_k$ is a convex subset of
$B_k$. This requires us to choose $r'$ small enough so that 
the orthogonal projection maps from $B$ to the $T_i$ are
well-behaved. By this we mean that for each $i$, there is
fiberwise starshaped (about $0$) set in the restriction to
$T_i\cap B$ of the normal bundle of $T_i$, which is carried
diffeomorphically onto $B$ by the exponential map. Then each
projection $B\to T_i$ has image in $B\cap T_i$, and the
projection
may be realized by a deformation retraction along geodesics. The
retraction is distance non-increasing since $T_i$ is
totally geodesic and $\Mhat$ has sectional curvature
$\leq0$. Because the $T_j$ are
orthogonal to $T_i$ for $j\neq i$, the track of the deformation retraction to
$T_i$ starting at a point outside $\cup_{j\neq i}T_j$
misses $\cup_{j\neq i}T_j$ entirely. Therefore the
deformation lifts to a deformation retraction from
$B_k-p_k^{-1}(\cup_{j\neq i}T_j)$ to
$\D_k-p_k^{-1}(\cup_{j\neq i}T_j)$. This extends to a
distance nonincreasing retraction $B_k\to\D_k$, which we will
also call orthogonal projection.

Now we prove by simultaneous induction that $B_k$ is \cat\
and that $\D_k$ is convex in $B_k$. The fact that
$B_0=B$ is \cat\ follows from its convexity in $\Mhat$ and the
fact that $\Mhat$ has sectional curvature~$\leq\x$. The convexity of
$\D_0=T_1\cap B$ in $B$ follows from the convexity of $T_1$ in
$V$. Now the inductive step is easy. If $B_k$ is \cat\ and
$\D_k$ is convex in $B_k$ then $B_{k+1}$ is \cat\ by
\ref{main-theorem}. 
In particular, geodesics in $B_{k+1}$ are unique.
Then if $\c$ is a
geodesic of $B_{k+1}$ with endpoints in $\D_{k+1}$,  the
orthogonal projection to $\D_{k+1}$ carries $\c$ to a path of
length $\leq\ell(\c)$ with the same endpoints. By the
uniqueness of geodesics, $\c$ lies in
$\D_{k+1}$, so we have proven that $\D_{k+1}$ is convex in
$B_{k+1}$. The theorem follows by induction.
\end{proof}

\begin{remark}{Remark}
We indicate here the additional work required to prove the
theorem when $\x>0$. The projection maps $B\to B\cap T_j$ may
increase distances in the presence of positive curvature. All
that is important for us is that the length of a path in $B$ {\it
with endpoints in $T_k$} does not increase under projection to
$T_k$. Even this is not true, but we only need the result for
paths of length $<2r'$. One should choose $r'$ small enough so
that any path in $B$ of length $<2r'$, with endpoints in $T_k$,
grows no longer under the projection to $T_k$. Presumably this
can be done but I have not checked the details.
\end{remark}

\Ref{thm-iterated-br-cover} has been widely believed,
but this seems to be the first proof.  As mentioned before, it is morally
contained in theorem~5.3 of
Charney and Davis
\cite{charney-davis:branched-covers-of-riemannian-manifolds},
who consider locally finite branched covers of Riemannian
manifolds over subsets more complicated than
mutually orthogonal submanifolds. Unfortunately there are  gaps
in their proof which I do not know how to bridge.
(Lemma~5.7 does
not seem to follow from lemma~5.6. Also, the techniques of \cite{ghys:sur-les-groupes-hyperboliques}
referred to in passing to finish the proof of theorem~5.3 use
properties of Riemannian manifolds, like continuous dependence
of sufficiently short geodesics on their endpoints, that are not
established for branched covers.)
Nevertheless 
their infinitesimal
\cat\ condition (condition~3 of  theorem~5.3) 
is very natural, and their theorem surely
holds and extends to the case of locally infinite branching.

% %% PROOF IN FULL GENERALITY

% What follows is a proof of \ref{main-theorem} in full
% generality. We sometimes write $[x,y]$ for a
% geodesic joining points $x$ and $y$.

