  Given an action of a complex reductive Lie group G on a normal variety X, we
show that every analytically Zariski-open subset of X admitting an analytic
Hilbert quotient with projective quotient space is given as the set of
semistable points with respect to some G-linearised Weil divisor on X.
  Applying this result to Hamiltonian actions on algebraic varieties we prove
that semistability with respect to a momentum map is equivalent to
GIT-semistability in the sense of Mumford and Hausen. It follows that the
number of compact momentum map quotients of a given algebraic Hamiltonian
G-variety is finite. As further corollary we derive a projectivity criterion
for varieties with compact Kaehler quotient.
  Additionally, as a byproduct of our discussion we give an example of a
complete Kaehlerian non-projective algebraic surface, which may be of
independent interest.
