\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{analytification}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{analytic_geometry}{}\paragraph*{{Analytic geometry}}\label{analytic_geometry}

[[!include analytic geometry -- contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Existence}{Existence and fully faithfulness (GAGA)}\dotfill \pageref*{Existence} \linebreak
\noindent\hyperlink{GeometricRealizationInA1Homotopy}{As geometric realization in $\mathbb{A}^1$-homotopy theory}\dotfill \pageref*{GeometricRealizationInA1Homotopy} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{complex_analytification}{Complex analytification}\dotfill \pageref*{complex_analytification} \linebreak
\noindent\hyperlink{ReferencesNonArchimedeanAnalytification}{Non-archimedean analytification}\dotfill \pageref*{ReferencesNonArchimedeanAnalytification} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Analytification is the process of universally turning an [[algebraic space]] into an [[analytic space]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $X \to Spec(\mathbb{C})$ be a [[scheme]] of [[locally finite type]] over the [[complex numbers]]. Its set $X(\mathbb{C})$ of ``complex points'' is the set of [[maximal ideals]], since $\mathbb{C}$ is an [[algebraically closed field]], e.g. \hyperlink{Neeman07}{Neeman 07, prop. 4.2.4}).

This set $X(\mathbb{C})$ canonically carries the [[complex analytic topology]]. As such it is a [[topological space]] written $X^{an}$. Equipped with the canonical [[structure sheaf]] $\mathcal{O}_{X^{an}}$ this is a [[complex analytic space]]. This $(X^{an}, \mathcal{O}_{X^{an}})$ is called the \emph{analytification} of $X$.

This construction extends to a [[functor]] from the [[category]] of [[schemes]] over $\mathbb{C}$ to that of [[complex analytic spaces]].

See e.g. (\hyperlink{Neeman07}{Neeman 07, section 4, p.71}, \hyperlink{Danilov91}{Danilov 91, chapter 3, paragraph 1, section 1.1 (p.61)}

Generalization to [[structured (infinity,1)-toposes]] is in (\hyperlink{Lurie08}{Lurie 08, remark 4.4.13}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The analytification of the [[projective space]] $\mathbb{P}^1$ is the [[complex projective space]] $(\mathbb{P}^1)^{an} \simeq \mathbb{C}\mathbb{P}^1$, hence the [[Riemann sphere]].

The analytification of an [[elliptic curve]] is the [[complex torus]].

see e.g. (\hyperlink{Danilov91}{Danilov 91, example in chapter 3, paragraph 1, section 1.1. (p. 61)})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Existence}{}\subsubsection*{{Existence and fully faithfulness (GAGA)}}\label{Existence}

The analytification of an [[algebraic space]] over the [[complex numbers]] which is

\begin{enumerate}%
\item [[locally of finite type]]


\item [[locally separated]]



\end{enumerate}
is a [[complex analytic space]].

Moreover, under suitable conditions analytification is a [[fully faithful functor]].

This is a classical result due to (\hyperlink{Artin70}{Artin 70, theorem 7.3}). A textbook account of the proof is in (\hyperlink{Neeman07}{Neeman 07, section 10}). Discussion in more general [[analytic geometry]] is in (\hyperlink{ConradTemkin09}{Conrad-Temkin 09, section 2.2}).

Generalization to [[algebraic stacks]]/[[Deligne-Mumford stacks]]/[[geometric stacks]] is in (\hyperlink{Lurie04}{Lurie 04}, \hyperlink{Hall11}{Hall 11}, \hyperlink{GeraschenkoZureickBrown12}{Geraschenko \& Zureick-Brown 12}).

\hypertarget{GeometricRealizationInA1Homotopy}{}\subsubsection*{{As geometric realization in $\mathbb{A}^1$-homotopy theory}}\label{GeometricRealizationInA1Homotopy}

For $k \hookrightarrow \mathbb{C}$ a field, then the functor that takes a smooth complex scheme to the the [[homotopy type]] underlying its analytification induces [[geometric realization]]

\begin{displaymath}
Sh_\infty(Sch^{sm}_k)
  \to 
  Sh_\infty(Sch^{sm}_k)^{\mathbb{A}^1}
  \to 
  \infty Grpd
\end{displaymath}
(\hyperlink{Isaksen01}{Isaksen 01}, \hyperlink{DuggerIsaksen05}{Dugger-Isaksen 05, theorem 5.2})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[GAGA]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{complex_analytification}{}\subsubsection*{{Complex analytification}}\label{complex_analytification}

Original articles include

\begin{itemize}%
\item [[Michael Artin]], \emph{Algebraization of formal moduli: II. Existence of modifications}, Annals of Math., 91 no. 1 (1970), pp. 88--135.


