Stability conditions and infinitesimal deformation of curves

TL;DR

This paper investigates how stability conditions on smooth projective curves behave under infinitesimal deformations, establishing an isomorphism between their stability condition spaces and linking derived autoequivalences.

Contribution

It demonstrates that the derived push-forward induces an isomorphism of stability condition spaces under infinitesimal deformations of curves.

Findings

Stability conditions are preserved under infinitesimal deformations.

The derived push-forward functor induces an isomorphism between stability condition spaces.

Autoequivalence groups act compatibly on deformed and undeformed derived categories.

Abstract

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.