The Weyl tube theorem for Kähler manifolds

As sharpened in terms of Alesker's theory of valuations on manifolds, a classic theorem of Weyl asserts that the coefficients of the tube polynomial of an isometrically embedded riemannian manifold M↪Rnconstitute a canonical finite dimensional subalgebra LK(M) of the algebra V(M)of all smooth valuations on M, isomorphic to the algebra of valuations on Euclidean space that are invariant under rigid motions. We construct an analogous, larger, canonical subalgebra KLK(M)⊂V(M)for Kähler manifolds M: i) if dimM=n, then KLK(M)≃ValU(n), the algebra of valuations on Cn invariant under the holomorphic isometry group, and ii) if M↪M~ is a Kähler embedding, then the restriction map V(M~)→V(M) induces a surjection KLK(M~)→KLK(M). This answers a question posed by Alesker in 2010 and gives a structural explanation for some previously known, but mysterious phenomena in hermitian integral geometry.