nLab stable vector bundle

Idea

A vector bundle (typically considered in complex-analytic geometry or algebraic geometry) is called (semi-)stable if it is a (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 slopes of the sub-bundles (see 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 Bridgeland stability condition.

Definition

Over a Riemann surface / over an algebraic curve

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

$$\mu(K) \lt \mu(E)$$

between their slopes holds, i.e. if the inequality

$$\frac{deg(K)}{rank(K)} \lt \frac{deg(E)}{rank(E)}$$

holds between the fractions of degree and rank of the vector bundles holds.

e.g. (Huybrechts-Lehn 96, bottom of p. 24)

Over a general (Noetherian) scheme

e.g. (Huybrechts-Lehn 96, def. 1.2.4, def. 1.2.12)

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. (Huybrechts-Lehn 96, example 1.2.10)

Properties

Relation to GIT-stability

The slope-(semi-)stable vector bundles are essentially the (semi-)stable points in the sense of geometric invariant theory in the moduli space of bundles. The precise statement is discussed in (King 94) reviewed for instance in (Saiz 09, section 2.3.

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.

Relation to Bridgeland stability conditions

Slope-stability of vector bundles is a special case of a Bridgeland stability condition, see there For review see e.g. (Engenhorst 14, sections 3 and 4) and see King 94.

Related concepts

• Harder-Narasimhan filtration

• moduli space of bundles

• stable point

• positive line bundle, ample line bundle

• Bridgeland stability condition

• geometric invariant theory

References

The concept was introduced in

• David Mumford, Geometric invariant theory, Ergebnisse Math. Vol 34 Springer (1965)

• F. Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972) 29-48 (euclid); II, 52 (1973) (euclid)

• David Mumford, John Fogarty, Frances Clare Kirwan, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, Springer-Verlag (1965)

Review is in

• Michael Atiyah, Raoul Bott, section 7 of 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 (jstor, lighning summary)

A textbook account is in

• Daniel Huybrechts, Manfred Lehn, The Geometry of the Moduli Spaces of Sheaves, 1996 (pdf)

More discussion with regards to geometric invariant theory and Bridgeland stability conditions is in

• Alastair King, Moduli of representations of finite dimensional algebras, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 (pdf)

• Jan Engenhorst, Bridgeland Stability Conditions in Algebra, Geometry and Physics, 2014 (pdf)

See also

• Paolo de Bartolomeis, Gang Tian, Stability of complex vector bundles, J. Diff. Geometry 43:2 (1996) (pdf)

• Wei-Ping Li, Zhenbo Qin, Stable vector bundles on algebraic surfaces (pdf)

• Alfonso Zamora Saiz, On the stability of vector bundles, Master thesis 2009 (pdf)

Discussion for equivariant vector bundles is in

• C. S. Seshadri, 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 (doi:10.1007/978-3-642-11015-3_5)

• Oscar García-Prada, Invariant connections and vortices, Commun.Math. Phys. (1993) 156: 527 (doi:10.1007/BF02096862)