  A holomorphic triple over a compact Riemann surface consists of two
holomorphic vector bundles and a holomorphic map between them. After fixing the
topological types of the bundles and a real parameter, there exist moduli
spaces of stable holomorphic triples. In this paper we study non-emptiness,
irreducibility, smoothness, and birational descriptions of these moduli spaces
for a certain range of the parameter. Our results have important applications
to the study of the moduli space of representations of the fundamental group of
the surface into unitary Lie groups of indefinite signature, which we explore
in a companion paper "Surface group representations in PU(p,q) and Higgs
bundles". Another application, that we study in this paper, is to the existence
of stable bundles on the product of the surface by the complex projective line.
  This paper, and its companion mentioned above, form a substantially revised
version of math.AG/0206012.
