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.

37 pages; minor corrections and improvements

References in corpus (2)

• Fourier-Mukai and autoduality for compactified Jacobians II
• A Proof of the Barsotti-Chevalley Theorem on Algebraic Groups

Cited by in corpus (4)

• A support theorem for the Hitchin fibration: the case of $S{L}_n$
• The dual Lagrangian fibration of known hyper-Kähler manifolds
• Ngô support theorem and polarizability of quasi-projective commutative group schemes
• Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations