Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

Abstract

A rational Lagrangian fibration f on an irreducible symplecitc variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov--Beauville square 0. This conjecture is proved in the case when V is the punctual Hilbert scheme of a generic algebraic K3 surface S. The construction of f uses a twisted Fourier--Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier--Mukai partner.

arXiv commentFinal version; minor changes

References in corpus (4)

• Abelian fibred holomorphic symplectic manifolds
• Lagrangian fibrations on generalized Kummer varieties
• Equivalences of twisted K3 surfaces
• The abelian fibration on the Hilbert cube of a K3 surface of genus 9

Cited by in corpus (10)

• A survey of Torelli and monodromy results for holomorphic-symplectic varieties
• MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations
• Projectivity and Birational Geometry of Bridgeland moduli spaces
• Brauer groups on K3 surfaces and arithmetic applications
• On the Beauville conjecture
• On the degree of irrationality of low genus $K3$ surfaces
• Fibrations on four-folds with trivial canonical bundles
• Moduli spaces of sheaves on K3 surfaces
• Fourier-Mukai transform on abelian surfaces
• New symplectic V-manifolds of dimension four via the relative compactified Prymian