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