= Holomorphic sections of a unitary flat bundle are parallel
{title2=$H^0(X,E_\rho)\cong V^{\pi_1X}$}
In a local parallel orthonormal frame a <holomorphic section> has holomorphic coefficients $f_j$. Its squared norm satisfies $\partial_z\partial_{\bar z}\sum_j|f_j|^2=\sum_j|f'_j|^2\ge0$. 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.
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