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
$\operatorname{Hom}(E,E)=\mathbb{C}$, $\operatorname{Ext}^1(E,E)=\mathbb{C}$, $\operatorname{Ext}^{\neq 0,1}(E,E)=0$
Is my understanding correct?
Is there an object $E$ in $\operatorname{D^bCoh}(\mathbb{C}P^2)$ with
$\operatorname{Hom}^0(E,E)=\mathbb{C}$, $\operatorname{Hom}^1(E,E)=\mathbb{C}$, $\operatorname{Hom}^{\neq 0,1}(E,E)=0$