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Consider a Riemann surface $\Sigma$ and a holomorphic vector bundle $\mathcal{E} \to \Sigma$ of rank $N$.

I just came across the remark that in order to endow $\mathcal{E}$ with a Hermitian metric it is necessary that the positive definite Hermitian matrices of rank $N$ constitute a convex set in GL($N,\mathbb{C})$.

I would like to understand why this is so, but I have trouble finding the right source where I can read about this - any hints as to why I need the convexity, or reference suggestions targeted to this aspect of complex manifold theory in case this question is too broad, would be very helpfull! many thanks !

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For any trivilization of the vector bundle, you can construct a Hermitian metric as a constant metric w.r.t the trivializtion; then you use partition of unity to add them up, so all you need to prove is that convex linear combination of positive definite Hermitian matrices is also positive definite. – Yuchen Liu Jul 8 '12 at 11:03

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