Stability conditions and infinitesimal deformation of curves

Let $\mathcal X$ be an infinitesimal deformation of a smooth projective curve $X_0$ over a field. We study stability conditions under such deformations and show that the derived push-forward functor associated with the inclusion $X_0 \to \mathcal X$ induces an isomorphism between the space of stability conditions on $\mathcal X$ and that on $X_0$. This yields a direct comparison between the deformed and undeformed settings. As an application, we prove that the autoequivalence group $\mathrm{Aut}{\mathbf D^b(\mathcal X)}$ naturally acts on $\mathbf D^b(X_0)$, providing a link between derived symmetries and the deformation structure.