Inmathematics, astable vector bundle is a (holomorphic oralgebraic)vector bundle that is stable in the sense ofgeometric invariant theory. Any holomorphic vector bundle may be built from stable ones usingHarder–Narasimhan filtration. Stable bundles were defined byDavid Mumford inMumford (1963) and later built upon byDavid Gieseker,Fedor Bogomolov,Thomas Bridgeland and many others.

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One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact,Moduli spaces of stable vector bundles can be constructed using theQuot scheme in many cases, whereas the stack of vector bundles${\displaystyle \mathbf{B}G{L}_n}$ is anArtin stack whose underlying set is a single point.

Here's an example of a family of vector bundles which degenerate poorly. If we tensor theEuler sequence of${\displaystyle {\mathbb{P}}^1}$ by${\displaystyle \mathcal{O}(1)}$ there is an exact sequence^[1]

${\displaystyle 0\to \mathcal{O}(-1)\to \mathcal{O}\oplus \mathcal{O}\to \mathcal{O}(1)\to 0}$^[2]

which represents a non-zero element${\displaystyle v\in {\text{Ext}}^1(\mathcal{O}(1),\mathcal{O}(-1))\cong k}$^[3] since the trivial exact sequence representing the${\displaystyle 0}$ vector is

${\displaystyle 0\to \mathcal{O}(-1)\to \mathcal{O}(-1)\oplus \mathcal{O}(1)\to \mathcal{O}(1)\to 0}$

If we consider the family of vector bundles${\displaystyle E_t}$ in the extension from${\displaystyle t\cdot v}$ for${\displaystyle t\in {\mathbb{A}}^1}$, there are short exact sequences

${\displaystyle 0\to \mathcal{O}(-1)\to E_t\to \mathcal{O}(1)\to 0}$

which haveChern classes${\displaystyle c_1=0,c_2=0}$ generically, but have${\displaystyle c_1=0,c_2=-1}$ at the origin. This kind of jumping of numerical invariants does not happen in moduli spaces of stable vector bundles.^[4]

Aslope of aholomorphic vector bundleW over a nonsingularalgebraic curve (or over aRiemann surface) is a rational numberμ(W) = deg(W)/rank(W). A bundleW isstable if and only if^[5]

for all proper non-zero subbundlesV ofW and issemistable if

${\displaystyle \mu (V)\le \mu (W)}$

for all proper non-zero subbundlesV ofW. Informally this says that a bundle is stable if it is "moreample" than any proper subbundle, and is unstable if it contains a "more ample" subbundle.

IfW andV are semistable vector bundles andμ(W) >μ(V), then there are no nonzero mapsW →V.

Mumford proved that the moduli space of stable bundles of given rank and degree over a nonsingular curve is aquasiprojectivealgebraic variety. Thecohomology of themoduli space of stable vector bundles over a curve was described byHarder & Narasimhan (1975) using algebraic geometry overfinite fields andAtiyah & Bott (1983) usingNarasimhan-Seshadri approach.

IfX is asmoothprojective variety of dimensionm andH is ahyperplane section, then a vector bundle (or atorsion-free sheaf)W is calledstable (or sometimesGieseker stable) if

for all proper non-zero subbundles (or subsheaves)V ofW, where χ denotes theEuler characteristic of an algebraic vector bundle and the vector bundleV(nH) means then-thtwist ofV byH.W is calledsemistable if the above holds with < replaced by ≤.

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 ofgeometric invariant theory, while μ-stability has better properties fortensor products,pullbacks, etc.

LetX be asmoothprojective variety of dimensionn,H itshyperplane section. Aslope of a vector bundle (or, more generally, atorsion-freecoherent sheaf)E with respect toH is a rational number defined as

${\displaystyle \mu (E):=\frac{c_1(E)\cdot H^{n-1}}{\mathrm{rk}(E)}}$

wherec₁ is the firstChern class. The dependence onH is often omitted from the notation.

A torsion-free coherent sheafE isμ-semistable if for any nonzero subsheafF ⊆E the slopes satisfy the inequality μ(F) ≤ μ(E). It'sμ-stable if, in addition, for any nonzero subsheafF ⊆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 bundleE the following chain of implications holds:E is μ-stable ⇒E is stable ⇒E is semistable ⇒E is μ-semistable.

LetE be a vector bundle over a smooth projective curveX. Then there exists a uniquefiltration by subbundles

${\displaystyle 0=E_0\subset E_1\subset \dots \subset E_{r+1}=E}$

such that theassociated graded componentsF_i :=E_i+1/E_i are semistable vector bundles and the slopes decrease, μ(F_i) > μ(F_i+1). This filtration was introduced inHarder & Narasimhan (1975) and is called theHarder-Narasimhan filtration. Two vector bundles with isomorphic associated grades are calledS-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.

