Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety? Ideally, I would also like the Chern character of this object to be non-zero.

In case $H^1(\mathcal{O}_X) =0$, any line bundle will do the job. On the other hand, if $H^1(\mathcal{O}_X) \neq 0$, the existence of the trace maps shows that the rank of such an object must be zero. I have some specific examples in mind (mostly structure sheaves of rigid subvarieties of some special varieties), but I would like to know if such objects exist in general on any smooth projective variety.