Does $\operatorname{D}(X)\cong \operatorname{D}(Y)$ imply $X\cong Y$? Let $X,Y$ be smooth projective varieties and by $\operatorname{D}(X)$ we mean the derived category of the category of bounded complexes of coherent sheaves on $X$. It is well-known that there might exist an equivalence $\operatorname{D}(X)\simeq \operatorname{D}(Y)$ even if $X\not \cong Y$. What if there is an isomorphism $\operatorname{D}(X) \cong \operatorname{D}(Y)$? Does it necessarily imply that $X\cong Y$?

By a theorem of Bondal and Orlov (http://arxiv.org/abs/alg-geom/9712029) the answer is yes for varieties with ample canonical or anticanonical bundle.
