Derived categories and birationality

We discuss the question of finding conditions on a derived equivalence between two smooth projective varieties $X$ and $Y$ that imply that $X$ and $Y$ are birational. The types of conditions we consider are in the spirit of finding categorical analogous of classical Torelli theorems. We study, in particular, a notion of strongly filtered derived equivalence and study cases where strongly filtered derived equivalence implies birationality. We also consider an open variant of our main question.Comment: Paper is withdrawn in order to break it into more manageable parts. In addition, Theorem 3.2 is false as stated. A fixed version of the relevant material is now found at arXiv:2208.1437