Lagrangian fibrations on hyperkähler fourfolds
arXiv:1110.2680 · doi:10.1070/IM2014v078n01ABEH002678
Abstract
Let X be a projective hyperkähler manifold containing a Lagrangian subtorus L. We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an almost holomorphic Lagrangian fibration with (strong) fibre L. In dimension four we prove that in the above situation there is always a holomorphic Lagrangian fibration with fibre L, thus answering a question of Beauville in this particular case.
10 pages, 1 figure; v2: added reference to arXiv:1110.2852; v3: exposition improved, bibliography updated, to appear in Izvestiya: Mathematics