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\begin{document}

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\section*{stable vector bundle}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - 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{OverARiemannSurface}{Over a Riemann surface / over an algebraic curve}\dotfill \pageref*{OverARiemannSurface} \linebreak
\noindent\hyperlink{over_a_general_noetherian_scheme}{Over a general (Noetherian) scheme}\dotfill \pageref*{over_a_general_noetherian_scheme} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToGITStability}{Relation to GIT-stability}\dotfill \pageref*{RelationToGITStability} \linebreak
\noindent\hyperlink{relation_to_connections}{Relation to connections}\dotfill \pageref*{relation_to_connections} \linebreak
\noindent\hyperlink{relation_to_bridgeland_stability_conditions}{Relation to Bridgeland stability conditions}\dotfill \pageref*{relation_to_bridgeland_stability_conditions} \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}

A [[vector bundle]] (typically considered in [[complex-analytic geometry]] or [[algebraic geometry]]) is called \emph{(semi-)stable} if it is a \emph{[[GIT-stable point|(semi-)stable point]]} in the [[moduli space of bundles]] in the sense of [[geometric invariant theory]].

Under suitable conditions this is equivalent to a certain [[inequality]] on the [[slope of a coherent sheaf|slopes]] of the sub-bundles (see \hyperlink{OverARiemannSurface}{below}), and this inequality is what tends to be stated as the definition of stability of vector bundles.

For more discussion (informal and formal) of this concept of stability see at \emph{[[Bridgeland stability condition]]}.

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

\hypertarget{OverARiemannSurface}{}\subsubsection*{{Over a Riemann surface / over an algebraic curve}}\label{OverARiemannSurface}

For $\Sigma$ a [[Riemann surface]], a [[complex vector bundle]] $E \to \Sigma$ over $\Sigma$ is called \textbf{(slope-)stable} if for all non-trivial subbundles $K \hookrightarrow E$ the [[inequality]]

\begin{displaymath}
\mu(K) \lt \mu(E)
\end{displaymath}
between their [[slope of a coherent sheaf|slopes]] holds, i.e. if the inequality

\begin{displaymath}
\frac{deg(K)}{rank(K)}
  \lt
  \frac{deg(E)}{rank(E)}
\end{displaymath}
holds between the fractions of [[degree of a coherent sheaf|degree]] and [[rank]] of the vector bundles holds.

e.g. (\hyperlink{HuybrechtsLehn96}{Huybrechts-Lehn 96, bottom of p. 24})

\hypertarget{over_a_general_noetherian_scheme}{}\subsubsection*{{Over a general (Noetherian) scheme}}\label{over_a_general_noetherian_scheme}

e.g. (\hyperlink{HuybrechtsLehn96}{Huybrechts-Lehn 96, def. 1.2.4, def. 1.2.12})

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

Every [[line bundle]] is slope-stable. The [[extension]] of a degree-0 line bundle by a degree-1 line bundle is stable.

e.g. (\hyperlink{HuybrechtsLehn96}{Huybrechts-Lehn 96, example 1.2.10})

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

\hypertarget{RelationToGITStability}{}\subsubsection*{{Relation to GIT-stability}}\label{RelationToGITStability}

The slope-(semi-)stable vector bundles are essentially the [[stable point|(semi-)stable points]] in the sense of [[geometric invariant theory]] in the [[moduli space of bundles]]. The precise statement is discussed in (\hyperlink{King94}{King 94}) reviewed for instance in (\hyperlink{Saiz09}{Saiz 09, section 2.3}.

\hypertarget{relation_to_connections}{}\subsubsection*{{Relation to connections}}\label{relation_to_connections}

The [[Narasimhan–Seshadri theorem]] identifies [[moduli spaces]] of stable vector bundles over [[complex curves]] with those of certain [[flat connections]].

The [[Donaldson-Uhlenbeck-Yau theorem]] relates semi-stable vector bundles over [[Kähler manifolds]] to [[Hermite-Einstein connections]].

Still more generally, the [[Kobayashi-Hitchin correspondence]] relates semi-stable vector bundles over [[complex manifolds]] to Hermite-Einstein connections.

