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