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In [[mathematics]], a '''stable vector bundle''' is a ([[holomorphic vector bundle|holomorphic]] or [[algebraic vector bundle|algebraic]]) [[vector bundle]] that is stable in the sense of [[geometric invariant theory]]. Any holomorphic vector bundle may be built from stable ones using '''Harder–Narasimhan filtration'''. Stable bundles were defined by [[David Mumford]] in {{harvtxt|Mumford|1963}} and later built upon by [[David Gieseker]], [[Fedor Bogomolov]], [[Thomas Bridgeland]] and many others.

== Motivation ==
One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, [[Moduli space]]s of stable vector bundles can be constructed using the [[Quot scheme]] in many cases, whereas the stack of vector bundles <math>\mathbf{B}GL_n</math> is an [[Artin stack]] whose underlying set is a single point.

Here's an example of a family of vector bundles which degenerate poorly. If we tensor the [[Euler sequence]] of <math>\mathbb{P}^1</math> by <math>\mathcal{O}(1)</math> there is an exact sequence<blockquote><math>0 \to \mathcal{O}(-1) \to \mathcal{O}\oplus \mathcal{O} \to \mathcal{O}(1) \to 0</math><ref>Note <math>\Omega^1_{\mathbb{P}^1} \cong \mathcal{O}(-2)</math> from the [[Adjunction formula]] on the canonical sheaf.</ref></blockquote>which represents a non-zero element <math>v \in \text{Ext}^1(\mathcal{O}(1),\mathcal{O}(-1)) \cong k</math><ref>Since there are isomorphisms<math>\begin{align}
\text{Ext}^1(\mathcal{O}(1),\mathcal{O}(-1)) &\cong \text{Ext}^1(\mathcal{O},\mathcal{O}(-2)) \\
&\cong H^1(\mathbb{P}^1,\omega_{\mathbb{P}^1})
\end{align}</math></ref> since the trivial exact sequence representing the <math>0</math> vector is<blockquote><math>0 \to \mathcal{O}(-1) \to \mathcal{O}(-1)\oplus \mathcal{O}(1) \to \mathcal{O}(1) \to 0</math></blockquote>If we consider the family of vector bundles <math>E_t</math> in the extension from <math>t\cdot v</math> for <math>t \in \mathbb{A}^1</math>, there are short exact sequences<blockquote><math>0 \to \mathcal{O}(-1) \to E_t \to \mathcal{O}(1) \to 0</math></blockquote>which have [[Chern class]]es <math>c_1 = 0, c_2=0</math> generically, but have <math>c_1=0, c_2 = -1</math> at the origin. This kind of jumping of numerical invariants does not happen in moduli spaces of stable vector bundles.<ref>{{Cite web|url=http://www.mathe2.uni-bayreuth.de/stoll/lecture-notes/vector-bundles-Faltings.pdf|title=Vector bundles on curves|last=Faltings|first=Gerd|url-status=live|archive-url=https://web.archive.org/web/20200304184453/http://www.mathe2.uni-bayreuth.de/stoll/lecture-notes/vector-bundles-Faltings.pdf|archive-date=4 March 2020}}</ref>

== Stable vector bundles over curves ==

A '''slope''' of a [[holomorphic vector bundle]] ''W'' over a nonsingular [[algebraic curve]] (or over a [[Riemann surface]]) is a rational number ''μ(W)'' = deg(''W'')/rank(''W''). A bundle ''W'' is '''stable''' if and only if

:<math>\mu(V) < \mu(W)</math>

for all proper non-zero subbundles ''V'' of ''W'' 
and is '''semistable''' if

:<math>\mu(V) \le \mu(W)</math>

for all proper non-zero subbundles ''V'' of ''W''. Informally this says that a bundle is stable if it is "more [[ample]]" than any proper subbundle, and is unstable if it contains a "more ample" subbundle.

If ''W'' and ''V'' are semistable vector bundles and ''μ(W)'' >''μ(V)'', then there are no nonzero maps ''W'' → ''V''.

[[David Mumford|Mumford]] proved that the moduli space of stable bundles of given rank and degree over a nonsingular curve is a [[quasiprojective]] [[algebraic variety]]. The [[cohomology]] of the [[moduli space]] of stable vector bundles over a curve was described by {{harvtxt|Harder|Narasimhan|1975}} using algebraic geometry over [[finite field]]s and {{harvtxt|Atiyah|Bott|1983}} using [[Narasimhan-Seshadri theorem|Narasimhan-Seshadri approach]].

==Stable vector bundles in higher dimensions==
If ''X'' is a [[smooth scheme|smooth]] [[projective variety]] of dimension ''m'' and ''H'' is a [[hyperplane section]], then a vector bundle (or a [[torsion-free module#Torsion-free quasicoherent sheaves|torsion-free]] sheaf) ''W'' is called '''stable''' (or sometimes '''[[David Gieseker|Gieseker]] stable''') if

:<math>\frac{\chi(V(nH))}{\hbox{rank}(V)} < \frac{\chi(W(nH))}{\hbox{rank}(W)}\text{ for }n\text{ large}</math>

for all proper non-zero subbundles (or subsheaves) ''V'' of ''W'', where χ denotes the [[Euler characteristic]] of an algebraic vector bundle and the vector bundle ''V(nH)'' means the ''n''-th [[Serre twist|twist]] of ''V'' by ''H''. ''W'' is called '''semistable''' if the above holds with &lt; replaced by ≤.

