Mathematics

 For smooth separated families over a noetherian regular affine scheme, we give an alternative description of the derived category of the generic fiber as a Verdier quotient. When the family is a proper effectivization of a formal deformation, the Verdier quotient is equivalent to the derived category of the general fiber introduced by Huybrechts--Macrì--Stellari. We also study the induced Fourier--Mukai transform on the generic fiber. If either of those families is locally projective, then general fibers are derived-equivalent if and only if so are the generic fiber. As an application, given a pair of derived-equivalent Calabi--Yau manifolds of dimension more than two, we show that the derived equivalence can be extended to the generic fiber of versal deformations. In this talk, after reviewing basic properties of Serre and Verdier quotients, we explain the key idea in the proof, which is inspired by Bondal--Van den Bergh (presumably in turn Neeman).