# Stable vector bundle

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

== Motivation ==
On a smooth projective variety, [line bundles](/source/Line_bundle) of given numerical invariants are parametrised over a well-behaved [moduli space](/source/moduli_space) (isomorphic to the [Picard variety)](/source/Picard_scheme). This space is, in particular, a [proper](/source/Proper_scheme) and [separated](/source/Separated_scheme) scheme of [finite type](/source/Finite_type_scheme), thus lending itself to algebro-geometric analysis. Similar considerations fail when naively parametrising vector bundles of higher rank. 

As an example, consider the moduli of vector bundles of rank <math>r=2</math> and first [Chern class](/source/Chern_class) <math>c_1=0</math> on the [complex projective line](/source/Projective_line) <math>\mathbb{P}^1</math>. If they were to form a separated moduli space, the [valuative criterion](/source/valuative_criterion) would imply that any family over the punctured line <math>\mathbb{C}^\ast</math> can be completed to at most one family over <math>\mathbb{C}</math>. But it is straightforward to construct a family that admits two non-isomorphic completions.

Indeed consider the constant family assigning each <math>t\in \mathbb{C}^\ast</math> to the bundle <math>V_t\simeq \mathcal{O}\oplus\mathcal{O} </math>. Tensoring the [Euler sequence](/source/Euler_sequence) of <math>\mathbb{P}^1</math> by <math>\mathcal{O}(1)</math> gives a non-split exact sequence<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](/source/Adjunction_formula) on the canonical sheaf.</ref> and hence this family can be described by saying <math>V_t</math> is the [extension](/source/Ext_group) corresponding to <math>1\in \mathbb{C}\simeq  \operatorname{Ext}^1(\mathcal{O}(1),\mathcal{O}(-1))</math>. Likewise, assigning <math>t\in \mathbb{C}^\ast</math> to the extension corresponding to <math>t\in \operatorname{Ext}^1(\mathcal{O}(1),\mathcal{O}(-1))</math> furnishes another family <math>W_t\simeq \mathcal{O}\oplus \mathcal{O}</math>. The two families are isomorphic, where the isomorphism <math>V_t\to W_t</math> is induced by the automorphism <math>\mathcal{O}(-1)\to \mathcal{O}(-1)</math> given by multiplication by <math>t</math> - that is, we have two equivalent descriptions of the same family.  

When naturally completed over <math>\mathbb{C}</math>, the first description continues to yield the extension <math>V_0=\mathcal{O}\oplus \mathcal{O}</math> corresponding to <math>1\in \operatorname{Ext}^1(\mathcal{O}(1),\mathcal{O}(-1))</math>. The second description, on the other hands, yields the split extension <math>W_0\simeq \mathcal{O}(-1)\oplus\mathcal{O}(1)</math> corresponding to <math>t=0\in \operatorname{Ext}^1(\mathcal{O}(1),\mathcal{O}(-1))</math>. The two bundles, and hence the two completions, are non-isomorphic. 

The notion of stability navigates this issue by restricting the class of bundles that may appear in the moduli space, with the upshot of preserving desirable algebro-geometric properties.

== Stable vector bundles over curves ==

A '''slope''' of a [holomorphic vector bundle](/source/holomorphic_vector_bundle) ''W'' over a nonsingular [algebraic curve](/source/algebraic_curve) (or over a [Riemann surface](/source/Riemann_surface)) is a rational number ''μ(W)'' = deg(''W'')/rank(''W''). A bundle ''W'' is '''stable''' if and only if<ref>Huybrechts 04, Definition 4.B.8</ref>

:<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](/source/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''.

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

==Stable vector bundles in higher dimensions==
If ''X'' is a [smooth](/source/smooth_scheme) [projective variety](/source/projective_variety) of dimension ''m'' and ''H'' is a [hyperplane section](/source/hyperplane_section), then a vector bundle (or a [torsion-free](/source/torsion-free_module) sheaf) ''W'' is called '''stable''' (or sometimes '''[Gieseker](/source/David_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](/source/Euler_characteristic) of an algebraic vector bundle and the vector bundle ''V(nH)'' means the ''n''-th [twist](/source/Serre_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](/source/geometric_invariant_theory), while μ-stability has better properties for [tensor products](/source/tensor_product_bundle), [pullbacks](/source/pullback_bundle), etc.

Let ''X'' be a [smooth](/source/smooth_scheme) [projective variety](/source/projective_variety) of dimension ''n'', ''H'' its [hyperplane section](/source/hyperplane_section). A '''slope''' of a vector bundle (or, more generally, a [torsion-free](/source/torsion-free_module) [coherent sheaf](/source/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](/source/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](/source/filtration_(mathematics)) by subbundles

:<math>0 = E_0 \subset E_1 \subset \ldots \subset E_{r+1} = E</math>
such that the [associated graded](/source/associated_graded_module) 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 grades are called [S-equivalent](/source/S-equivalence).

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](/source/Narasimhan%E2%80%93Seshadri_theorem) says that stable bundles on a projective nonsingular curve are the same as those that have projectively flat unitary irreducible [connections](/source/connection_(vector_bundle)). For bundles of degree 0 projectively flat connections are [flat](/source/flat_vector_bundle) and thus stable bundles of degree 0 correspond to [irreducible](/source/irreducible_representation) [unitary representation](/source/unitary_representation)s of the [fundamental group](/source/fundamental_group).

[Kobayashi](/source/Shoshichi_Kobayashi) and [Hitchin](/source/Nigel_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](/source/Hermitian%E2%80%93Einstein_connection).

==Generalizations==

It's possible to generalize (μ-)stability to [non-smooth](/source/Singular_point_of_an_algebraic_variety) projective [schemes](/source/scheme_(mathematics)) and more general [coherent sheaves](/source/coherent_sheaf) using the [Hilbert polynomial](/source/Hilbert_series_and_Hilbert_polynomial). Let ''X'' be a [projective scheme](/source/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](/source/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 category](/source/Quotient_of_an_abelian_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](/source/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 [Bridgeland](/source/Thomas_Bridgeland)'s [stability condition](/source/Bridgeland_stability_condition)s.

One may define [stable principal bundle](/source/stable_principal_bundle)s in analogy with stable vector bundles.

== See also==
* [Kobayashi–Hitchin correspondence](/source/Kobayashi%E2%80%93Hitchin_correspondence)
* [Corlette–Simpson correspondence](/source/Simpson_correspondence)
*[Quot scheme](/source/Quot_scheme)

== Literature ==

*{{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 | bibcode=1983RSPTA.308..523A }}
*{{Citation | last1=Donaldson | first1=S. K. |author-link=Simon Donaldson | 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](/source/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](/source/Mathematische_Annalen) | issn=0025-5831 | volume=212 | pages=215–248 | issue=3}}
*{{cite book |last1=Huybrechts |first1=Daniel |author-link=Daniel Huybrechts |url=https://link.springer.com/book/10.1007/b137952 |title=Complex geometry: An introduction |publisher=[Springer Science+Business Media](/source/Springer_Science%2BBusiness_Media) |year=2004-11-18 |isbn=978-3540212904 |series=Universitext |volume= |pages= |language=en |doi= |issn= |mr=}}
*{{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](/source/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](/source/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](/source/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}}

==References==
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Adapted from the Wikipedia article [Stable vector bundle](https://en.wikipedia.org/wiki/Stable_vector_bundle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Stable_vector_bundle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.