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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface of Picard number one we find the following: The Kummer variety is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if is a perfect square. And this is the case if and only if carries a divisor with vanishing Beauville-Bogomolov square.
Let be a compact hyperkähler manifold containing a complex torus as a Lagrangian subvariety. Beauville posed the question whether admits a Lagrangian fibration with fibre . We show that this is indeed the case if is not projective. If is projective we find an almost holomorphic Lagrangian fibration with fibre under additional assumptions on the pair , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian...
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