We introduce a general method to induce Bridgeland stability conditions on
semiorthogonal decompositions. In particular, we prove the existence of
Bridgeland stability conditions on the Kuznetsov component of the derived
category of many Fano threefolds (including all but one deformation type of
Picard rank one), and of cubic fourfolds. As an application, in the appendix,
written jointly with Xiaolei Zhao, we give a variant of the proof of the
Torelli theorem for cubic fourfolds by Huybrechts and Rennemo.Comment: 52 pages. Appendix about the Torelli theorem for cubic fourfolds by
A. Bayer, M. Lahoz, E. Macri', P. Stellari, and X. Zhao. v2: main results
also for characteristic p; updated discussion about related wor