Narasimhan–Seshadri theorem says that stable bundles on a projective nonsingular curve are the same as those that have projectively flat unitary irreducibleconnections. For bundles of degree 0 projectively flat connections areflat and thus stable bundles of degree 0 correspond toirreducibleunitary representations of thefundamental group.

Kobayashi andHitchin conjectured an analogue of this in higher dimensions. It was proved for projective nonsingular surfaces byDonaldson (1985), who showed that in this case a vector bundle is stable if and only if it has an irreducibleHermitian–Einstein connection.

It's possible to generalize (μ-)stability tonon-smooth projectiveschemes and more generalcoherent sheaves using theHilbert polynomial. LetX be aprojective scheme,d a natural number,E a coherent sheaf onX with dim Supp(E) =d. Write the Hilbert polynomial ofE asP_E(m) =Σ^d
_i=0 α_i(E)/(i!)mⁱ. Define thereduced Hilbert polynomialp_E :=P_E/α_d(E).

A coherent sheafE issemistable if the following two conditions hold:^[6]

• E is pure of dimensiond, i.e. allassociated primes ofE have dimensiond;
• for any proper nonzero subsheafF ⊆E the reduced Hilbert polynomials satisfyp_F(m) ≤p_E(m) for largem.

A sheaf is calledstable if the strict inequalityp_F(m) <p_E(m) holds for largem.

Let Coh_d(X) be the full subcategory of coherent sheaves onX with support of dimension ≤d. Theslope of an objectF in Coh_d may be defined using the coefficients of the Hilbert polynomial as${\displaystyle {\hat{\mu }}_d(F)={\alpha }_{d-1}(F)/{\alpha }_d(F)}$ if α_d(F) ≠ 0 and 0 otherwise. The dependence of${\displaystyle {\hat{\mu }}_d}$ ond is usually omitted from the notation.

A coherent sheafE with${\displaystyle \dim \mathrm{Supp}(E)=d}$ is calledμ-semistable if the following two conditions hold:^[7]

• the torsion ofE is in dimension ≤d-2;
• for any nonzero subobjectF ⊆E in thequotient category Coh_d(X)/Coh_d-1(X) we have${\displaystyle \hat{\mu }(F)\le \hat{\mu }(E)}$.

E isμ-stable if the strict inequality holds for all proper nonzero subobjects ofE.

Note that Coh_d is aSerre subcategory for anyd, 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 ford =n are equivalent.

There are also other directions for generalizations, for exampleBridgeland'sstability conditions.

One may definestable principal bundles in analogy with stable vector bundles.

• Kobayashi–Hitchin correspondence
• Corlette–Simpson correspondence
• Quot scheme

• Atiyah, Michael Francis;Bott, Raoul (1983), "The Yang-Mills equations over Riemann surfaces",Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences,308 (1505):523–615,Bibcode:1983RSPTA.308..523A,doi:10.1098/rsta.1983.0017,ISSN 0080-4614,JSTOR 37156,MR 0702806
• Donaldson, S. K. (1985), "Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles",Proceedings of the London Mathematical Society, Third Series,50 (1):1–26,doi:10.1112/plms/s3-50.1.1,ISSN 0024-6115,MR 0765366
• Friedman, Robert (1998),Algebraic surfaces and holomorphic vector bundles, Universitext, Berlin, New York:Springer-Verlag,ISBN 978-0-387-98361-5,MR 1600388
• Harder, G.; Narasimhan, M. S. (1975), "On the cohomology groups of moduli spaces of vector bundles on curves",Mathematische Annalen,212 (3):215–248,doi:10.1007/BF01357141,ISSN 0025-5831,MR 0364254
• Huybrechts, Daniel (2004-11-18).Complex geometry: An introduction. Universitext.Springer Science+Business Media.ISBN 978-3540212904.{{cite book}}: CS1 maint: year (link)
• Huybrechts, Daniel; Lehn, Manfred (2010),The Geometry of Moduli Spaces of Sheaves, Cambridge Mathematical Library (2nd ed.),Cambridge University Press,ISBN 978-0521134200
• Mumford, David (1963), "Projective invariants of projective structures and applications",Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 526–530,MR 0175899
• Mumford, David; Fogarty, J.; Kirwan, F. (1994),Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34 (3rd ed.), Berlin, New York:Springer-Verlag,ISBN 978-3-540-56963-3,MR 1304906 especially appendix 5C.
• Narasimhan, M. S.; Seshadri, C. S. (1965), "Stable and unitary vector bundles on a compact Riemann surface",Annals of Mathematics, Second Series,82 (3), The Annals of Mathematics, Vol. 82, No. 3:540–567,doi:10.2307/1970710,ISSN 0003-486X,JSTOR 1970710,MR 0184252

• Algebraic geometry

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