% \begin{proof}{Proof that $\Yhat$ is $D_\x$-geodesic}
% Suppose $x,y\in\Yhat$ lie at distance~$<D_\x$. If they satisfy
% $d(x,y)=d(x,\D)+d(\D,y)$ then one can mimic the corresponding
% part of the proof of Reshetnyak's lemma. 
% Otherwise, we can join $x$ and $y$ by a path $\c$ of length less
% than $d(x,\D)+d(\D,y)$. We also suppose
% $\ell(\c)<D_\x$. Consider the set $A$ of points of $\c$ that are
% joined to $x$ by a geodesic of $\Yhat$  projecting to a
% geodesic of $\Xhat$. 
% Any geodesic from $x$ to a point of $\c$ misses $\D$, because
% such a geodesic, together with the rest of $\c$, has
% length bounded by $\ell(\c)$. Using this and the continuous
% dependence of geodesics in $\Xhat$ with respect to their
% endpoints, one proves that
% $A$ is open and closed. Since $A$ contains $x$, it also contains
% $y$. 
% \end{proof}

% \begin{lemma}{short-path-lying-in-cone}
% Let $T$ be a geodesic triangle in a \cat\ space, with vertices
% $x$, $y_0$ and $y_1$ and  perimeter~$<2D_\x$. 
% Suppose $p_0$ (resp. $p_1$) lies on the edge $[x,y_0]$
% (resp. $[x,y_1]$), and let $D$ be the distance between
% the corresponding points in a comparison triangle for
% $T$. Then there is a path joining $p_0$ and $p_1$ of
% length~$\leq D$ that lies entirely in the cone on $[y_0,y_1]$
% based at $x$. 
% %(That is, the union of the geodesics from $x$ to
% %the various points of $[y_0,y_1]$.)
% \end{lemma}

% \begin{proof}{Proof}
% We denote the points of $[y_0,y_1]$ by $y_t$ for $t\in[0,1]$ in
% the obvious way and write $\c[t]$ for the geodesic from $x$ to
% $y_t$. For each $N$, there are points $p({1\over
% N}),\ldots,p({N-1\over N})$ of $\c[{1\over
% N}],\ldots,\c[{N-1\over N}]$ such that
% $$
% \sum_{i=0}^{N-1}d\(p(\textstyle{i\over N}),
% p(\textstyle{i+1\over N})\)\leq D\;,
% \equation{stepping-stones-1}
% $$
% where we take $p(0)=p_0$ and $p(1)=p_1$. To construct these
% points, chop $[y_0,y_1]$ into $N$ equal pieces and $T$ into the
% corresponding $N$ triangles, place comparison triangles for
% these triangles next to each other in the obvious way, construct
% the geodesic $\a$ in the union of these triangles that joins the
% points corresponding to $p_0$ and $p_1$, look at where $\a$
% crosses from each triangle into the next, and take the
% $p({i\over N})$ to be the corresponding points on the
% $\c[{i\over N}]$. The inequality \eqref{stepping-stones-1}
% follows from Alexandrov's lemma
% \ecite{paulin:hyperbolic_groups_via_hyperbolizing_polyhedra}{lemma~2.12}
% applied repeatedly. By restricting to those $N$ that are powers
% of two and applying a diagonal argument, we can construct points
% $p(t)$ for all dyadic rationals $t$ in $[0,1]$, with $p(0)=p_0$
% and $p(1)=p_1$, such that for all $0=t_0<\cdots<t_n=1$ we have
% $$
% \sum_{i=0}^{n-1}d\(p(t_i),p(t_{i+1})\)\leq D\;.
% \equation{stepping-stones-2}
% $$
% Morally, we just extend $p$ to $[0,1]$ by continuity to complete
% the proof. Unfortunately, it can happen that $p$ does not extend
% continuously. This is only a minor inconvenience because the
% gaps can be bridged by subsegments of the $\c_t$ for various
% (countable many) $t\in[0,1]$. The fact that the resulting path
% has length~$\leq D$ follows from \eqref{stepping-stones-2}.
% \end{proof}

