2
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ 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, 2014 at 17:33

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
8
  • $\begingroup$ But I think you must add the condition $M^s=M^{ss}$? $\endgroup$
    – Daniel
    Nov 29, 2014 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. $\endgroup$ Nov 29, 2014 at 14:54
  • $\begingroup$ Here has been written mathoverflow.net/questions/6316/… $\endgroup$
    – Daniel
    Nov 29, 2014 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 $\endgroup$ Nov 29, 2014 at 15:01
  • $\begingroup$ I can not see it in page 84 $\endgroup$
    – Daniel
    Nov 29, 2014 at 15:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.