# nLab degree of a coherent sheaf

Contents

### Context

#### Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## 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).

## References

Last revised on October 3, 2018 at 10:22:19. See the history of this page for a list of all contributions to it.