Lagrangian submanifolds in hyperkahler manifolds, Legendre transformation

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperk<(a)double over dot>hler 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 hyperk<(a)double over dot>hler manifolds. We give a Pl<(u)double over dot>cker type formula for Lagrangian intersections under this transformation.