% \begin{lemma}{edge-in-branch-locus}
% Under the hypotheses of \ref{main-theorem}, every geodesic triangle $T$ of
% $\Yhat$ with an edge in $\D$ and perimeter~$<2D_\x$
% satisfies \cat.
% \end{lemma}

% \begin{proof}{Proof}
% Let $y_0$ and $y_1$ be the endpoints of the edge lying in $\D$,
% and let $x$ be the third vertex. Suppose $p_0$ and $p_1$ lie on
% two edges of $T$ and that $\bar p_0$ and $\bar p_1$ are the
% corresponding points in a comparison triangle $\Tbar$. If either
% $p_i$ lies in $[y_0,y_1]$ then $d(p_0,p_1)=d(\pi p_0,\pi
% p_1)\leq d(\bar p_0,\bar p_1)$, where we use the fact that
% $\Tbar$ is also a comparison triangle for $\pi T$. It suffices
% now to treat the case $p_0\in[x,y_0]$ and $p_1\in[x,y_1]$. By
% \ref{short-path-lying-in-cone} there is a path $\a$ in $\Xhat$
% of length~$\leq d(\bar p_0,\bar p_1)$ from $\pi p_0$ to $\pi
% p_1$ that lies in the cone on $[\pi y_0, \pi y_1]$ from $\pi
% x$. If $\a$ meets $\D$ then it obviously lifts to a path joining
% $p_0$ and $p_1$. Otherwise, it has a unique lift to a path
% beginning at $p_0$. Using the fact that the cone on $\a$ from
% $\pi x$ does not meet $\D$ (by the convexity of $\D$), one shows
% that the final endpoint of this lift is $p_1$. In either case we
% have $d(p_0,p_1)\leq d(\bar p_0,\bar p_1)$.
% \end{proof}

% \begin{lemma}{geodesics-missing-D-project-to-geodesics}
% Under the hypotheses of \ref{main-theorem}, a geodesic $\c$ of $\Yhat$ of
% length~$<D_\x$ that misses $\D$ projects to a geodesic of $\Xhat$.
% \end{lemma}

% \begin{proof}{Proof}
% Suppose the endpoints of $\c$ are $x$ and $z$, and that $\pi\c$
% is not a geodesic. Consider the homotopy $\Gamma$ from $\pi\c$
% to the constant path at $\pi x$ given by retraction along
% geodesics. If $\Gamma$ misses $\D$ then the geodesic in $\Xhat$
%  from $\pi x$ to $\pi z$ lifts to a path in $\Yhat$ joining $x$
% and $z$, of length less than that of $\c$, which is a
% contradiction. Otherwise, some track of $\Gamma$ meets $\D$, say
% the track of $\pi y$, where $y$ lies on $\c$. This track lifts to
% a path joining $x$ and $y$, of length less than that of the
% subsegment of $\c$ from $x$ to $y$, which is again a
% contradiction. 
% \end{proof}

% \begin{lemma}{geodesics-unique}
% Under the hypotheses of \ref{main-theorem}, geodesics in $\Yhat$ of
% length~$<D_\x$ are unique.
% \end{lemma}

% \begin{proof}{Proof}
% If two points of $\Yhat$ are joined by a geodesic of
% length~$<D_\x$ that misses $\D$, then it follows immediately
%  from \ref{geodesics-missing-D-project-to-geodesics} that this is the unique geodesic joining them. All
% that remains to prove is that if $x,z\in\Yhat$ are joined by
% geodesics $\c_1$ and $\c_2$ of length~$<D_\x$, meeting $\D$ at points $y_1$ and
% $y_2$ respectively, then $\c_1=\c_2$. To see this, subdivide the
% bigon formed by $\c_1$ and $\c_2$ into two triangles along the
% segment $[y_0,y_1]$. Both triangles satisfy \cat\ by \ref{edge-in-branch-locus}, and
% then Alexandrov's lemma forces the $\c_i$ to coincide.
% \end{proof}

% \begin{proof}{Proof of \ref{main-theorem}}
% We have already proven that $\Yhat$ is $D_\x$-geodesic. Suppose $T$ is
% a geodesic triangle with vertices $A$, $B$ and $C$ and
% perimeter~$<2D_\x$.  Then $T$ is treated by one of the following
% cases, most of which rely on Alexandrov's lemma. We define an
% altitude to be a geodesic joining a vertex of $T$ to a point on
% the opposite side.

