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.Comment: 33 pages, 1 figure; improved exposition, many of the results are now
proven for complete and not only for projective quotients, examples showing
the necessity of the assumptions made in the main results added; to appear in
Advances in Mathematic