  We obtain new general results on the structure of the space of translation
invariant continuous valuations on convex sets (a version of the hard Lefschetz
theorem). Using these and our previous results we obtain explicit
characterization of unitarily invariant translation invariant continuous
valuations. It implies new integral geometric formulas for real submanifolds in
Hermitian spaces generalizing the classical kinematic formulas in Euclidean
spaces due to Poincar\'e, Chern, Santal\'o, and others.
