= Unitary flat holomorphic vector bundle
A <unitary representation> of the <fundamental group> of a connected <complex manifold> gives a bundle $(\widetilde X\times V)/\pi_1X$ with locally constant unitary transition matrices. Its induced flat <unitary connection> has $(0,1)$ part equal to its <Dolbeault operator>, so it is the <Chern connection> for the induced Hermitian metric. On a <compact Riemann surface>, zero curvature gives degree zero; the subbundle curvature inequality makes the bundle semistable.
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