Hermitian Geometry of Complex Multivectors, Determinants and Orientations

Abstract

Two geometric interpretations for complex multivectors and determinants are presented: a little known one in terms of square roots of volumes, and a new one using fractions of volumes. The fraction is determined by a new holomorphy index, which measures the lack of holomorphy of real subspaces of $\mathbb{C}^n$ via generalized Kähler angles or a disjointness angle, and relates real and complex exterior products. The interpretations are completed with a natural but uncommon concept of complex orientation, related to elementary complex transformations. We also propose graphical representations for complex blades (decomposable multivectors) as fractions of parallelotopes, and discuss how Clifford algebras relate (or not) to Hermitian geometry.