Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference
in the literature…

Let X be a smooth variety (in our case X=Ms(r) coarse moduli space of stable rank r vector bundles with trivial determinant over a curve C , so X is not compact !) and
let L be a line bundle over X. Then we have the standard exact sequence

0OXDiff1(L)TX0

now take the long exact sequence of cohomology

...H1(OX)H1(Diff1(L))H1(TX)H2(OX)...

Since Pic(X)=Z , we obtain H1(OX)=0. Do we also have H2(OX)=0, so that the
middle arrow is an isomorphism ?
It is natural to expect that

dim(H1(TX))=dim(H1(C,TC))=3g3,

so that any infinitesimal deformation of the moduli
comes from a deformation of the curve, but I can find the result only for the coprime case (Narasimhan, Ramanan) and not trivial determinant.

Is this true for trivial determinant as well?

Answer

Attribution
Source : Link , Question Author : IMeasy , Answer Author : Community

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