Dima Arinkin

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

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

Singular support of coherent sheaves and the geometric Langlands conjecture

^\tau

Autoduality of compactified Jacobians for curves with plane singularities

(X/B) to the derived category of X. Here Pic

When is the self-intersection of a subvariety a fibration?

^\tau

Fourier transform and middle convolution for irregular D-modules

(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.