TITLE: Derived category of a hypersurface
QUESTION [5 upvotes]: Let $X$ be a smooth projective variety over $\mathbb{C}$, and $H \subset X$ be a smooth hypersurface.
Many properties of an ambient variety $X$ could somehow inherit to the hypersurface $H$, I was wondering if there is any know result dealing with the relation between the derived categories of (bounded) complexes of quasi-coherent sheaves (or coherent sheaves) on $X$ and $H$?

REPLY [7 votes]: In terms of semiorthogonal decompositions, there is an answer to this question, as it has been mentioned above.
More precisely, if one has a so called Lefschetz decomposition of the bounded derived category of coherent sheaves on a smooth projective variety (such a decomposition depends also on the choice of a projective embedding) then one has an induced semiorthogonal decomposition of the derived category of a hyperplane section, with pieces coming from the ambient variety and then an additional component. One can think of this as some sort of a Lefschetz hyperplane theorem result. More details here: Kuznetsov`s Main HPD Paper
Then, if you have a degree d hypersurface H in P^n, this hypersurface can be seen as a linear section of P^n with respect to the degree d Veronese embedding.
In this direction, the paper mentioned above (by Ballard, Deliu, Favero, Isik and Katzarkov) gives an answer to what the extra information is, either in terms of LG-models (matrix factorization categories) or in terms of the derived category of sheaves of modules over a sheaf of A_infty algebras (there is a nice, simple formula describing the sheaf of A_infty algebras).