There is a theorem that (stable, topologically trivial) holomorphic $G$-bundles are in one-to-one correspondence to flat $K$-bundles (with the appropriate corresponding condition), where $K$ is the maximal compact subgroup of the Lie group $G$. This gives compatible symplectic and complex structures and thus a Kähler structure.
There exists an analogous theorem (Mehta-Seshadri?) for the case of Riemann surface with marked points giving a correspondence between flat $K$-bundles on the punctured Riemann surface and holomorphic $G$-bundles with a choice of a point in the flag variety at each marked point (and some extra data).
For physics reasons relating to the Chern-Simons theory, I need to deal with a situation where there is a bit of extra data at each puncture beyond that. Choose a smooth, odd, function $f: \mathbb{R} \to [-1, 1]$ such $f(x) = 1$ for all $x \geq \epsilon$ for some small $\epsilon >0$. Furthermore, let $X$ be a Riemann surface with $n$ marked points $x_1, \ldots, x_n$ and local coordinate charts at each marked point (of size at least $\epsilon$ in the local coordinate). Then, let $\mathcal{M}$ be the moduli space of $K$-bundles on $X$ with unitary connection and a choice of trivialization at each marked point such that the connection is flat away from the marked points and of the form $$A = J_i f(r_i) d\theta_i$$ near a marked point $x_i$ for some lie algebra element $J_i$. The variables $r_i$ and $\theta_i$ are the polar coordinates for the local coordinate chart.
$\mathcal M$ is equivalent to a $K$-bundle with flat connection on $X - \{x_1, \ldots, x_n\}$, a choice of logarithm of the monodromy around each puncture, and a choice of group element for each puncture. (This group element depends on a choice of angle at the puncture, where changing the angle shifts the group element by $J$.) $\mathcal M$ has a canonical symplectic structure where symplectic reduction by the action of $G^{\times n}$ on the choice of trivializations at the marked points recovers the original space of flat connections. Furthermore, forgetting the non-holomorphic part of the connection gives a holomorphic $G$-bundle with a choice of trivialization at each marked point.
My actual questions are the following: is this map from $\mathcal M$ to the space of holomorphic $G$-bundles on $X$ with trivialization at the marked points $x_1, \cdots, x_n$ an isomorphism? Locally an isomorphism? Are the resulting complex and symplectic structures compatible?
Really, it is only the potential existence of some Kähler space which reduces under symplectic reduction by the action of $G^{\times n}$ to the space of flat $K$-bundles which is important to me.
