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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.

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    $\begingroup$ For the first part of your question : It is due to holomorphic slice theorem, see p.18 and Theorem 2.6 of arxiv.org/pdf/alg-geom/9304004.pdf . This gives a right answer to the first part of your question. It seems that Sjamaar first solved this question in 1993 . About part 2 .in your case , symplectic quotiens are birational together due to V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math. 97 (1989), 485–522 $\endgroup$
    – user21574
    Commented Dec 5, 2017 at 1:40

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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!

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  • $\begingroup$ Dear Macbeth, many thanks for these references! $\endgroup$
    – aglearner
    Commented Jun 16, 2017 at 9:28

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