论文标题
Kähler歧管的Weyl Tube定理
The Weyl tube theorem for Kähler manifolds
论文作者
论文摘要
正如阿莱斯克（Alesker）对流形的估值理论的尖锐性一样，韦尔（Weyl）的经典理论断言，同一个嵌入的riemannian歧管$ m \ hookrightArrow \ hookrightarrow \ mathbb r^n $构成了属于规范的有限量下级$ sula $ naber y MathBra $ ka） $ \ MATHCAL {V}（M）$ M $平滑估值的$，同构成欧几里得空间的估值代数，这些估值在僵化的动作下是不变的。我们构造了一个类似的，更大的，规范的subalgebra $ \ Mathcal {klk}（m） \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）$诱导滤波$ \ MATHCAL {klk}（\ tilde m）\ to \ Mathcal {klk}（klk}（m）$。这回答了Alesker在2010年提出的一个问题，并为Hermitian积分几何形状中一些以前已知但神秘的现象提供了结构性解释。
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 Kähler 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 Kähler 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.
