We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperk¨ahler 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¨ahler manifolds. We give a Pl¨ucker type formula for Lagrangian intersections under this transformation.

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses K¨ahler identities, the Rumin-de Rham complex and spectral geometry.

We verify a conjecture of Lutwak, Yang, and Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version.

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