Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration

Abstract

Let $X$ be an abelian scheme over a scheme $B$. The Fourier–Mukai transform gives an equivalence between the derived category of $X$ and the derived category of the dual abelian scheme. We partially extend this to certain schemes $X$ over $B$ (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 ${Pic}^{\tau }(X/B)$ to the derived category of $X$. Here ${Pic}^{\tau }(X/B)$ 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.