% \cA\
% Suppose no altitude from $A$ meets $\D$. It follows from \ref{geodesics-missing-D-project-to-geodesics}
% that the altitudes from $A$ vary continuously, and then
% Alexandrov's patchwork argument \ecite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}{p.~199} shows that
% $T$ satisfies \cat.

% \cB\
% Suppose that the only altitude from $A$ meeting $\D$ is
% $\geo{AB}$. We may suppose $B\neq C$, for otherwise $T$
% degenerates to a segment by the uniqueness of geodesics. Take a
% sequence of points $B_n$ in the interior of $[B,C]$ that
% approach $B$. By case \cA, each triangle $\tri{AB_nC}$ satisfies
% \cat. It follows from \ref{geodesics-missing-D-project-to-geodesics} that the geodesics $[A,B_n]$
% converge uniformly to a geodesic, which by uniqueness must be
% $[A,B]$. As a uniform limit of triangles that satisfy \cat, $T$
% does also.

% \cC\
% Suppose $\D$ contains two vertices of $T$. This is
% \ref{edge-in-branch-locus}.

% \cD\
% Suppose $\D$ contains a vertex of $T$ and also a point of an
% opposite side. Subdivide $T$ along the geodesic joining them and apply
% case~\cC\ to each of the resulting triangles.

% \cE\
% Suppose $\D$ contains a vertex (say $B$) of $T$. 
% Consider the set of points $P$ of $[B,C]$ such that $[A,P]$
% meets $\D$, and let $B'$ be the point of this set closest to
% $C$. We subdivide $T$ along $[A,B']$ and apply case \cB\ to
% $\tri{AB'C}$ and case \cD\ to $\tri{ABB'}$.

% \cF\
% Suppose $\D$ contains a point of $T$. There is a geodesic joining
% this point with a vertex opposite it. Subdivide along it 
% and apply case \cE\ to each of the resulting
% triangles.

% \cG\
% Suppose an altitude of $T$ meets $\D$. Subdivide $T$ along
% it and apply case \cF\ to each of the resulting
% triangles. 
% \end{proof}

%% END OF PROOF IN FULL GENERALITY

\end{section}

\begin{section}{sec-apps}{Asphericity of Moduli Spaces}

In this section we solve the problems which motivated our
investigation, concerning the asphericity of certain moduli
spaces. By using known models for the moduli spaces of cubic
surfaces in $\cp^3$ and of Enriques surfaces we will show that  
these spaces have contractible universal covers. In both
cases the main ingredient is the following theorem, which is a sort
of global version of \ref{thm-iterated-br-cover}.

\begin{theorem}{thm-hyperplane-complements-aspherical}
Let $\Mhat$ be a complete simply connected Riemannian manifold
with sectional curvature bounded above by $\x\leq0$. Let $\H$ be
the union of a family of complete submanifolds which is normal in the sense
of \ref{sec-br-covers}. Then the metric completion $\Nhat$ of the universal
cover $N$ of $\Mhat-\H$ is \cat, and $N$ and $\Nhat$ are
contractible. 
\end{theorem}

We will conform
to the notation of \ref{sec-br-covers} by writing $M$ for
$\Mhat-\H$, $\pi$
for the covering map $N\to M$ and its completion, and
$\Htil$ for $\pi^{-1}(\H)\sset\Nhat$.

