Mathematics – Algebraic Geometry
Scientific paper
2008-12-15
Adv. Math. 224 (2010), no. 2, 401-431
Mathematics
Algebraic Geometry
33 pages, 1 figure; improved exposition, many of the results are now proven for complete and not only for projective quotients
Scientific paper
10.1016/j.aim.2009.11.013
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.
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