Construction of moduli space with hands.

Question

I make this question because I'd like to have an explicit example of moduli space. First of all, we consider $G=SL(2,\mathbb{C})$ and $X$ a compact Riemann surface with genus $2$. So we can build $M^2$: the moduli space of principal stable $SL(2)$-bundles on $X$. In order to do this, I cite this answer Principal stable $SL(2)$-bundles on a genus $2$ compact Riemann surface., in order to have a model of principal $SL(2)$-bundle on $X$. Now, how can I build ''with hands'' the moduli space $M^2$?

Answer 1

First of all, I will try to interpret your question to the best of my abilities: $SL(2)$ should be $SL(2, \mathbb C)$, since notion of stability makes sense only for holomorphic principal bundles and I would not know how to define it in the case of $SL(2, \mathbb R)$-bundles, which do not have obvious holomorphic structure. Secondly, the moduli space of stable bundles I will interpret as the moduli space of holomorphic bundles: The spaces are almost the same, you just have to identify bundles which are semistable but not stable. This should be very doable as it this locus should be biholomorphic to the space $J^0(X)$ of degree 0 line bundles over your complex curve. Lastly, I will interpret "with hands" as "identify it with some familiar space".

With this interpretation, the answer to this question is essentially contained in this MO (Mathoverflow) post: https://mathoverflow.net/questions/104254 . Namely, the moduli space of (semistable) principal $SL(2, \mathbb C)$-bundles over $X$ is naturally biholomorphic to the space of rank 2 holomorphic vector bundles with trivial determinant over $X$. This correspondence is via the associated bundle constriction: If $V={\mathbb C}^2$, $G=SL(2, {\mathbb C})$ acting on $V$ in the standard fashion and $E$ is a holomorphic principal $G$-bundle (stable or not is irrelevant) over $X$, one has the canonical holomorphic $V$-bundle $E_V$ over $X$. The condition that we started with $SL(2, {\mathbb C})$-bundle rather than $GL(2, {\mathbb C})$-bundle translates to the property that the determinant line bundle $det(E_V)$ over $X$ is trivial (this triviality assumption helps a lot). Now, the moduli space $M$ of rank 2 holomorphic bundles with trivial determinant over (smooth projective) genus 2 curves was described in the paper

[NS] M.S. Narasimhan, S. Ramanan, Moduli of vector bundles on a compact Riemann surface. Ann. of Math. (2) 89 (1969) p. 14–51.

I got this reference from the MO post above, I was not aware of it before (I am not an algebraic geometer and my knowledge of the subject is quite limited). Now, [NS] describe the moduli space $M$ as follows: It is biholomorphic to $\mathbb PH^0(J^1, L^2)$. Here $W=H^0(\cdot)$ is a 4-dimensional complex vector space consisting of sections of a certain holomorphic line bundle $L^2$ over $J^1$, the tensor square of another line bundle $L\to J^1$ which I will define below. The manifold $J^1=Pic^1(X)$ is the space of degree 1 holomorphic line bundles over $X$ (geometrically speaking, it is a torus of complex dimension $2$). The line bundle $L\to J^1$ is the bundle of the divisor $\Theta$ in $J^1$ given by the natural embedding $X\to J^1$, sending each $x\in X$ to the divisor $(x)$ on $X$). You will find many more details in [NS]. Lastly, $\mathbb P W$ stands for the complex projective space of $W$, in other words, $\mathbb C \mathbb P^3$, which is as familiar as space to a topologist or geometer as I know.

Hope, it helps.