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For a holomorphic vector bundle $E$, $\mathcal{O}(E)(U)$ is its space of holomorphic sections on $U$, with their restrictions. It is the kernel of the bundle Dolbeault operator on smooth sections.

For a holomorphic vector bundle $E$, this sheaf assigns to $U$ the smooth $E$-valued differential forms of type (p, q) on $U$. Its Dolbeault operator acts coefficientwise in a holomorphic local frame and squares to zero.

A nowhere-zero smooth section $s$ of a holomorphic line bundle writes its Dolbeault operator as ${\bar{\partial }}_{L}s=a\otimes s$. Its integrability gives $\bar{\partial }a=0$. If $H^{0,1}(X)=0$, then $a=\bar{\partial }f$ globally. The section $e^{-f}s$ is nowhere zero and holomorphic. This argument requires no compactness and explicitly constructs the holomorphic trivialization.

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.