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.



Bimeromorphic equivalence of reduced spaces for Kähler $S^1$-actions
I have not read it, but I think the article Heinzner-Loose, "Reduction of complex Hamiltonian G-spaces" (1994, [link](http://www.ams.org/mathscinet-getitem?mr=1274117)) 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](http://www.ams.org/mathscinet-getitem?mr=1397979)). 


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!


