Higher-Dimensional Calabi-Yau Manifolds and Dimensional Saturation

This paper studies Calabi-Yau manifolds in complex dimensions four and higher, extending the classical theory established for dimensions one through three. We prove existence of a compact CYₙ for every n ≥ 1, survey their Hodge theory, moduli spaces, and mirror symmetry, and investigate how the structure of the Calabi-Yau landscape changes as n increases. The central contribution is the Dimensional Saturation Program: a framework for understanding the asymptotic behaviour of the CY landscape. We propose the Dimensional Saturation Conjecture that beyond a critical dimension n*, the statistical distributions of CY invariants (Hodge numbers, Euler characteristics, moduli dimensions) converge to universal limiting laws. Four growth models for N(n), the number of distinct CYₙ types, are developed and compared. Supporting tools include entropy models for the landscape, reflexive polytope combinatorics, and Monte Carlo experiments at dimensions n = 4,..., 20.