The Weyl tube theorem for Kähler manifolds

Abstract

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 $M\hookrightarrow {\mathbb{R}}^n$ constitute a canonical finite dimensional subalgebra $\mathcal{L}\mathcal{K}(M)$ of the algebra $\mathcal{V}(M)$ 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 $\mathcal{K}\mathcal{L}\mathcal{K}(M)\subset \mathcal{V}(M)$ for Kähler manifolds $M$: i) if $\dim M=n$, then $\mathcal{K}\mathcal{L}\mathcal{K}(M)\simeq {\mathrm{Val}}^{\mathrm{U}(n)}$, the algebra of valuations on ${\mathbb{C}}^n$ invariant under the holomorphic isometry group, and ii) if $M\hookrightarrow \tilde{M}$ is a Kähler embedding, then the restriction map $\mathcal{V}(\tilde{M})\to \mathcal{V}(M)$ induces a surjection $\mathcal{K}\mathcal{L}\mathcal{K}(\tilde{M})\to \mathcal{K}\mathcal{L}\mathcal{K}(M)$. 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.

62 pages; minor changes