  We develop the foundation of the complex symplectic geometry of Lagrangian
subvarieties in a hyperkahler 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
hyperkahler manifolds. We give a Plucker type formula for Lagrangian
intersections under this transformation.
