TITLE: Bimeromorphic equivalence of reduced spaces for Kähler $S^1$-actions
QUESTION [7 upvotes]: Let $(X,\omega)$ be a smooth Kähler manifold (not necessarily compact) with an isometric $S^1$-action with a Hamiltonian $H$. It is a well known fact that 
1) The reduced spaces $X(c)=H^{-1}(c)/S^1$ have a complex analytic structure. 
2) Such spaces are bimeromorphic for $c$ satisfying $\min(H)<c<\max(H)$. 
I was not able to find a reference for these facts and would be very grateful if you can give me one.
PS. In fact I'll be happy with a user-friendly reference for the case when $X$ is a smooth projective  complex manifold.

REPLY [2 votes]: I have not read it, but I think the article Heinzner-Loose, "Reduction of complex Hamiltonian G-spaces" (1994, link) may be the original reference for your first question.
In the compact (but not necessarily projective) case, both questions are treated in Fujiki, "Kähler quotient and equivariant cohomology" (1994, link). 
I would also be interested in seeing a reference for the second fact in the non-compact case -- let me know if you find one!