Perverse-Hodge complexes for Lagrangian fibrations

TL;DR

This paper explores the properties of perverse-Hodge complexes in the context of Lagrangian fibrations, proposing a symmetry that generalizes known identities and connects to various geometric structures.

Contribution

It introduces a conjectural symmetry for perverse-Hodge complexes in Lagrangian fibrations, extending the 'Perverse = Hodge' identity and relating to several key geometric theories.

Findings

Conjectural symmetry verified in several cases

Connections established with variations of Hodge structures

Links made to Hilbert schemes and Lie algebras

Abstract

Perverse-Hodge complexes are objects in the derived category of coherent sheaves obtained from Hodge modules associated with Saito's decomposition theorem. We study perverse-Hodge complexes for Lagrangian fibrations and propose a symmetry between them. This conjectural symmetry categorifies the "Perverse = Hodge" identity of the authors and specializes to Matsushita's theorem on the higher direct images of the structure sheaf. We verify our conjecture in several cases by making connections with variations of Hodge structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.