Base Manifolds for Lagrangian Fibrations on Hyperkähler Manifolds

Lagrangian Submanifolds in Hyperkähler manifolds, Legendre transformation

Families of Lagrangian fibrations on hyperkähler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with b2 > 7 admits only finitely many deformation types of holomorphic Lagrangian fibrations.

Algebraic Structures on Hyperkähler Manifolds Algebraic Structures on Hyperkähler Manifolds

Let M be a compact hyperkähler manifold. The hy-perkähler structure equips M with a set R of complex structures parametrized by CP 1 , called the set of induced complex structures. It was known previously that induced complex structures are non-algebraic, except may be a countable set. We prove that a countable set of induced complex structures is algebraic, and this set is dense in R. A more g...

Kobayashi pseudometric on hyperkähler manifolds

The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincaré disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperkähler manifold if it admits a deformation with a Lagrangian fibration, and its...

Mirror Symmetry for hyperkähler manifolds

We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form, also known as complex manifolds of hyperkähler type. We obtain that a complex manifold of hyperkähler type is mirror dual to itself. The Mirror Conjecture is stated (following Kontsevich, ICM talk) as the equivalence of certain algebraic structures related to variations of Hodge structures. We c...