# Tangent space of a Moduli space

Let $X$ be a compact Riemann surface with genus $2$ and $M^2$ the moduli space of stable principal $SL(2)$-bundles of rank $r$. We know that $M^2$ is a complex projective variety of dimention $r^2(g-1)+1=r^2+1$. I have to prove that if $p \in M^2$ is a smooth point then $T_pM^2 \simeq H^1(X,\mathfrak{sl}(2))$. With $\mathfrak{sl}(2)$ we mean the adjoint bundle. I suppose that $H^1(X,\mathfrak{sl}(2))$ is the sheaf cohomology with coefficients the holomorphic sections of $\mathfrak{sl}(2)$ and they can be regarded as an infinite dimentional Lie algebra. How can I prove the isomorphism $T_pM^2 \simeq H^1(X,\mathfrak{sl}(2))$? If I have $M$ the moduli space of stable principal $G$-bundles ($G$ is a simple Lie group) over a compact Riemann surface $X$, how can I prove, in general setting, that $T_pM \simeq H^1(X,\mathfrak{g})$?

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why do you happen to have to prove that? – Fernando Muro Oct 22 '13 at 9:02
@FernandoMuro Because I read it but I don't know how can I do it... – Ste Oct 22 '13 at 9:29
The cohomology should be on $X$, not on $M^2$. Also the cohomology group is finite dimensional. Finally, the cohomology group does not (obviously) have any nontrivial structure of Lie algebra. It might have a structure of module over the Lie algebra $H^0(X,\mathfrak{sl}_{2,E})$, where $E$ is the original bundle and where $\mathfrak{sl}_{2,E}$ is the associated adjoint bundle. But the stability condition means this Lie algebra will just be $\mathbb{C}$, so the module structure is pretty useless. – Jason Starr Oct 22 '13 at 12:44
@JasonStarr I'm sorry... So I have some questions: 1) How can I prove that $T_pM \simeq H^1(X,\mathfrak{g})$? Why the stability condition means that $H^0(X,\mathfrak{sl}_{2,E}) \simeq \mathbb{C}$? – Ste Oct 22 '13 at 13:06
This question is clearly stated as if it were an exercise. I think it's not, but the person who asked chose to offer no motivation after being asked about it. I think this is against the phylosophy of this forum. – Fernando Muro Oct 22 '13 at 13:22

## 1 Answer

This is the general method of deformation theory. The two immediate papers that will give you sufficient detail are:

Holomorphic Vector Bundles on a Compact Riemann Surface (by Narasimhan-Seshadri)
Stable Principal Bundles on a Compact Riemann Surface (by Ramanathan)

I am more familiar with the specific scenario of the moduli space of flat or anti-self-dual connections on a principal bundle, where everything is translated into an elliptic chain complex, and the Zariski tangent space is the kernel of the linearization of some operator modulo the infinitesimal action.

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