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 $$H^0(X,E\otimes E^\*\otimes K\_X)\rightarrow \mathcal M\_X(r,0,K\_X)$$



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> Is this map a closed immersion? Is it proper? 
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Thanks



Higgs bundles and stable vector bundle
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.


