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Let $X$ be a smooth, projective curve of genus at least $2$ over $\mathbb{C}$ and $E$ be a vector bundle on $X$ of rank at least $2$. Given any point $x \in X$, denote by $S_x$ the image of the natural morphism from $H^0(\mathcal{E}nd(E))$ to $\mathcal{E}nd(E)_x$, where $\mathcal{E}nd(E)_x$ denotes the fiber over $x$ for the vector bundle. We define $$N_x:=\{\phi \in \mathcal{E}nd(E)_x| \mbox{ for all } s \in S_x \mbox{ there exists } s' \in S_x \mbox{ such that } \phi \circ s = s' \circ \phi\}$$ $$C_x:=\{\phi \in \mathcal{E}nd(E)_x| \mbox{ for all } s \in S_x, \phi \circ s = s \circ \phi\}.$$ Is there any known condition on $E$ under which, there exists a closed point $x \in X$ such that $N_x$ or $C_x$ is the entire $\mathcal{E}nd(E)_x$?

Any idea or reference regarding this question is most welcome.

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  • $\begingroup$ For a simple bundle (for example, stable) $E$, $H^0(\mathcal{E}nd(E))$ is just the scalars, so your condition would be trivially satisfied and you have $C_x=N_x=\mathcal{E}nd(E)_x$. $\endgroup$
    – Mohan
    Commented Apr 17, 2017 at 15:55
  • $\begingroup$ @Mohan Thanks, is there any other possibilities? $\endgroup$
    – user43198
    Commented Apr 17, 2017 at 15:58
  • $\begingroup$ What do you mean? $\endgroup$
    – Mohan
    Commented Apr 17, 2017 at 15:59
  • $\begingroup$ @Mohan meaning, are there examples when $E$ is not simple but gives a positive answer to the question. $\endgroup$
    – user43198
    Commented Apr 17, 2017 at 16:00
  • $\begingroup$ Take $E$ to be the trivial bundle, $\mathcal{O}_X^n$. $\endgroup$
    – Mohan
    Commented Apr 17, 2017 at 16:04

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