On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces

We study holomorphic (n + 1)-chains En → En−1 → ⋯ → E0 consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on n real parameters was introduced by the first two authors and moduli spaces were constructed by the third author. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples). Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface.