Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space ${\text{splcpx}}_{X\mathrm{/}k}\text{uet}$ of the objects $E$ in $D^b(\mathrm{C}\mathrm{o}\mathrm{h}(X))$ satisfying ${\text{Ext}}_X^{-1}(E,E)=0$ and $\text{Hom}(E,E)\cong k$. Then we can prove that ${\text{splcpx}}_{X\mathrm{/}k}\text{uet}$ is smooth and has a symplectic structure.