\item [[Alexander Grothendieck]], [[SGA]] I, Expos\'e{} XII



\end{itemize}
A review of that is in

\begin{itemize}%
\item Yan Zhao, \emph{G\'e{}om\'e{}trie alg\'e{}brique et g\'e{}om\'e{}trie analytique}, 2013 (\href{http://pub.math.leidenuniv.nl/~jinj/2013/efg/gaga.pdf}{pdf})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Amnon Neeman]], \emph{Algebraic and analytic geometry}, London Math. Soc. Lec. Note Series \textbf{345}, 2007 (\href{http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry}{publisher})


\item [[Vladimir Danilov]], chapter 3 of \emph{Cohomology of algebraic varieties}, in I. Shafarevich (ed.), \emph{Algebraic Geometry II}, volume 35 of \emph{Encyclopedia of mathematical sciences}, Springer 1991 (\href{http://books.google.de/books?id=ZhzXJHUgcRUC&lpg=PA67&ots=aVQoeMkBwc&dq=analytification&pg=PA61&redir_esc=y#v=onepage&q=analytification&f=false}{GoogleBooks}))



\end{itemize}
Discussion for [[real analytic spaces]] includes

\begin{itemize}%
\item [[Johannes Huisman]], section 2 of \emph{The exponential sequence in real algebraic geometry and Harnack's Inequality for proper reduced real schemes}, Communications in Algebra, Volume 30, Issue 10, 2002 (\href{http://pageperso.univ-brest.fr/~huisman/rech/publications/exphi.pdf}{pdf})

\end{itemize}
Generalizations to [[higher geometry]] are in

\begin{itemize}%
\item [[Brian Conrad]], M. Temkin, \emph{Non-Archimedean analytification of algebraic spaces}, J. Algebraic Geom. 18 (2009), no. 4, 731--788 (\href{http://arxiv.org/abs/0706.3441}{arXiv:0706.3441})


\item [[Jacob Lurie]], \emph{[[Tannaka duality for geometric stacks]]}, (\href{http://arxiv.org/abs/math/0412266}{arXiv:math.AG/0412266})


\item [[Jacob Lurie]], \emph{[[Structured Spaces]]}, 2008


\item [[Jack Hall]], \emph{Generalizing the GAGA Principle} (\href{http://arxiv.org/abs/1101.5123}{arXiv:1101.5123})


\item [[Anton Geraschenko]], David Zureick-Brown, \emph{Formal GAGA for good moduli spaces} (\href{http://arxiv.org/abs/1208.2882}{arXiv:1208.2882})



\end{itemize}
See also

\begin{itemize}%
\item [[Walter Gubler]]. \emph{Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros)} (\href{http://arxiv.org/abs/1303.7364}{arXiv:1303.7364})

\end{itemize}
Discussion in the context of [[hypercovers]] and [[A1-homotopy theory]] is in

\begin{itemize}%
\item [[Daniel Isaksen]], \emph{\'E{}tale realization of the $\mathbb{A}^1$-homotopy theory of schemes}, 2001 (\href{http://www.math.uiuc.edu/K-theory/0495/}{K-theory archive})


\item [[Daniel Dugger]] and [[Daniel Isaksen]], \emph{Hypercovers in topology}, 2005 (\href{http://www.math.uiuc.edu/K-theory/0528/hypercover.pdf}{pdf}, \href{http://www.math.uiuc.edu/K-theory/0528/}{K-Theory archive})



\end{itemize}
\hypertarget{ReferencesNonArchimedeanAnalytification}{}\subsubsection*{{Non-archimedean analytification}}\label{ReferencesNonArchimedeanAnalytification}

Discussion in more general [[rigid analytic geometry]] is in

\begin{itemize}%
\item [[Brian Conrad]], Michael Temkin, \emph{Non-archimedean analytification of algebraic spaces} (\href{http://arxiv.org/pdf/0706.3441}{arXiv:0706.3441})

\end{itemize}
[[!redirects analytifications]]



\end{document}