Stability manifolds of Kuznetsov components of prime Fano threefolds

Let $X$ be a cubic threefold, quartic double solid or Gushel–Mukai threefold, and $\mathcal{K}u(X)\subset {\mathrm{D}}^b(X)$ be its Kuznetsov component. We show that a stability condition $\sigma$ on $\mathcal{K}u(X)$ is Serre-invariant if and only if its homological dimension is at most $2$. As a corollary, we prove that all Serre-invariant stability conditions on $\mathcal{K}u(X)$ form a contractible connected component of the stability manifold.