Hermitian Geometry of Complex Multivectors, Determinants and Orientations

We present two geometric interpretations for complex multivectors and determinants: a little known one in terms of square roots of volumes, and a new one which uses fractions of volumes and allows graphical representations. The fraction corresponds to a holomorphy index, which measures the lack of holomorphy of real subspaces of $\mathds{C}^n$ via generalized K\"ahler angles or a disjointness angle. Their interpretations are completed with an unorthodox concept of complex orientation, linked to elementary complex transformations. We also discuss how Clifford algebras relate (or not) to the geometry of Hermitian spaces.