Fourier-Mukai transforms and the decomposition theorem for integrable systems

TL;DR

This paper explores the relationship between Fourier-Mukai transforms and the decomposition theorem in integrable systems, proposing a conjecture about their interplay and providing evidence through various cases and geometric properties.

Contribution

It introduces a conjecture linking Fourier-Mukai transforms of sheaves of differentials to Hodge modules in integrable systems, unifying several key results and proving it in specific cases.

Findings

Fourier-Mukai images are Cohen-Macaulay sheaves with middle-dimensional support.

The conjecture holds for smooth integrable systems.

Support of sheaves is governed by higher discriminants.

Abstract

We study the interplay between the Fourier-Mukai transform and the decomposition theorem for an integrable system $\pi: M \rightarrow B$. Our main conjecture is that the Fourier-Mukai transform of sheaves of K\"ahler differentials, after restriction to the formal neighborhood of the zero section, are quantized by the Hodge modules arising in the decomposition theorem for $\pi$. For an integrable system, our formulation unifies the Fourier-Mukai calculation of the structure sheaf by Arinkin-Fedorov, the theorem of the higher direct images by Matsushita, and the "perverse = Hodge" identity by the second and the third authors. As evidence, we show that these Fourier-Mukai images are Cohen-Macaulay sheaves with middle-dimensional support on the relative Picard space, with support governed by the higher discriminants of the integrable system. We also prove the conjecture for smooth integrable systems and certain 2-dimensional families with nodal singular fibers. Finally, we sketch the proof when cuspidal fibers appear.