Degree of a holomorphic vector bundle ($\mathrm{deg}E=\mathrm{deg}\det E=\frac{i}{2\pi }{\int }_X\mathrm{tr}{F}_E$)

On a compact Riemann surface, the degree of a holomorphic vector bundle is the degree of a line bundle of its determinant line bundle. A Hermitian metric on a holomorphic vector bundle and its Chern connection express this degree by the trace-curvature integral. The value is independent of the metric because the normalized trace curvature represents the First Chern class.

The slope of a nonzero holomorphic vector bundle on a compact Riemann surface is its degree of a holomorphic vector bundle divided by its rank of a vector bundle. It measures degree per fibre dimension. Comparing subbundle slopes defines semistable holomorphic vector bundles.

Fix a Hermitian metric on a holomorphic vector bundle $E$ over a compact Riemann surface and a positive area form $\omega$. Write $i{F}_E=K\omega$ for its Chern curvature. If every eigenvalue of the Hermitian endomorphism $K$ is at most $C$, the curvature formula for a holomorphic subbundle gives $\mathrm{deg}F\le \mathrm{rank}(F)C{\int }_X\omega /(2\pi )$. The second-fundamental-form term is nonpositive. Compactness supplies a finite $C$ independent of the subbundle. If $F_E=0$, every subbundle has nonpositive degree.

A positive-rank holomorphic vector bundle on a compact Riemann surface is semistable when every nonzero proper holomorphic subbundle has slope at most its own. Equivalently one tests coherent subsheaves: saturation on a smooth curve preserves rank, increases degree, and gives a subbundle with torsion-free quotient. Strict inequality defines stability, which is stronger.