\hypertarget{relation_to_bridgeland_stability_conditions}{}\subsubsection*{{Relation to Bridgeland stability conditions}}\label{relation_to_bridgeland_stability_conditions}

Slope-stability of vector bundles is a special case of a [[Bridgeland stability condition]], see \href{Bridgeland+stability+condition#SlopeStabilityOfVectorBundles}{there} For review see e.g. (\hyperlink{Engenhorst14}{Engenhorst 14, sections 3 and 4}) and see \hyperlink{King94}{King 94}.

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

\begin{itemize}%
\item [[Harder-Narasimhan filtration]]


\item [[moduli space of bundles]]


\item [[stable point]]


\item [[positive line bundle]], [[ample line bundle]]


\item [[Bridgeland stability condition]]


\item [[geometric invariant theory]]



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

The concept was introduced in

\begin{itemize}%
\item [[David Mumford]], \emph{Geometric invariant theory}, Ergebnisse Math. Vol 34 Springer (1965)


\item F. Takemoto, \emph{Stable vector bundles on algebraic surfaces}, Nagoya Math. J. 47 (1972) 29-48 (\href{http://projecteuclid.org/euclid.nmj/1118798682}{euclid}); \emph{II}, 52 (1973) (\href{http://projecteuclid.org/euclid.nmj/1118794885}{euclid})


\item [[David Mumford]], John Fogarty, Frances Clare Kirwan, \emph{Geometric invariant theory}, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) \textbf{34}, Springer-Verlag (1965)



\end{itemize}
Review is in

\begin{itemize}%
\item [[Michael Atiyah]], [[Raoul Bott]], section 7 of \emph{The Yang-Mills equations over Riemann surfaces}, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (\href{http://www.jstor.org/stable/37156}{jstor}, \href{http://math.stackexchange.com/a/295505/58526}{lighning summary})

\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Daniel Huybrechts]], [[Manfred Lehn]], \emph{The Geometry of the Moduli Spaces of Sheaves}, 1996 ([[HuybrechtsLehn.pdf:file]])

\end{itemize}
More discussion with regards to [[geometric invariant theory]] and [[Bridgeland stability conditions]] is in

\begin{itemize}%
\item [[Alastair King]], \emph{Moduli of representations of finite dimensional algebras}, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.623.649&rep=rep1&type=pdf}{pdf})


\item Jan Engenhorst, \emph{Bridgeland Stability Conditions in Algebra, Geometry and Physics}, 2014 (\href{https://www.freidok.uni-freiburg.de/fedora/objects/freidok:9595/datastreams/FILE1/content}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Paolo de Bartolomeis, Gang Tian, \emph{Stability of complex vector bundles}, Journal of Differential Geometry, Vol. 43, No. 2 (1996) (\href{http://www.intlpress.com/JDG/archive/1996/43-2-231.pdf}{pdf})


\item Wei-Ping Li, Zhenbo Qin, \emph{Stable vector bundles on algebraic surfaces} (\href{http://arxiv.org/pdf/math/9411233.pdf}{pdf})


\item Alfonso Zamora Saiz, \emph{On the stability of vector bundles}, Master thesis 2009 ([[SaizStableBundles.pdf:file]])



\end{itemize}
Discussion for [[equivariant vector bundles]] is in

\begin{itemize}%
\item [[C. S. Seshadri]], \emph{Moduli of $\pi$-Vector Bundles over an Algebraic Curve}, In: Marchionna E. (eds) Questions on Algebraic Varieties. C.I.M.E. Summer Schools, vol 51. Springer, Berlin, Heidelberg (\href{https://doi.org/10.1007/978-3-642-11015-3_5}{doi:10.1007/978-3-642-11015-3\_5})


\item Oscar García-Prada, \emph{Invariant connections and vortices}, Commun.Math. Phys. (1993) 156: 527 (\href{https://doi.org/10.1007/BF02096862}{doi:10.1007/BF02096862})



\end{itemize}
[[!redirects stable vector bundles]]

[[!redirects stable coherent sheaf]] [[!redirects stable coherent sheaves]]



\end{document}