Log-Concavity and Fundamental Gaps on Complex Projective Domains

We prove quantitative log-concavity and fundamental-gap estimates for Dirichlet Schrödinger operators on complex projective domains. With holomorphic sectional curvature four, the boundary condition II≥√44/5\,g suffices in every complex dimension under Hess V≥2|∇ V|g and oscV≤4n. It yields Hess(-logϕ₁)>g and λ₂-λ₁>max+2+√n²+5n+11,\,2n+14/3\. The main difficulty is to control the distinguished Hopf direction together with a transverse Hessian block whose dimension grows with n. We extend modified-Hessian first-contact arguments by retaining the Hopf term and using longitudinal and transverse rotational tests. A trace minimization reduces all gradient orientations to three extreme directions. A dimension-scaled logarithmic barrier then compares their contact costs with a single limiting profile. A radial spectral estimate supplies the required quadratic-in-dimension spectral threshold from a dimension-independent boundary curvature bound. For n≥3 these are, to our knowledge, the first log-concavity estimates for domains in CPⁿ without symmetry assumptions. We also prove a sharp diameter bound and Hessian levels linear in n under larger dimension-free curvature bounds.