Valuations and Curvature Measures on Complex Spaces

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 4λ, and the dependence of the formulas on the parameter λ. We will also describe Wannerer’s recent proof of local additive kinematic formulas for unitarily invariant area measures.