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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 $H$?

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    $\begingroup$ You cannot expect much: think of a hypersurface $H\subset\mathbb{P}^n$, say of degree $>n$. The derived category of $\mathbb{P}^n$ is quite simple, while that of $H$ is extremely complicated. $\endgroup$ – abx Apr 22 '14 at 14:56
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    $\begingroup$ There are some results within the framework of homological projective duality. $\endgroup$ – Sasha Apr 22 '14 at 15:54
  • $\begingroup$ Kuznetsov has a whole theory about that called homological projective duality. The catch is that you have to start with an appropriate exceptional collection on X. Here are some handy notes. cf.ac.uk/maths/subsites/logvinenko/talks/… $\endgroup$ – bananastack Apr 22 '14 at 15:56
  • $\begingroup$ ah, whoopsie. I hadn't noticed the comment before mine... $\endgroup$ – bananastack Apr 22 '14 at 15:56
  • $\begingroup$ Thanks! I will try to learn something about homological projective duality! $\endgroup$ – Li Yutong Apr 23 '14 at 1:43
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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).

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Maybe it's worth spelling out the easy case of projective space (and Veronese embedding). Take the standard exceptional collection $<O,O(1),...,O(n)>.$

The derived category of a hypersurface X of degree d has then a semi-orthogonal decomposition $<A_X, O, O(1), ..., O(n-d)>$. We might think of $A_X$ as being the "interesting" part of the derived category of X. We can also intersect further and $D(X_1 \cap X_2) = <A_{X_1 \cap X_2}, O,\ldots, O(n-2d)>$.

Take the universal hypersurface of degree d, call it $H$. This sits inside $P^n \times P^N$, where $N$ is the ambient dimension of the degree d Veronese embedding.

The category $H$ (being itself a hypersurface) has a decomposition where we call the interesting part just $A$.

For each point $p \in P^N$ we have a hypersurface $H_p$ and thus an interesting category $A_{H_p}$.

The "homological dual" variety is a $Y$, mapping to the dual $P^N$, such that $D(Y)$ is the total $A$. In some sense, $Y$ is parameterezing all the categories $A_{H_p}$.

The point of HPD is then the compatibility between passing to base loci and base changing Y.

If $L < P^N$ is some linear system, then the derived category of the corresponding base locus has a decomposition and the interesting part is the interesting part of $D(Y \times_{P^N} L$.

(this is perhaps too sloppy to be useful but I tried)

The Viennese school of Katzarkov, Ballard, Favero etc. has shown that in this context (of degree d Veronese embedding) there is a (sort of tautological) dual. Unfortunately this dual is a highly non-commutative variety, in fact it's the dual $P^N$ with a sheaf of $A_\infty$-algebras over it.

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