The Weyl tube theorem for K\"ahler manifolds

TL;DR

This paper extends Weyl's tube theorem to Kähler manifolds, constructing a canonical subalgebra of valuations that generalizes Euclidean invariants and explains phenomena in hermitian integral geometry.

Contribution

It introduces a new canonical subalgebra of valuations for Kähler manifolds, generalizing classical Euclidean invariants and answering a question posed by Alesker in 2010.

Findings

Constructs a larger canonical subalgebra (M) for Ke4hler manifolds.

Shows (M) is isomorphic to valuations invariant under the holomorphic isometry group.

Demonstrates restriction maps induce surjections between these subalgebras for embeddings.

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