Lagrangian Submanifolds in Hyperkähler Manifolds, Legendre Transformation

Abstract

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkähler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection.

We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperkähler manifolds. We give a Plücker type formula for Lagrangian intersections under this transformation.