Let $G$ be a compact connected Lie group and $\mathfrak g^*$ be dual of Lie algebra $\mathfrak g$. Let $M$ be a compact projective variety and $G$ act on $M$ freely and $M$ is $G$ equivariant, and $\mu:M\to \mathfrak g^*$ be a moment map, then the symplectic quotient $M_\lambda$, $\lambda\in \mathfrak g^*$ is still projective?

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    $\begingroup$ If $M$ is compact then a Hamiltonian action of an non-trivial connected group cannot be free: $\mu$ must have critical points (certainly it may act freely \emph{on a level set} but this is different). $\endgroup$ – Jeremy Nov 30 '14 at 17:33

I believe that the answer is yes.

First note that the complexification $G_{\mathbb{C}}$ of $G$ is reductive and contains $G$ as a maximal compact subgroup. Secondly, $G_{\mathbb{C}}$ acts algebraically on $M$ via an extension of the original $G$-action. Your symplectic quotient is then homeomorphic to the projective Geometric Invariant Theory quotient of $M$ by $G_{\mathbb{C}}$.

A useful reference is Chapter 8 of Kirwan's thesis, Cohomology of Quotients in Symplectic and Algebraic Geometry.

  • $\begingroup$ But I think you must add the condition $M^s=M^{ss}$? $\endgroup$ – Daniel Nov 29 '14 at 14:50
  • $\begingroup$ I don't believe this is necessary. I think you need only assume that the $G$-stabilizers are finite. However, you took care of this by requiring the $G$-action to be free. $\endgroup$ – Peter Crooks Nov 29 '14 at 14:54
  • $\begingroup$ Here has been written mathoverflow.net/questions/6316/… $\endgroup$ – Daniel Nov 29 '14 at 14:58
  • $\begingroup$ A reference for the specific fact to which I'm referring is page 84 of cms.zju.edu.cn/UploadFiles/AttachFiles/201096185821505.pdf $\endgroup$ – Peter Crooks Nov 29 '14 at 15:01
  • $\begingroup$ I can not see it in page 84 $\endgroup$ – Daniel Nov 29 '14 at 15:05

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