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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.

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    $\begingroup$ The Bondal-Orlov reconstruction theorem requires that $\pm K$ is ample, so that implies projectivity. But I'll let an expert comment about the other results. $\endgroup$
    – Donu Arapura
    Commented May 30, 2023 at 17:28
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    $\begingroup$ I believe that "projectivity" should be understood as an extra datum. If your construction does not depend on the choice of an ample line bundle, I would be happy to learn obstructions of being proper in place of projective. $\endgroup$
    – Z. M
    Commented May 30, 2023 at 17:43
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    $\begingroup$ Re 3 (again, not from an expert): I think that projective is already hard enough. You run into question of how the result changes when you vary the (very) ample line bundle, but at least there is one to begin with! In addition, many 'naturally occurring' smooth proper varieties are actually projective — you kind of have to go out of your way to construct non-projective ones. I suppose that 'most' proper varieties are probably not projective, but this is hard to make precise because moduli spaces of higher-dimensional varieties are often organised by their polarisations. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 30, 2023 at 20:10

1 Answer 1

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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.

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    $\begingroup$ Note that Orlov’s formula doesn’t hold for arbitrary blowups. $\endgroup$
    – crystalline
    Commented Jun 3, 2023 at 1:10
  • $\begingroup$ Thank your for your answer and comment. For my own record, let me mention that I noticed Orlov also explains this point of view in Remark 4.5 of this paper arxiv.org/pdf/1402.7364.pdf $\endgroup$
    – P. Usada
    Commented Jun 5, 2023 at 23:19

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