The moduli space of holomorphic chains composed of line bundles over a compact Riemann surface

A holomorphic chain on a compact Riemann surface is a tuple of vector bundles together with homomorphisms between them. Holomorphic chains naturally occur in the study of Higgs bundles. We show that the moduli space of holomorphic chains composed of line bundles of any length can be identified with a fiber product of projective space bundles. We compute the Euler characteristic of the moduli space. The stability of chains involves real vector parameters. We also show that the variation of parameters corresponds to the characters of Gm.