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

TL;DR

This paper extends the Fourier--Mukai transform to degenerate abelian schemes with singular fibers, establishing a functorial relationship that applies to integrable systems like Hitchin systems.

Contribution

It introduces a partially extended Fourier--Mukai transform for degenerate abelian schemes and connects it to algebraically integrable systems, including Hitchin systems.

Findings

Established a fully faithful functor from a twist of the derived category of Pic$^ au$(X/B) to D(X)

Showed that algebraically integrable systems induce degenerate abelian schemes

Discussed applications to Hitchin systems

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.