Fourier-Mukai transforms and the decomposition theorem for integrable systems

Abstract

We study the interplay between the Fourier-Mukai transform and the decomposition theorem for an integrable system π : M → B. Our main conjecture is that the Fourier-Mukai transform of sheaves of Kähler differentials, after restriction to the formal neighborhood of the zero section, are quantized by the Hodge modules arising in the decomposition theorem for π. 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.

Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds

“…Note the very striking symmetry in this procedure: on one side, we start from a perverse sheaf and take the associated graded with respect to the Hodge filtration (which involves holomorphic forms); on the other side, we start from the sheaf of holomorphic forms and take the associated graded with respect to the perverse filtration. Interestingly, the BGG correspondence is a sort of "poor man's" version of the Fourier-Mukai transform (whose existence for arbitrary Lagrangian fibrations is unfortunately still a conjecture), and in [MSY23], Maulik, Shen and Yin had suggested relating P i and Ω n+i M with the help of the Fourier-Mukai transform.…”

“…1. The purpose of this paper is to prove two beautiful conjectures about the Hodge theory of Lagrangian fibrations on holomorphic symplectic manifolds [SY22a,MSY23]. In general, not much is known about the singular fibers of Lagrangian fibrations.…”