MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 2 down vote accepted

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

share|cite|improve this answer
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 $\lambda_0X_0^2 + \cdots + \lambda_{2g+1}X_{2g+1}^2$. – Nathaniel Bottman Mar 8 '13 at 21:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.