Let $\mathbb{C}P^2$ denote the projective plane.

From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf which surveys Gieseker stable sheaves, I have understood that there are no Gieseker stable bundles $E$ with

$\operatorname{Hom}(E,E)=\mathbb{C}$, $\operatorname{Ext}^1(E,E)=\mathbb{C}$, $\operatorname{Ext}^{\neq 0,1}(E,E)=0$

  1. Is my understanding correct?

  2. Is there an object $E$ in $\operatorname{D^bCoh}(\mathbb{C}P^2)$ with

$\operatorname{Hom}^0(E,E)=\mathbb{C}$, $\operatorname{Hom}^1(E,E)=\mathbb{C}$, $\operatorname{Hom}^{\neq 0,1}(E,E)=0$


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