Valuations and curvature measures on complex spaces

TL;DR

This paper surveys recent advances in hermitian integral geometry, focusing on valuations, curvature measures, and kinematic formulas on complex spaces, highlighting differences from Euclidean cases and recent proofs of local formulas.

Contribution

It provides a comprehensive overview of valuations and curvature measures on complex space forms and details recent results on local and global kinematic formulas, including Wannerer's proof.

Findings

Local formulas contain more information than global ones in hermitian case

Dependence of formulas on holomorphic curvature parameter is essential

Recent proof of local additive kinematic formulas for unitarily invariant measures

Abstract

We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. $\mathbb C^n$, it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature $4\lambda$, and the dependence of the formulas on the parameter $\lambda$. We will also describe Wannerer's recent proof of local additive kinematic formulas for unitarily invariant area measures.