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Kawamata conjectured that

Let $X$ and $Y$ be birationally equivalent smooth projective varieties. Then the following are equivalent. We denote by $D^b(Coh(X))$ the derived category of bounded complexes of coherent sheaves on $X$

(1) There exists an equivalence of triangulated categories $D^b(Coh(X)) \cong D^b(Coh(Y) )$.

(2) There exists a smooth projective variety $Z$ and birational morphisms $f : Z\to X$ and $ g : Z \to Y$ such that $f ^∗K_X ∼ g^∗K_Y $ (this means $X$ is a Fourier-Mukai partner of $Y$)

The main idea of the conjecture of "$K$-equivalence is equivalent with $D$-equivalence"

was that the Serre functor is invariant in the derived category of triangulated coherent sheaves and Serre functor is related to canonical divisor also, and so Kawamata gave such natural conjecture

In $dim=2 $ K-equivalency===> D-equivalency holds by a theorem of Zariski but for D-equivalency===>K-equivalency, Uehara gave a counterexample to this conjecture of Kawamata: Uehara, Hokuto: An example of Fourier-Mukai partners of minimal elliptic surfaces. Math. Res. Lett. 11 (2004), no. 2-3, 371–375

Now let $D^b(Coh(X)) \cong D^b(Coh(Y) )$ and $X$ admit Kähler-Einstein metric of negative Ricci curvature , then $Y$ admit also Kähler-Einstein metric of negative Ricci curvature (due to Theorem 1.4)

I:Now assume that $X$ and $Y$ are two smooth projective variety, $D^b(Coh(X)) \cong D^b(Coh(Y) )$ and $X$ is $K$-stable, then $Y$ is $K$-stable? i.e does K-stability is invariant under D-equivalency?

II: From Kawamata we know $D$-equivalence implies the $K$-equivalence for general type varieties, so my conjecture is that for Fano K-stable varieties $D$-equivalence implies the $K$-equivalence

Motivation: D-equivalency gave a lot of same information For example: Kawamata showed, If $K_X$ (resp. $−K_X$) is nef, then $K_Y$ (resp. $−K_Y$ ) is also nef, and an equality on the numerical Kodaira dimension $ν(X) = ν(K_Y )$ (resp. $ν(X, −K_X) = ν(Y, −K_Y )$) holds

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  • $\begingroup$ Uehara's example is in dimension 2, I guess. $\endgroup$ – Sasha Oct 21 '17 at 20:35
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    $\begingroup$ @Sasha , I think his conjecture is reduced to $K$-equivalency ===> $D$-equivalency and this holds for dimension 2 by Zariski theorem $\endgroup$ – user21574 Oct 21 '17 at 20:46
  • $\begingroup$ arxiv.org/abs/1710.07370v1 is a recent survey, by Kawamata himself. I haven't read all the details yet, but it might give some answers, or pointers to the literature. $\endgroup$ – pbelmans Oct 23 '17 at 4:30
  • $\begingroup$ Isn't K-stability typically a property of Fano varieties? In the Fano case derived category determines X. $\endgroup$ – Evgeny Shinder Oct 24 '17 at 7:01
  • $\begingroup$ @EvgenyShinder , there is two $K$ here, of course, K-stability I mean for Fano varieties, but for Sasakian manifolds also we have this notion(which in my opinion is misleading since for Sasakian manifolds $Ric^T(\omega)=\Phi\omega^T$ where $\Phi$ is the leafwise constant and such $\Phi$ is a function which may not be constant) $\endgroup$ – user21574 Oct 24 '17 at 11:55

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