= Subbundle slope bound from ambient Chern curvature
{title2=$\mu(F)\le C\operatorname{area}(X)/(2\pi)$}
Fix a <Hermitian metric on a holomorphic vector bundle> $E$ over a <compact Riemann surface> and a positive area form $\omega$. Write $iF_E=K\omega$ for its <Chern curvature>. If every eigenvalue of the Hermitian endomorphism $K$ is at most $C$, the <curvature formula for a holomorphic subbundle> gives $\deg F\le\operatorname{rank}(F)C\int_X\omega/(2\pi)$. 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.
Back to article page
