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Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K n A is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K n A carries a divisor with vanishing Beauville-Bogomolov square.

Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske (2013)

Annales scientifiques de l'École Normale Supérieure

Let  X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L . We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair ( X , L ) , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian...

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