Monodromy and birational geometry of O'Grady's sixfolds

TL;DR

This paper establishes that the bimeromorphic class of certain hyperk"ahler manifolds, specifically O'Grady's sixfolds, is uniquely determined by their Hodge structure, and explores implications for their geometric structures.

Contribution

It proves the monodromy group is maximal for these manifolds and links the existence of square zero divisors to rational Lagrangian fibrations.

Findings

Monodromy group is maximal for O'Grady's sixfolds.

Bimeromorphic class determined by Hodge structure.

Square zero divisors imply rational Lagrangian fibrations.

Abstract

We prove that the bimeromorphic class of a hyperk\"ahler manifold deformation equivalent to O'Grady's six dimensional one is determined by the Hodge structure of its Beauville-Bogomolov lattice by showing that the monodromy group is maximal. As applications, we give the structure for the K\"ahler and the birational K\"ahler cones in this deformation class and we prove that the existence of a square zero divisor implies the existence a rational lagrangian fibration with fixed fibre types.