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\begin{document}

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\section*{Oka principle}

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

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

[[!include complex geometry - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{StatementInComplexAnalyticGeometry}{Statement in higher complex analytic geometry}\dotfill \pageref*{StatementInComplexAnalyticGeometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{Oka-Grauert principle} states that for any [[Stein manifold]] $X$ the holomorphic and the topological classification of [[complex vector bundles]] on $X$ coincide. The original reference is (\hyperlink{Grauert58}{Grauert 58}).

The principle should maybe better be called the \emph{Oka-Grauert-Gromov principle/theory}. Gromov viewed it in his book on partial differential relations as one of the examples of [[h-principle]].

\hypertarget{StatementInComplexAnalyticGeometry}{}\subsection*{{Statement in higher complex analytic geometry}}\label{StatementInComplexAnalyticGeometry}

In (\hyperlink{Larusson01}{Larusson 01}, \hyperlink{Larusson03}{Larusson 03}) this is formulated in terms of [[higher complex analytic geometry]] of [[complex analytic infinity-groupoids]].

Say that a [[complex manifold]] $X$ is an \emph{[[Oka manifold]]} if for every [[Stein manifold]] $\Sigma$ the canonical morphism

\begin{displaymath}
Maps_{hol}(\Sigma, X) \longrightarrow Maps_{top}(\Sigma, X)
\end{displaymath}
from the [[mapping space]] of [[holomorphic functions]] to that of [[continuous functions]] (both equipped with the [[compact-open topology]]) is a [[weak homotopy equivalence]].

\begin{theorem}
\label{}\hypertarget{}{}
This is the case precisely if $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ satisfies [[descent]] with respect to [[finite set|finite]] [[covers]].

\end{theorem}
(\hyperlink{Larusson01}{Larusson 01, theorem 2.1})

\begin{theorem}
\label{}\hypertarget{}{}
The category of [[complex manifolds]] and [[holomorphic maps]] can be embedded into a Quillen [[model category]] such that:

\begin{itemize}%
\item a holomorphic map is a weak equivalence in the ambient model category if and only if it is a [[homotopy equivalence]] in the usual topological sense.


\item a holomorphic map is a fibration if and only if it is an Oka map. In particular, a complex manifold is fibrant if and only if it is an [[Oka manifold]].


\item a [[complex manifold]] is cofibrant if and only if it is [[Stein manifold|Stein]].


\item a Stein inclusion is a cofibration.



\end{itemize}
\end{theorem}
(\hyperlink{Larusson03}{Larusson 03})

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

\begin{itemize}%
\item [[Oka manifold]].

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

Original articles include

\begin{itemize}%
\item K. Oka, \emph{Sur les fonctions des plusieurs variables. III: Deuxi\`e{}me probl\`e{}me de Cousin}, J. Sc. Hiroshima Univ. \textbf{9}, 7--19 (1939)


\item [[Hans Grauert]], \emph{Analytische Faserungen \"u{}ber holomorph-vollst\"a{}ndigen R\"a{}umen}, Math. Ann. \textbf{135}, 263---273 (1958) \href{http://dx.doi.org/10.1007/BF01351803}{doi}



\end{itemize}
\begin{itemize}%
\item [[Mikhail Gromov]], \emph{Oka's principle for holomorphic sections of elliptic bundles}, J. Amer. Math. Soc. \textbf{2} (1989), 851---897.


\item [[Franc Forstnerič]], \emph{The Oka principle for sections of stratified fiber bundles}, Pure Appl. Math. Quarterly (Special Issue in honor of Joseph J. Kohn), 6 (2010), no. 3, 843--874, \href{http://arxiv.org/abs/0705.0591}{arxiv/0705.0591}



\end{itemize}
Surveys and reviews include

\begin{itemize}%
\item [[Finnur Lárusson]], \emph{What is an Oka manifold}, Notices AMS, \href{http://www.ams.org/notices/201001/rtx100100050p.pdf}{pdf}


\item [[Franc Forstnerič]], [[Finnur Lárusson]], \emph{Survey of Oka theory}, \href{http://arxiv.org/abs/1009.1934}{arxiv/1009.1934}


\item [[Franc Forstnerič]], section 5.3 of \emph{Stein manifolds and holomorphic mappings -- The homotopy principle in complex analysis}, Springer 2011



\end{itemize}
Discussion in terms of [[higher complex analytic geometry]] and [[complex analytic infinity-groupoids]] is in

\begin{itemize}%
\item [[Finnur Lárusson]], \emph{Excision for simplicial sheaves on the Stein site and Gromov's Oka principle} (\href{http://arxiv.org/abs/math/0101103}{arXiv:math/0101103})


\item [[Finnur Lárusson]], \emph{Model structures and the Oka principle}, \href{http://arxiv.org/abs/math/0303355}{math.CV/0303355}



\end{itemize}
This construction stems from some observations from Jardine, and uses his intermediate model structure from

\begin{itemize}%
\item [[John Frederick Jardine]], \emph{Intermediate model structures for simplicial presheaves}, Canad. Math. Bull. \textbf{49} (2006), no. 3, 407--413, \href{http://www.ams.org/mathscinet-getitem?mr=2252262}{MR2007d:18021}

\end{itemize}
Some other articles on Oka principle:

\begin{itemize}%
\item Tyson Ritter, \emph{A strong Oka principle for embeddings of some planar domains into $C\times C^*$}, \href{http://arxiv.org/abs/1011.4116}{arxiv/1011.4116}

\end{itemize}
Related MO discussion: \href{http://mathoverflow.net/a/60053/381}{by Georges Elencwajg}

[[!redirects Oka theory]] [[!redirects Oka's theory]] [[!redirects Oka's principle]]

[[!redirects Oka-Grauert principle]]



\end{document}