Lagrangian fibrations on hyperkähler manifolds: intersections of Lagrangians, L-reduction, and fourfolds

Details Lagrangian fibrations on hyperkähler manifolds: intersections of Lagrangians, L-reduction, and fourfolds Lagrangian fibrations on hyperkähler manifolds: intersections of Lagrangians, L-reduction, and fourfolds

2011-10-12

arxiv.org/abs/1110.2680v2

Mathematics

Algebraic Geometry

12 pages, 1 figure; v2: added reference to arXiv:1110.2852

Scientific paper

Let X be a projective hyperk\"ahler 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.

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