Let $X$ be the total space of the cotangent sheaf on $\mathbb{P}^{2}$ and $i \colon \mathbb{P}^{2} \hookrightarrow X$ be thezero section. Suppose that $E$ is a coherent sheaf on $X$ which is set-theoretically supported on the zero section. If $\mathrm{Hom}(E,E)=\mathbb{C}$, must $E$ be isomorphic to $i_{*}E_{0}$ for some coherent sheaf $E_{0}$ on $\mathbb{P}^{2}$?
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$\begingroup$ We have $X = {\rm Spec}({\rm Sym}(T_{\mathbb{P}^2}))$, and the ideal of the zero section is cut out by the first grading i.e. $T_{\mathbb{P}^2}$. Now, the tangent sheaf on $\mathbb{P}^2$ is globally generated. Thus, if $E$ is supported set theoretically on the zero section, then $\Gamma(\mathbb{P}^2, T_{\mathbb{P}^2})$ acts on it by nilpotent endomorphisms, and if ${\rm Hom}(E, E) = \mathbb{C}$, then all of these have to be zero, i.e. $E$ is (scheme-theoretically) supported on the zero section, $E = i_* i^* E$. $\endgroup$– Piotr AchingerCommented Nov 15 at 17:08
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$\begingroup$ @ Piotr Achinger, Thank you for your explanation! $\endgroup$– Tianle MaoCommented Nov 21 at 8:17
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