Reference for openness of stable locus of holomorphic family of vector bundles on a compact riemann surface.

Question

I am looking for a reference (or an easy explanation) for the openness of the stable locus of a holomorphic family of (holomorphic) vector bundles on a compact Riemann surface parametrized by a (compact) complex manifold. For me, a holomorphic family of vector bundles on a compact Riemann surface $X$ parametrized by a (compact) complex manifold is just a holomorphic vector $E$ bundle over $T \times X$.

There is a proof in Narasimhan and Seshadri's paper "Stable and Unitary Vector Bundles on a Compact Riemann Surface" but the proof there depends on their proof of the unrelated theorem that describes stable vector bundles on a compact riemann surface via certain unitary representations of suitably defined fuchsian groups.

If there is a simpler proof in the case that all parametrized bundles are of degree zero I would like to know it.

EDIT: Note that I do not want to assume that $T$ and the vector bundle $E$ over $T \times X$ are algebraic. So e.g. Huybrechts-Lehn or Le Potier's "Lectures on Vector Bundles" aren't of any help to me, I think.

Answer 1

I think this question is discussed in Kobayashi's "Differential geometry of complex vector bundles", at least the result follows implicitly.

One way to see is as follows: For simplicity, consider holomorphic rank 2 bundles $V$ of degree 0. They are not stable if there exists a holomorphic $f\colon L\to V$ of a holomorphic line bundle $L$ of degree $0.$ If you think of a holomorphic bundle as given by a holomorphic structure $\bar\partial,$ then the result follows from the observation, that, for a family of Fredholm operators (like the $\bar\partial$ on $L^*\otimes V,$ where you vary the holomorphic structures on $L$ and $V$) the minimal kernel dimension is attained on an open subset.