Nonseparation in the moduli of complex vector bundles

Abstract

Let X be a compact complex analytic space with structure sheaf (9 and let Hi(X, GL(r, (9)) denote the cohomology set parameterizing isomorphism classes of rank r analytic vector bundles on X. If r > 1, Ha(X, GL(r, (9)) may not naturally form an analytic space; in fact, the natural topology on the cohomology set is not generally Hausdorff. In this paper we provide computable criteria for determining the topological non-separation of points in the cohomology set. For bundles ~o and ~p admitting a nonzero homomorphism h :~0~p, we show that under suitable hypotheses, it is possible to simultaneously deform q~, ~/~ and h in analytic oneparameter families ~o(t), ~p(t) and h(t) such that, for t near zero, h(t) defines an isomorphism between ~o(t) and W(t) converging as t ~ 0 to the homomorphism h from ~o = ~0(0) to ~0 =~p(0). In case X is a Riemann surface, this method yields necessary and sufficient conditions for non-separation of rank two bundles.