
> 
> A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which
>  acts on $M$ and preserves complex structure then If $M$ has Kaehler
>  manifold, then the symplectic quotient of $M$, i.e, $M//G$ has kaehler
>  structure?
> 
> 
> 


What about hyper-kaehler? 



> 
> B) If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler
>  structure?
> 
> 
> 



If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler structure?
If $(X,\omega)=(M,I,\omega)$ is Kaehler, and $G$ is a compact Lie group, which acts freely on $M$ and preserves *both* $I$ and $\omega$, then the symplectic quotient *is* Kaehler. More precisely, $M//G\simeq X^{st}/G^c$, where $G^c$ is the complexification of $G$, and $X^{st}\subset X$ are the stable points for the action. See, e.g., [HKLR, Theorem 3.1](http://projecteuclid.org/euclid.cmp/1104116624) 
and Hitchin's [Sem. Bourbaki 748](http://archive.numdam.org/ARCHIVE/SB/SB_1991-1992__34_/SB_1991-1992__34__137_0/SB_1991-1992__34__137_0.pdf), section 3.4.


In the hyperkaehler case, there is again compatibility with the symplectic quotient, but it is more subtle. Your suggestion in B) cannot possibly work, because
$\dim\_R M-2\dim\_R G$ need not be divisible by $4$. The actual relation is as follows. Suppose $M$ is hyperkaehler, and $G$ (acts freely in a hamiltonian way by isometries and) preserves "the" three Kaehler forms $\omega\_I$, $\omega\_J$, $\omega\_K$. Then
we have a moment map
$$
\mu=(\mu\_I,\mu\_J,\mu\_K): M\to \mathbb{R}^3\otimes\mathfrak{g}^\vee,
$$
and for a regular value $\xi$, the quotient $\mu^{-1}(\xi)/G$ is hyperkaehler (see HKLR, Theorem 3.2 and SB 748, Theorem 3.) **added:** provided $\xi$ is $G$-invariant. 


If you want to link this with the symplectic or Kaehler quotient, you fix one of the complex structures, say $I$, and break the three moment maps into a real moment map $\mu\_I$ and a complex one, $\mu\_c=\mu\_J+i\mu\_K$. Then, if
$\xi=(\xi\_1,\xi\_2,\xi\_3)$, we have
$$
\mu^{-1}(\xi) = \mu\_I^{-1}(\xi\_1)\bigcap\mu\_c^{-1}(z)
$$
where $z=\xi\_2+i\xi\_3$.
The hyperkaehler quotient is the symplectic reduction of $\mu\_c^{-1}(z)$, or equivalently, the Kaehler quotient
$\mu\_c^{-1}(z)^{st}/G^c$.