==Slope stability==

For bundles on curves the stability defined by slopes and by growth of Hilbert polynomial coincide. In higher dimensions, these two notions are different and have different advantages. Gieseker stability has an interpretation in terms of [[geometric invariant theory]], while μ-stability has better properties for [[tensor product bundle|tensor products]], [[pullback bundle|pullbacks]], etc.

Let ''X'' be a [[smooth scheme|smooth]] [[projective variety]] of dimension ''n'', ''H'' its [[hyperplane section]]. A '''slope''' of a vector bundle (or, more generally, a [[torsion-free module#Torsion-free quasicoherent sheaves|torsion-free]] [[coherent sheaf]]) ''E'' with respect to ''H'' is a rational number defined as

:<math>\mu(E) := \frac{c_1(E) \cdot H^{n-1}}{\operatorname{rk}(E)}</math>

where ''c''<sub>1</sub> is the first [[Chern class]]. The dependence on ''H'' is often omitted from the notation.

A torsion-free coherent sheaf ''E'' is '''μ-semistable''' if for any nonzero subsheaf ''F'' ⊆ ''E'' the slopes satisfy the inequality μ(F) ≤ μ(E). It's '''μ-stable''' if, in addition, for any nonzero subsheaf ''F'' ⊆ ''E'' of smaller rank the strict inequality μ(F) < μ(E) holds. This notion of stability may be called slope stability, μ-stability, occasionally Mumford stability or Takemoto stability.

For a vector bundle ''E'' the following chain of implications holds: ''E'' is μ-stable ⇒ ''E'' is stable ⇒ ''E'' is semistable ⇒ ''E'' is μ-semistable.

==Harder-Narasimhan filtration==
{{main|Harder–Narasimhan stratification}}
Let ''E'' be a vector bundle over a smooth projective curve ''X''. Then there exists a unique [[filtration (mathematics)|filtration]] by subbundles

:<math>0 = E_0 \subset E_1 \subset \ldots \subset E_{r+1} = E</math>
such that the [[associated graded module|associated graded]] components ''F''<sub>''i''</sub> := ''E''<sub>''i''+1</sub>/''E''<sub>''i''</sub> are semistable vector bundles and the slopes decrease, μ(''F''<sub>''i''</sub>) > μ(''F''<sub>''i''+1</sub>). This filtration was introduced in {{harvtxt|Harder|Narasimhan|1975}} and is called the '''Harder-Narasimhan filtration'''. Two vector bundles with isomorphic associated gradeds are called [[S-equivalence|S-equivalent]].

On higher-dimensional varieties the filtration also always exist and is unique, but the associated graded components may no longer be bundles. For Gieseker stability the inequalities between slopes should be replaced with inequalities between Hilbert polynomials.

==Kobayashi–Hitchin correspondence==

{{main article|Kobayashi–Hitchin correspondence}}
[[Narasimhan–Seshadri theorem]] says that stable bundles on a projective nonsingular curve are the same as those that have projectively flat unitary irreducible [[connection (vector bundle)|connections]]. For bundles of degree 0 projectively flat connections are [[flat vector bundle|flat]] and thus stable bundles of degree 0 correspond to [[irreducible representation|irreducible]] [[unitary representation]]s of the [[fundamental group]].

[[Shoshichi Kobayashi|Kobayashi]] and [[Nigel Hitchin|Hitchin]] conjectured an analogue of this in higher dimensions.  It was proved for projective nonsingular surfaces by {{harvtxt|Donaldson|1985}}, who showed that in this case a vector bundle  is stable if and only if it has an irreducible [[Hermitian–Einstein connection]].

==Generalizations==

It's possible to generalize (μ-)stability to [[Singular point of an algebraic variety|non-smooth]] projective [[scheme (mathematics)|schemes]] and more general [[coherent sheaf|coherent sheaves]] using the [[Hilbert series and Hilbert polynomial#Generalization to coherent sheaves|Hilbert polynomial]]. Let ''X'' be a [[projective scheme]], ''d'' a natural number, ''E'' a coherent sheaf on ''X'' with dim Supp(''E'') = ''d''. Write the Hilbert polynomial of ''E'' as ''P''<sub>''E''</sub>(''m'') = {{larger|Σ}}{{sup sub | ''d'' |''i''{{=}}0}} α<sub>''i''</sub>(''E'')/(''i''!) ''m''<sup>''i''</sup>. Define the '''reduced Hilbert polynomial''' ''p''<sub>''E''</sub> := ''P''<sub>''E''</sub>/α<sub>''d''</sub>(''E'').

