Let $B$ be a smooth projective complex variety and $\pi:X\to B$ a smooth projective map whose fibres $X\_b$ are abelian varieties. Let $\psi:Y\to B$ be the naturally associated bundle such that the fibres $Y\_b$ are dual to $X\_b$ (more precisely, $Y$ is the relative Picard scheme $Pic^0(X/B)$). Is it always true that the bounded derived categories of $X$ and $Y$ are equivalent?



Derived equivalence of families of dual abelian varieties
Yes if $X$ is an abelian scheme over $B$: the Fourier-Mukai functor provides an equivalence of the derived categories. This is Theorem 1.1 in Mukai's *Fourier functor and its application to the moduli of bundles on an abelian variety*, in Algebraic geometry, Sendai, 1985, pp. 515-550; Adv. Stud. Pure Math., 10, North-Holland (1987). Note that you need not assume that $B$ is projective. 