\begin{lemma}{thm-neighborhoods-described} 
Suppose $\xtil\in\Nhat$ lies over $x\in\Mhat$ and $V$ is
an open ball about $x$ that meets none of the submanifolds
except those passing  through $x$. If $\Vtil$ is the ball of the
same radius about $\xtil$, then $\Vtil-\Htil$ is a copy of the
universal cover of $V-\H$.
\end{lemma}

\begin{proof}{Proof}
Since $\Vtil-\Htil$ is connected (\ref{punctured-balls-are-path-connected}),
all we must show is that $\pi_1(V-\H)$ injects into
$\pi_1(\Mhat-\H)$. Writing $\I$ for the union in $\Mhat$ of the
submanifolds that pass through $x$, we have
homomorphisms 
$$ 
\pi_1(V-\I)
\to\pi_1(\Mhat-\H)
\to\pi_1(\Mhat-\I)
\to\pi_1(V-\I)
\;,
$$
where the first two maps are induced by inclusions and the third
by a retraction of $\Mhat-\I$ into $V-\I$ along
geodesics from $x$. The composition is obviously the identity
map, so the first map is injective.
\end{proof}

\begin{lemma}{thm-weak-homotopy-equiv}
The inclusion $N\to\Nhat$ is a weak homotopy equivalence.
\end{lemma}

\begin{proof}{Proof}
First we show that for each $\xtil\in\Nhat$ there is a homotopy of
$\Nhat$ to itself that \rom1 carries some neighborhood of $x$ into
$N$, \rom2 carries $N$ into itself, and \rom3 fixes each point of
$\Nhat-N$ that doesn't get pushed into $N$.  We write $n$ for the
number of submanifolds passing through $x=\pi(\xtil)$. By the previous
lemma there is a closed neighborhood $\Vtil$ of $\xtil$ which is
homeomorphic to $(\Atil)^n\times D$, where $\Atil$ is the metric
completion of the universal cover of a punctured disk and $D$ is a
closed Euclidean ball. It is easy to see that $\Atil$ is homeomorphic
to a wedge in the plane, by which we mean
$$ 
\Atil\isomorphism
\{(0,0)\}\;\cup\;
\set{(x,y)\in\R^2}{|y|<x\hbox{ and }x^2+y^2\leq1}
\;.
$$
There is obviously a homotopy of $\Atil$ into $\Atil-\{(0,0)\}$
which is supported on a small neighborhood of $(0,0)$. Using this it
is easy to construct a homotopy of $\Nhat$ satisfying \rom1--\rom3.

Now, if $f:S^k\to\Nhat$ represents any element of the homotopy group
$\pi_k(\Nhat)$ then we may cover $f(S^k)$ with finitely many open
sets, each of which is carried into $N$ by some homotopy of $\Nhat$
that satisfies \rom2 and \rom3.  Applying these homotopies one after
another shows that $f$ is homotopic to a map $S^k\to N$. Therefore
$\pi_k(N)$ surjects onto $\pi_k(\Nhat)$.  The same argument applied to
balls rather than spheres shows that $\pi_k(N)$ also injects.  By
elaborating this argument one can show that $N\to\Nhat$ is actually a
homotopy equivalence, but it is easier to prove
\ref{thm-hyperplane-complements-aspherical} first and then apply
the contractibility of both spaces.
\end{proof}

\begin{proof}{Proof of \ref{thm-hyperplane-complements-aspherical}}
By \ref{thm-neighborhoods-described}, $\Nhat$ is a standard
branched cover of $\Mhat$ over the normal family $\H_0$. Since
$\Mhat$ has sectional curvature $\leq\x\leq0$,
\ref{thm-iterated-br-cover} shows that $\Nhat$ is
locally \cat. Since $N$ is simply connected,
\ref{thm-weak-homotopy-equiv} implies that $\Nhat$ is
also. The Cartan-Hadamard theorem for Alexandrov spaces
\ecite{bridson-haefliger:metric-spaces-of-nonpositive-curvature}{p.~193} 
implies that $\Nhat$
is \cat\ and hence contractible. In particular, all of its
homotopy groups vanish, and by another application of
\ref{thm-weak-homotopy-equiv} the same is true of $N$. As
a manifold all of whose homotopy groups vanish, $N$ is
contractible.
\end{proof}

