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 $H$?



Derived category of a hypersurface
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](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_2007__105_/PMIHES_2007__105__157_0/PMIHES_2007__105__157_0.pdf)


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](http://arxiv.org/abs/1306.3957)) 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).


