Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

Abstract

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 fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birational to X on which the fibration is holomorphic.

28 pages; v2: minor corrections, added Thm 6.7; v3: implemented referees' comments, added slightly more detailed argument in Sect. 6.3.3, accepted for publication by Annales scientifiques de l'École normale supérieure

References in corpus (3)

• New outlook on the Minimal Model Program, I
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• On deformations of Q-factorial symplectic varieties

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• Strictly nef divisors on K-trivial fourfolds
• Subvarieties of moduli spaces of sheaves via finite coverings
• The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces
• Lagrangian fibrations on hyperkähler fourfolds