Now we turn to moduli spaces.
The set $\cal C$ of cubic surfaces in $\cp^3$ may be identified
with $\cp^{19}$, because there are~20 cubic monomials in~4
variables. The smooth surfaces form an open
subset ${\cal C}_0$, and it is known that $\PGL(4,\C)$ acts properly on
${\cal C}_0$. Therefore the moduli space $\moduli={\cal
C}_0/\PGL(4,\C)$ carries the natural structure of a complex
analytic orbifold. The main result of \cite{allcock:ch4-cubic-moduli} shows that
$\moduli$ is orbifold-isomorphic to $(\ch^4-\H)/P\Gamma$, where
$\ch^4$ is  complex hyperbolic 4-space, $\H$ is the union of an
infinite family of complex hyperplanes and $\Gamma$ is a certain
discrete group. To state this more precisely, let $\w$ be
a primitive cube root of unity and let $\E$ be the discrete
subring $\Z[\w]$ of $\C$. Let $\Lambda$ be the lattice $\E^5$ equipped with the
Hermitian inner product
$$ 
h(x,y)=-x_0\bar y_0+x_1\bar y_1+\cdots+x_4\bar y_4\;. 
$$
Then the complex hyperbolic space 
$\ch^4$ is the set of lines in $\C^5$ on
which $h$ is negative-definite, $\H$ is the union of the hyperplanes in $\ch^4$
which are the orthogonal complements of those $r\in\Lambda$ with
$h(r,r)=1$, and $\Gamma$ is the unitary group of $\Lambda$, which
is obviously discrete in $\U(4,1)$. 

\begin{corollary}{cubic-surfaces}
$\moduli$ has contractible orbifold universal cover.
\end{corollary}

\begin{proof}{Proof}
Since $\ch^4-\H$ covers $(\ch^4-\H)/P\Gamma$ and $\ch^4$ has
negative sectional curvature, all we have to prove is the
normality of the family of hyperplanes. Suppose $r,r'\in\Lambda$
satisfy $h(r,r)=h(r',r')=1$. If $r^\bot$ meets $r'{}^\bot$ then
$r$ and $r'$ span a positive-definite sublattice of
$\Lambda$. This requires $|h(r,r')|<1$, and since $h(r,r')\in\E$
we must have $h(r,r')=0$, so that $r^\bot$ and $r'{}^\bot$ meet
orthogonally. The local finiteness of the family of hyperplanes
follows from a standard argument: if $x\in\C^5$ represents a
point of $\ch^4$ and $N$ is given, then there are only finitely
many $r\in\Lambda$ satisfying $h(r,r)=1$ and $|h(r,x)|\leq N$.
\end{proof}

Enriques surfaces are smooth compact complex surfaces that satisfy
certain cohomological conditions; see for example Horikawa
\cite{horikawa:periods-of-enriques-surfaces-I}, \cite{horikawa:periods-of-enriques-surfaces-II}. Horikawa's global Torelli theorem 
identifies the set of isomorphism classes of these surfaces with
$(\DD-\H)/\Gamma$, where $\DD$ is the Hermitian
symmetric space for $O(2,10)$, $\Gamma$ is a certain discrete
subgroup and $\H$ is the union of an infinite family of complex
hyperplanes. Namikawa \cite{namikawa:periods-of-enriques-surfaces} refined Horikawa's work, and
according to the rephrasing of these results in \cite{allcock:period-lattice-for-enriques-surfaces},
$\Gamma$ may be taken to be the
isometry group of the lattice $L$ which is $\Z^{12}$ equipped
with the inner product
$$ 
x\cdot y=x_1y_1+x_2y_2-x_3y_3-\cdots-x_{12}y_{12}\;. 
$$
A concrete model for $\DD$ is the set of $v\in P(L\tensor\C)$
satisfying $v\cdot v=0$ and $v\cdot\bar{v}>0$, and $\H$ may be
taken to be the union of the (complex) hyperplanes in $\DD$ which are
the orthogonal complements of the norm $-1$ vectors of
$L$. We regard the moduli space of Enriques surfaces as being
the orbifold $(\DD-\H)/\Gamma$.

