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?
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$\begingroup$ Are $X$ and $Y$ projective? It is quite rare to have families of Abelian varieties over a base $B$ that is projective (rather than merely quasi-projective). $\endgroup$– Jason StarrCommented Dec 21, 2015 at 19:57
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$\begingroup$ I'm mainly interested in the projective case, but comments to the nonprojective case would be apreciated as well. $\endgroup$– DominikCommented Dec 21, 2015 at 20:05
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1 Answer
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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.
