Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

Details Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

2005-09-15

arxiv.org/abs/math/0509346v3

Manuscripta Math. 120 (2006), no. 2, 131--150

Mathematics

Algebraic Geometry

Final version; minor changes

Scientific paper

10.1007/s00229-006-0631-4

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.

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