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Geometric structures on complex manifolds
October 4, 2011 12:40, Moscow
 


Lagrangian fibrations on hyperkaehler manifolds

Sönke Rollenske

Johannes Gutenberg-Universität Mainz
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Sönke Rollenske


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Abstract: Hyperkaehler (also called irreducible holomorphic symplectic) manifolds form an important class of manifolds with trivial canonical bundle. One fundamental aspect of their structure theory is the question whether a given hyperkaehler manifold admits a Lagrangian fibration. I will report on a joint project with Daniel Greb and Christian Lehn investigating the following question of Beauville: if a hyperkaehler manifold contains a complex torus $T$ as a Lagrangian submanifold, does it admit a (meromorphic) Lagrangian fibration with fibre $T$? I will describe a complete positive answer to Beauville's Question for non-algebraic hyperkaehler manifolds, and give explicit necessary and sufficient conditions for a positive solution in the general case using the deformation theory of the pair $(X,T)$.

Language: English

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