# Symplectic quotient of projective variety is projective?

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?

• 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). – Jeremy Nov 30 '14 at 17:33

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 $G_{\mathbb{C}}$.
• But I think you must add the condition $M^s=M^{ss}$? – Daniel Nov 29 '14 at 14:50
• 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. – Peter Crooks Nov 29 '14 at 14:54