\begin{corollary}{enriques-surfaces}
The moduli space $(\DD-\H)/\Gamma$ has contractible orbifold
universal cover.
\end{corollary}

\begin{proof}{Proof}
The proof that hyperplanes that meet do so orthogonally is the same as
before. Local finiteness follows by essentially the same standard
argument: any $v\in\DD$ defines a 2-dimensional positive-definite
subspace $V$ of $L\tensor\R$, namely the projection to $L\tensor\R$ of
the complex line it represents. For each $N$ there are only finitely
many $r\in L$ with $r\cdot r=-1$ having projection to $V$ of
norm~$\leq N$. This proves local finiteness and hence normality.  Then
we use the fact that $\DD$ has nonpositive sectional curvature and
appeal to \ref{thm-hyperplane-complements-aspherical}.
\end{proof}

\end{section}

\begin{bibliography}{99}
% computer-maintained bibliography entries
\bibitem{allcock:ch4-cubic-moduli}
D.~Allcock, J.~Carlson, and D.~Toledo.
The complex hyperbolic geometry of the moduli space of cubic
  surfaces.
To appear in {\it J. Alg. Geom.}

\bibitem{allcock:period-lattice-for-enriques-surfaces}
D.~J. Allcock.
The period lattice for {E}nriques surfaces.
{\it Math. Ann.}, 317:483--488, 2000.

\bibitem{bridson-haefliger:metric-spaces-of-nonpositive-curvature}
M.~R. Bridson and A.~Haefliger.
{\it Metric Spaces of Non-positive Curvature}.
Springer, 1999.

\bibitem{charney-davis:branched-covers-of-riemannian-manifolds}
R.~Charney and M.~Davis.
Singular metrics of nonpositive curvature on branched covers of
  {R}iemannian manifolds.
{\it Am. J. Math.}, 115:929--1009, 1993.

\bibitem{charney-davis:K-pi-1-problem-for-hyperplane-complements}
R.~Charney and M.~W. Davis.
The ${K}(\pi,1)$-problem for hyperplane complements associated to
  infinite reflection groups.
{\it J.A.M.S.}, 8:597--627, 1995.

\bibitem{ghys:sur-les-groupes-hyperboliques}
E.~Ghys and P.~de~la Harpe, editors.
{\it Sur les Groupes Hyperboliques d'apr{\`e}s {M}ikhael {G}romov},
  volume~83 of {\it Progress in Mathematics}. Birk{\"a}user, 1990.

\bibitem{gromov:hyperbolic-groups}
M.~Gromov.
Hyperbolic groups.
In S.~M. Gersten, editor, {\it Essays in Group Theory}, volume~8 of
  {\it MSRI Publications}, pages 75--263. Springer-Verlag, 1987.

\bibitem{gromov-thurston:pinching-constants-for-hyperbolic-manifolds}
M.~Gromov and W.~Thurston.
Pinching constants for hyperbolic manifolds.
{\it Invent. Math.}, 89(1):1--12, 1987.

\bibitem{horikawa:periods-of-enriques-surfaces-I}
E.~Horikawa.
On the periods of {E}nriques surfaces. {I}.
{\it Math. Ann.}, 234:73--88, 1978.

\bibitem{horikawa:periods-of-enriques-surfaces-II}
E.~Horikawa.
On the periods of {E}nriques surfaces. {I}{I}.
{\it Math. Ann.}, 235:217--246, 1978.

\bibitem{namikawa:periods-of-enriques-surfaces}
Y.~Namikawa.
Periods of {E}nriques surfaces.
{\it Math. Ann.}, 270:201--222, 1985.
% end of computer-maintained bibliography entries
\end{bibliography}
\bye