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Let $X$ be a complex manifold. Let $E$ be a hermitian vector bundle with a given hermitian metric over $X$. On a local trivialization open subset, is there a smooth orthonormal local frame? is there a holomorphic orthonormal local frame?

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Yes, there's always a (local) smooth unitary frame, just as in the real orthonormal case, as you can do Gram-Schmidt. Since the only holomorphic functions of constant magnitude are constants, the only unitary frames that can be holomorphic are constant frames on a trivial bundle.

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  • $\begingroup$ Thank you ! Maybe I haven't understand your answer yet. Please excuse my stupidity. Does the Gram-Schmidt process break the smoothness of the original section? Does the holomorphic local orthonomal frame exist or not on earth (i.e., does a constant frame exist locally? ) $\endgroup$ – jack lion Mar 6 '20 at 3:40
  • $\begingroup$ No, Gram-Schmidt is perfectly smooth, but not holomorphic. Unless you have a globally trivial (or flat) bundle, having a constant frame in one trivialization will not translate to a constant frame in another, overlapping trivialization. $\endgroup$ – Ted Shifrin Mar 6 '20 at 3:45

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