The geometry of Lagrangian fibres

If the generic fibre f−1(c) of a Lagrangian fibration f : X → B on a complex Poisson– variety X is smooth, compact, and connected, it is isomorphic to the compactification of a complex abelian Lie–group. For affine Lagrangian fibres it is not clear what the structure of the fibre is. Adler and van Moerbeke developed a strategy to prove that the generic fibre of a Lagrangian fibration is isomorphic to the affine part of an abelian variety.rnWe extend their strategy to verify that the generic fibre of a given Lagrangian fibration is the affine part of a (C∗)r–extension of an abelian variety. This strategy turned out to be successful for all examples we studied. Additionally we studied examples of Lagrangian fibrations that have the affine part of a ramified cyclic cover of an abelian variety as generic fibre. We obtained an embedding in a Lagrangian fibration that has the affine part of a C∗–extension of an abelian variety as generic fibre. This embedding is not an embedding in the category of Lagrangian fibrations. The C∗–quotient of the new Lagrangian fibration defines in a natural way a deformation of the cyclic quotient of the original Lagrangian fibration.

Hyperbolic Fibrations and PDE

2010 Mathematics Subject Classification: 35L10, 35L90.

A new construction of Calabi–Yau manifolds: Generalized CICYs

We present a generalization of the complete intersection in products of projective space (CICY) construction of Calabi–Yau manifolds. CICY three-folds and four-folds have been studied extensively in the physics literature. Their utility stems from the fact that they can be simply described in terms of a ‘configuration matrix’, a matrix of integers from which many of the details of the geometries can be easily extracted. The generalization we present is to allow negative integers in the configuration matrices which were previously taken to have positive semi-definite entries. This broadening of the complete intersection construction leads to a larger class of Calabi–Yau manifolds than that considered in previous work, which nevertheless enjoys much of the same degree of calculational control. These new Calabi–Yau manifolds are complete intersections in (not necessarily Fano) ambient spaces with an effective anticanonical class. We find examples with topology distinct from any that has appeared in the literature to date. The new manifolds thus obtained have many interesting features. For example, they can have smaller Hodge numbers than ordinary CICYs and lead to many examples with elliptic and K3-fibration structures relevant to F-theory and string dualities.