2
$\begingroup$

I am looking for the Picard group of the moduli space of principal $G$-bundles for a connected reductive complex algebraic group $G$.

Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\mathrm{Sp}(2n,\mathbb{C})?$

Is there any reference for this?

$\endgroup$

1 Answer 1

4
$\begingroup$

Theorem A in:

S. Kumar and M. S. Narasimhan. Picard group of the moduli spaces of G-bundles. Math. Ann., 308(1):155-173, 1997,

shows that when $G$ is a simple simply-connected connected complex affine algebraic group, $C$ is a complex smooth irreducible projective curve of genus at least 2, and $M$ is the moduli space of semistable principal $G$-bundles on $C$, then $\mathrm{Pic}(M)\cong \mathbb{Z}.$

$\endgroup$
2
  • $\begingroup$ Thank you for your answer. Is there any specific result when G is not simple? $\endgroup$
    – yors
    Jun 15, 2021 at 15:14
  • 1
    $\begingroup$ I do not know of any references that specifically address the non-simple case. I suspect that if $G$ is complex, reductive (and connected) then the above result can be used to determine what happens when the derived subgroup $DG$ is simply-connected. That intuition comes from my paper with C. Manon that does the computation for $G$-character varieties of punctured surfaces: arxiv.org/pdf/1504.01210.pdf (corollary 2.4). But of course that is a different moduli space (although a related one). $\endgroup$ Jun 15, 2021 at 15:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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