If $\Sigma\_g$ is a genus-$g$ surface, $g \geq 2$, then let $\mathcal{M}(\Sigma\_g)$ be its twisted SU(2) representation variety, i.e. $$\mathcal{M}(\Sigma\_g) := \{ (A\_1, B\_1, \ldots, A\_g, B\_g) \in SU(2)^{2g} \:|\: [A\_1,B\_1]\cdots[A\_g,B\_g] = -I \}/SO(3),$$ where $SO(3) = SU(2)/\{\pm 1\}$ acts on these tuples by conjugating each component. Then $\mathcal{M}(\Sigma\_g)$ is a smooth, compact $(6g-6)$-dimensional symplectic manifold, and it's symplectomorphic to the moduli space of stable bundles over a genus-$g$ Riemann surface with rank $2$ and determinant equal to some fixed line bundle of odd degree (this moduli space is a Kaehler manifold).


In his 1968 paper "Stable bundles of rank 2 and odd degree over a curve of genus 2", Peter Newstead showed that $\mathcal{M}(\Sigma\_2)$ is isomorphic to the intersection of two generic quadric hypersurfaces in $\mathbb{P}^5$. In [a recent paper](http://arxiv.org/abs/1006.1099), Ivan Smith used this characterization to get a lot of information about the Fukaya category of $\mathcal{M}(\Sigma\_2)$ in terms of the Fukaya category of $\Sigma\_2$. (My impression is that people care a lot about the Lagrangian intersection theory of $\mathcal{M}(\Sigma\_g)$ because if you understand that, you understand the instanton Floer homology of 3-manifolds.) My question is:



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> are there any similarly elementary characterizations of $\mathcal{M}(\Sigma\_g)$ for $g \geq 3$?
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I know that bits and pieces about $\mathcal{M}(\Sigma\_g)$ are known, e.g. its symplectic volume and characteristic classes (maybe even its cohomology?); I would really like to know about more elementary things that are known to be isomorphic to it as symplectic/Kaehler manifolds.



How many "elementary" characterizations of twisted SU(2) representation varieties are known?
I think Nate might know this by now, but in case anyone else is curious, there is a generalization of the intersection-of-quadrics-in-$\mathbb{P}^5$ picture to general genus, proved first in this paper, I believe



> 
> Classification of Vector Bundles of
>  Rank 2 on Hyperelliptic Curves, U.V.
>  Desale and S. Ramanan, Inventiones
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The story is that $\mathcal{M}\_g$ is homeomorphic (in fact, symplectomorphic), to the space of $g-2$ dimensional linear subspaces in the intersection of two quadrics in $\mathbb{P}^{2g+1}$.


