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When two varieties have equivalent derived categories of coherent sheaves are the stable $\infty$-categories of coherent sheaves also equivalent?

Are the stable $\infty$-categories of varieties "discrete" in some sense?

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  • $\begingroup$ I think this paper sums up well the state of the art. Everything you'll find will be DG. Algebraic geometers are not that much interested in non linear enhancements arxiv.org/abs/2101.04404 $\endgroup$ Sep 18, 2021 at 6:49
  • $\begingroup$ That is great but I expected it would be true without input on the level of Orlov's theorem. Is there a more trivial argument? $\endgroup$
    – tiapal
    Sep 18, 2021 at 7:14
  • $\begingroup$ @tiapal Why do you expect it to have a trivial argument? It does not look like a trivial result to me... $\endgroup$ Sep 18, 2021 at 7:52
  • $\begingroup$ @tiapal And if a "trivial argument" exists, it would probably imply Orlov's theorem (since enhanced functors are essentially the same as Fourier-Mukai functors). $\endgroup$
    – Sasha
    Sep 18, 2021 at 9:24
  • $\begingroup$ No one seems to have addressed your second question. When the canonical bundle of $X$ is ample, Bondal-Orlov says that you can reconstruct $X$ from its (bounded coherent) derived category $D(X)$, so presumably the answer is no, $D(X)$ (and its $\infty$ version) "should have moduli". I don't know what the precise statement would be. $\endgroup$ Sep 18, 2021 at 16:21

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Orlov proved that any derived equivalence (of smooth projective varieties) is realized by a Fourier-Mukai functor; this should imply equivalence on $\infty$-level.

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The answer of this question in the language of $(\infty,1)$-categories can be found in the following paper by Benjamin Antieau, On the uniqueness of infinity-categorical enhancements of triangulated categories. Where it is shown that derived categories admit unique $\infty$-categorical enhancements ( by stable $(\infty,1)$-categories ). See Theorem 6.4 and Corollary 6.5.

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