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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, 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:

are there any similarly elementary characterizations of $\mathcal{M}(\Sigma_g)$ for $g \geq 3$?

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.

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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

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 $\mathbb{P}^{2g+1}$.

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  • $\begingroup$ Thanks Sam! This is so cool. Can someone with enough rep change $\mathbb{P}^{2g+2}$ to $\mathbb{P}^{2g+1}$? Here's how you get the two quadrics in $\mathbb{P}^{2g+1}$: let $\Sigma_g$ double-cover $\mathbb{P}^1$ with branch points $\lambda_0, \ldots, \lambda_{2g+1}$. Then the two quadrics are cut out by $X_0^2 + \cdots + X_{2g+1}^2$ and $\lambda_0X_0^2 + \cdots + \lambda_{2g+1}X_{2g+1}^2$. $\endgroup$
    – Nathaniel Bottman
    Commented Mar 8, 2013 at 21:44

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