The Weyl tube theorem for Kähler manifolds

Bernig, Andreas; Fu, Joseph H G; Solanes, Gil; Wannerer, Thomas

doi:10.2140/gt.2026.30.2713

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ISSN (electronic): 1364-0380

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The Weyl tube theorem for Kähler manifolds

Andreas Bernig, Joseph H G Fu, Gil Solanes and Thomas Wannerer

Geometry & Topology 30 (2026) 2713–2772

DOI: 10.2140/gt.2026.30.2713

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

Keywords

smooth valuations, Kaehler manifolds, hermitian integral geometry

Mathematical Subject Classification

Primary: 32Q15, 53A07, 53A55, 53C65

References

Publication

Received: 3 June 2025

Revised: 29 January 2026

Accepted: 27 March 2026

Published: 13 August 2026

Proposed: Gang Tian

Seconded: Urs Lang, Leonid Polterovich

Authors

Andreas Bernig
Institut fur Mathematik Goethe-Universität Frankfurt Frankfurt Germany

Joseph H G Fu
Department of Mathematics University of Georgia Athens, GA United States

Gil Solanes
Departament de Matemàtiques Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica Campus de Bellaterra Bellaterra Spain

Thomas Wannerer
Fakultät für Mathematik und Informatik Institut für Mathematik Friedrich-Schiller-Universität Jena Jena Germany

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Volume 30, issue 7 (2026)