nLab degree of a coherent sheaf

Idea

The concept of degree of a coherent sheaf is a generalization of the concept of first Chern class of a line bundle, to which it reduces for coherent sheaves of sections of holomorphic line bundles over Riemann surfaces in complex analytic geometry (see at first Chern class – In complex analytic geometry).

Definition

Discussion of degrees of coherent sheaves over complex curves is in e.g. (Huybrechts-Lehn 96, def. 1.2.11).

On a compact Kähler manifold $(X,\omega)$ of complex dimension $n$, then the degree of a torsion-free holomorphic vector bundle $E$ is the integral of differential forms

$$deg(E) = \int_X c_1(E) \wedge \omega^{\wedge_{n-1}}$$

of the de Rham representative of the actual first Chern class of $E$ wedge with copies of the given symplectic form (e.g. Scheinost-Schottenloher 96, def. 1.14).

In complex analytic geometry

Related concepts

• rank of a coherent sheaf

• slope of a coherent sheaf

• stable coherent sheaf, Bridgeland stability condition

References

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

• Peter Scheinost, Martin Schottenloher, Metaplectic quantization of the moduli spaces of flat and parabolic bundles, J. reine angew. Mathematik, 466 (1996) (web)