A coherent sheaf ''E'' is '''semistable''' if the following two conditions hold:<ref>{{cite book|author1=Huybrechts, Daniel |author2=Lehn, Manfred |title=The Geometry of Moduli Spaces of Sheaves|year=1997|url=https://ncatlab.org/nlab/files/HuybrechtsLehn.pdf}}, Definition 1.2.4</ref>
* ''E'' is pure of dimension ''d'', i.e. all [[associated primes]] of ''E'' have dimension ''d'';
* for any proper nonzero subsheaf ''F'' ⊆ ''E'' the reduced Hilbert polynomials satisfy ''p''<sub>''F''</sub>(''m'') ≤ ''p''<sub>''E''</sub>(''m'') for large ''m''.
A sheaf is called '''stable''' if the strict inequality ''p''<sub>''F''</sub>(''m'') < ''p''<sub>''E''</sub>(''m'') holds for large ''m''.

Let Coh<sub>''d''</sub>(X) be the full subcategory of coherent sheaves on ''X'' with support of dimension ≤ ''d''. The '''slope''' of an object ''F'' in Coh<sub>''d''</sub> may be defined using the coefficients of the Hilbert polynomial as <math>\hat{\mu}_d(F) = \alpha_{d-1}(F)/\alpha_d(F)</math> if α<sub>''d''</sub>(''F'') ≠ 0 and 0 otherwise. The dependence of <math>\hat{\mu}_d</math> on ''d'' is usually omitted from the notation.

A coherent sheaf ''E'' with <math>\operatorname{dim}\,\operatorname{Supp}(E) = d</math> is called '''μ-semistable''' if the following two conditions hold:<ref>{{cite book|author1=Huybrechts, Daniel |author2=Lehn, Manfred |title=The Geometry of Moduli Spaces of Sheaves|year=1997|url=https://ncatlab.org/nlab/files/HuybrechtsLehn.pdf}}, Definition 1.6.9</ref>
*the torsion of ''E'' is in dimension ≤ ''d''-2;
*for any nonzero subobject ''F'' ⊆ ''E'' in the [[Quotient of an abelian category|quotient category]] Coh<sub>''d''</sub>(X)/Coh<sub>''d-1''</sub>(X) we have <math>\hat{\mu}(F) \leq \hat{\mu}(E)</math>.
''E'' is '''μ-stable''' if the strict inequality holds for all proper nonzero subobjects of ''E''.

Note that Coh<sub>''d''</sub> is a [[Serre subcategory]] for any ''d'', so the quotient category exists. A subobject in the quotient category in general doesn't come from a subsheaf, but for torsion-free sheaves the original definition and the general one for ''d'' = ''n'' are equivalent.

There are also other directions for generalizations, for example [[Thomas Bridgeland|Bridgeland]]'s [[Bridgeland stability condition|stability condition]]s.

One may define [[stable principal bundle]]s in analogy with stable vector bundles.

== See also==
* [[Kobayashi–Hitchin correspondence]]
* [[Simpson correspondence|Corlette–Simpson correspondence]]
*[[Quot scheme]]

==References==
{{reflist}}
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Bott | first2=Raoul | author2-link=Raoul Bott | title=The Yang-Mills equations over Riemann surfaces | jstor=37156 | mr=702806  | year=1983 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=308 | issue=1505 | pages=523–615 | doi=10.1098/rsta.1983.0017}}
*{{Citation | last1=Donaldson | first1=S. K. | title=Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles | mr=765366  | year=1985 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=50 | issue=1 | pages=1–26|doi=10.1112/plms/s3-50.1.1 }}
*{{Citation | last1=Friedman | first1=Robert | title=Algebraic surfaces and holomorphic vector bundles | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-0-387-98361-5 | mr=1600388  | year=1998}}
*{{Citation | last1=Harder | first1=G. | last2=Narasimhan | first2=M. S. | title=On the cohomology groups of moduli spaces of vector bundles on curves | doi=10.1007/BF01357141 | mr=0364254  | year=1975 |  journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=212 | pages=215–248 | issue=3}}
*{{Citation | last1=Huybrechts | first1=Daniel | author1-link=Daniel Huybrechts | last2=Lehn | first2=Manfred | title=The Geometry of Moduli Spaces of Sheaves | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Mathematical Library | isbn=978-0521134200 | year=2010}}
*{{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Proc. Internat. Congr. Mathematicians (Stockholm, 1962) | publisher=Inst. Mittag-Leffler | location=Djursholm | mr=0175899  | year=1963 | chapter=Projective invariants of projective structures and applications | pages=526–530}}
*{{Citation | last1=Mumford | first1=David | author1-link=David Mumford | last2=Fogarty | first2=J. | last3=Kirwan | first3=F. | title=Geometric invariant theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)] | isbn=978-3-540-56963-3 | mr=1304906  | year=1994 | volume=34}} especially appendix 5C.
*{{Citation | last1=Narasimhan | first1=M. S. | last2=Seshadri | first2=C. S. | title=Stable and unitary vector bundles on a compact Riemann surface | jstor=1970710 | mr=0184252  | year=1965 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=82 | pages=540–567 | doi=10.2307/1970710 | issue=3 | publisher=The Annals of Mathematics, Vol. 82, No. 3}}

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