Let $\mathcal M_X(r,0,K_X)$ be tha moduli space of semistable Higgs bundles over a smooth irreducible algebraic curve over $\mathbb C$. And let $E$ be a stable vector bundle of rank $r$ and degree $0$. Then clearly we have an injection $$H^0(X,E\otimes E^*\otimes K_X)\rightarrow \mathcal M_X(r,0,K_X)$$

Is this map a closed immersion? Is it proper?


  • 1
    $\begingroup$ It the coarse moduli space, not the stack $\endgroup$
    – Z.A.Z.Z
    Oct 14, 2016 at 14:13

1 Answer 1


This is a partial answer to my question!

We can deduce easly that if this map is closed immersion, then $E$ is very stable vector bundle (because in this case, the Hitchin map is finite). Hence if $E$ is stable non very stable vector bundle (there exist of course such vector bundles) then we deduce that the above map couldn't be closed immersion.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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