Reference for openness of stable locus of holomorphic family of vector bundles on a compact riemann surface. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:29:07Z http://mathoverflow.net/feeds/question/69132 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69132/reference-for-openness-of-stable-locus-of-holomorphic-family-of-vector-bundles-on Reference for openness of stable locus of holomorphic family of vector bundles on a compact riemann surface. bavajee 2011-06-29T18:18:33Z 2011-07-14T08:47:08Z <p>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 $T \times X$.</p> <p>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. </p> <p>If there is a simpler proof in the case that all parametrized bundles are of degree zero I would like to know it.</p> <p>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.</p> http://mathoverflow.net/questions/69132/reference-for-openness-of-stable-locus-of-holomorphic-family-of-vector-bundles-on/69133#69133 Answer by David Steinberg for Reference for openness of stable locus of holomorphic family of vector bundles on a compact riemann surface. David Steinberg 2011-06-29T18:25:45Z 2011-06-29T18:25:45Z <p>Proposition 2.3.1 of "The Geometry of the Moduli Space of Sheaves" by Huybrechts and Lehn provides a proof of what you are looking for.</p> http://mathoverflow.net/questions/69132/reference-for-openness-of-stable-locus-of-holomorphic-family-of-vector-bundles-on/70305#70305 Answer by Sebastian for Reference for openness of stable locus of holomorphic family of vector bundles on a compact riemann surface. Sebastian 2011-07-14T08:47:08Z 2011-07-14T08:47:08Z <p>I think this question is discussed in Kobayashi's "Differential geometry of complex vector bundles", at least the result follows implicitly. </p> <p>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.</p>