Lagrangian fibrations on irreducible symplectic manifolds. Deformations of Lagrangian subvarieties and affine structures.

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The topic of the research project is the study of Lagrangian fibrations on irreducible symplectic manifolds. The classification theory of complex manifolds (Kodaira classification) and the study of manifolds with vanishing first Chern class suggest to focus on irreducible symplectic manifolds. Lagrangian fibrations are a geometric mean to understand them. The concrete tasks of our project are concerned with deformation theory and affine structures. Some terminology: a variety is a set defined by equations, a manifold is a variety, which locally looks like Euclidean space. A fibration of a manifold is a decomposition of it into subvarieties (fibers), parametrised by a base variety. The large part of the fibers are manifolds, but in the set-up under consideration there are always degenerations. We speak of singular fibers of the fibration. These are at the heart of the study of Lagrangian fibrations. They themself and also as a whole carry a great deal of information. The Lagrangian property allows us to control their deformations. Here are interesting and deep links to moduli- and Hodge theory which we want to understand them for the most important models of singular fibers. It has been noticed, that there are strong restrictions to the base of a fibration. For example, it has an affine structure. We want to use the latter to reproduce the fibration from it. The well-known method from real geometry breaks down in complex geometry and we are looking for the necessary additional element permitting us to carry out the analogous construction. The long term goal is one the one hand to contribute to the classification theory and on the other hand to obtain a systematic method to construct examples.