Universal Hermitian Projective Calculus for CH2

Abstract

We develop an algebraic invariant calculus for complex hyperbolic two-space $\mathbb{C}\mathrm{H}^2$ in its Hermitian projective model. Starting from a three-dimensional Hermitian vector space of signature $(2,1)$, we treat $\mathbb{C}\mathrm{H}^2$ as the projectivized negative cone and its boundary as the projectivized null cone. The resulting calculus replaces metric and angular quantities by projectively invariant algebraic data: Hermitian pair moduli, determinant quadrance, polar spread, triple-product phase, cross-ratio invariants, contact boundary forms, normalized ball kernels, and finite Hermitian incidence laws. We prove denominator-cleared identities for three-point Gram determinants, triangle trigonometry, complex geodesics, bisectors, spinal spheres, projective-unitary covariance, CR/contact atlas transitions, and kernel boundary limits. The triple-product phase supplies the algebraic source of the Cartan angular invariant, while pair invariants recover the usual Bergman distance and polar-spread data over $\mathbb{C}$. The same formulas are also organized over starred fields, including finite Hermitian geometries, giving a field-uniform projective calculus for complex hyperbolic geometry.