nLab degree of a coherent sheaf




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).


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,ω)(X,\omega) of complex dimension nn, then the degree of a torsion-free holomorphic vector bundle EE is the integral of differential forms

deg(E)= Xc 1(E)ω n1 deg(E) = \int_X c_1(E) \wedge \omega^{\wedge_{n-1}}

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

In complex analytic geometry


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