Stability conditions on threefolds

We investigate a subspace of Bridgeland stability conditions on ${\text{PP}}^n$ satisfying the so-called Li condition. These are the stability conditions whose restriction to a smooth projective subvariety $X\subset {\text{PP}}^n$ is again a stability condition. We then show that, when $X$ is a threefold, the restricted stability conditions coincide with those obtained via the double-tilt construction introduced by Bayer-Macrì-Toda. As an application, we prove the weak BMT conjecture.