Unitary flat holomorphic vector bundle

A unitary representation of the fundamental group of a connected complex manifold gives a bundle $(\tilde{X}\times V)/{\pi }_{1}X$ 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.

In a local parallel orthonormal frame a holomorphic section has holomorphic coefficients $f_j$. Its squared norm satisfies ${\partial }_z{\partial }_{\bar{z}}{\sum }_j|f_j{|}^2={\sum }_j|f_j^{\prime }{|}^2\ge 0$. On a compact connected complex curve, integration of the Laplacian makes the right side zero. Thus every coefficient is locally constant and the section is parallel. A global parallel section is determined by a vector fixed by all monodromy operators, and every such vector gives a global section.