Let be an abelian scheme over a scheme . The Fourier–Mukai transform gives an equivalence between the derived category of and the derived category of the dual abelian scheme. We partially extend this to certain schemes over (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular.
Our main result provides a fully faithful functor from a twist of the derived category of to the derived category of . Here is the algebraic space classifying fiberwise numerically trivial line bundles.
Next, we show that every algebraically integrable system gives rise to a degenerate abelian scheme and discuss applications to Hitchin systems.
"Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration." Duke Math. J. 165 (15) 2991 - 3042, 15 October 2016. https://doi.org/10.1215/00127094-3645223