I have a log pair $(X,D)$ which is purely log terminal and $D$ is a projective $\mathbb{Q}$-Cartier divisor ($X$ may not be projective). Moreover, $D$ is a variety of Fano type. Is there a space of finite type parametrizing first order deformation theory of the pair $(X,D)$ which are isotrivial along $D$? i.e. the induced flat deformation on $D$ is isotrivial.
My expectation, is that such deformation theory is determined by the corresponding deformation of the normal bundle of $D$ on $X$.
