We know several amazing techniques about the derived category $Perf (X)$ of a smooth projective variety such as the whole theory of Fourier-Mukai transforms. On the other hand, from a dg-categorical point of view, it is natural to work with smooth proper varieties instead of smooth projective varieties. Indeed, (dg) derived categories do not see projectivity of a variety and some natural birational geometry operations can produce proper Fourier-Mukai partners(<https://mathoverflow.net/a/369756/177839>). However, as far as I know, nearly all of the papers on derived categories of smooth varieties assume projectivity. So my question is whether there are specific reasons people do not work with proper varieties. More specifically


1. Are there any fundamental results such as ones coming from Bondal, Orlov, Kawamata, Bridgeland, Mukai etc. (I’m not trying to be exhaustive at all) that have counter-examples for proper cases?
2. Is there any fundamental result that is known to work for proper cases?
3. Are there technical/philosophical reasons why people seem to avoid proper cases?


I would really appreciate any comment to any of the questions.



Derived categories of smooth proper varieties?
The study of derived categories is a special case of the more general study of semiorthogonal components of derived categories. By Chow lemma for any proper variety $X$ there is a blow up $\pi \colon X' \to X$ such that $X'$ is projective, and by Orlov's blowup formula the derived category of $X$ is a semiorthogonal component of the derived category of $X'$. So, from this point of view, the story of proper varieties is a part of the story of projective varieties.


