  We consider the self-dual vortex equations on a positive line bundle L --> M
over a compact Kaehler manifold of arbitrary dimension. When M is simply
connected, the moduli space of vortex solutions is a projective space. When M
is an abelian variety, the moduli space is the projectivization of the
Fourier-Mukai transform of L. We extend this description of the moduli space to
the abelian GLSM, i.e. to vortex equations with a torus gauge group acting
linearly on a complex vector space. After establishing the Hitchin-Kobayashi
correspondence appropriate for the general abelian GLSM, we give explicit
descriptions of the vortex moduli space in the case where the manifold M is
simply connected or is an abelian variety. In these examples we compute the
Kaehler class of the natural L^2-metric on the moduli space. In the simplest
cases we compute the volume and total scalar curvature of the muduli space.
Finally, we note that for abelian GLSM the vortex moduli space is a
compactification of the space of holomorphic maps from M to toric targets, just
as in the usual case of M being a Riemann surface. This leads to various
natural conjectures, for instance explicit formulae for the volume of the space
of maps CP^m --> CP^n.
