Moduli space of semistable bundles

Question

It is well-known that the space of $S$-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 Riemann surface $M$ is $CP^3$ (more concretely $PH^0(Jac(M),L(2\theta)$). Especially, the points corresponding to semistable (and not stable) bundles are smooth points. On the open dense subspace consisting of points corresponding to stable bundles, there is a natural symplectic structure, compatible with the natural Riemannian metric. It defines a Kaehler structure. It should be true that this Kaehler structure extends to the semistable points (and therefore $CP^3$ is equipped with its natural (only) Kaehler structure). Does anyone know a reference, or a proof of this? Thank you.

Answer 1

Disclaimer: This answer is rewritten in response to Chris Woodward's insightful comments.

The moduli space of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 surface $\Sigma$ is homeomorphic to the character variety $$\mathfrak{X}_\Sigma(SU(2)):=\mathrm{Hom}(\pi_1(\Sigma),SU(2))/SU(2).$$ This homeomorphism restricts to a diffeomorphism between stable bundles and irreducible representations.

In the case of fixed determinant Higgs bundles on $\Sigma$ there is a similar homeomorphic correspondence with $$\mathfrak{X}_\Sigma(SL(2,\mathbb{C})):=\mathrm{Hom}(\pi_1(\Sigma),SL(2,\mathbb{C}))/\!/SL(2, \mathbb{C}).$$ Carlos Simpson has shown under this latter correspondence all points, even singular points, have isomorphic corresponding étale neighborhoods (Isosingularity Theorem). In short, they are locally isomorphic. However, it is known these moduli spaces are not even biholomorphic let alone biregular, since the complex structure of the Higgs moduli space depends on the complex structure of $\Sigma$ as a Riemann surface whereas the complex structure on the character variety only depends on the group $SL(2,\mathbb{C})$.

The moduli space of holomorphic vector bundles naturally embeds into the moduli space of Higgs bundles as those with trivial Higgs field. Likewise, $\mathfrak{X}_\Sigma(SU(2))$ embeds into $\mathfrak{X}_\Sigma(SL(2,\mathbb{C}))$, and the homeomorphisms above respect these embeddings.

So it is natural to think that the stratified analytic smooth structures on the moduli space of vector bundles and that of the semi-algebraic space $\mathfrak{X}_\Sigma(SU(2))$ also correspond; as they do generically and also do on their "complexifications". However, as pointed out by Chris Woodward, Johannes Huebschmann proves in Smooth Structures on Certain Moduli Spaces for Bundles on a Surface that this is not the case. In particular, in Section 8 he explicitly addresses the case of a genus 2 surface where the moduli spaces in question are homeomorphic to $\mathbb{C}P^3$, showing explicitly that the natural smooth structure on $\mathfrak{X}_\Sigma(SU(2))$ differs from that of $\mathbb{C}P^3$.

That is sufficient to answer the original question as: NO, if one interprets the question as "Is $\mathfrak{X}_\Sigma(SU(2))$ Kähler isomorphic to $\mathbb{C}P^3$?"

Remark: With regard to the (singular) symplectic structure on $\mathfrak{X}_\Sigma(SU(2))$. The Goldman Poisson structure, a Lie algebra structure and derivation, is defined globally on the coordiante ring of the character variety and makes sense even in a singular setting. In general, such a Poisson structure gives a foliation of the smooth locus by symplectic sub-manifolds. In this case there is one leaf (since the surface in question is closed).

Remark: For a direct proof, not using the theory of holomorphic bundles, that the character variety is $\mathbb{C}P^3$ see here. We can see the singularities directly in Choi's construction.