Abstract.I will present work done in collaboration with Zhiyu Liu and Songtao Kenneth Ma.  

Let X be a cubic threefold, quartic double solid or Gushel--Mukai threefold, and Ku(X) be its Kuznetsov component. We show that a stability condition on Ku(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 Ku(X) form a contractible connected component of the stability manifold.