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 $\mathrm{II}\ge\sqrt{44/5}\,g$ suffices in every complex dimension under $\operatorname{Hess} V\ge2|\nabla V|g$ and $\operatorname{osc}V\le4n$. It yields $\operatorname{Hess}(-\logϕ_1)>g$ and $λ_2-λ_1>\max\{n+2+\sqrt{n^2+5n+11},\,2n+14/3\}$. 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 $n\ge3$ these are, to our knowledge, the first log-concavity estimates for domains in $\mathbb{CP}^n$ without symmetry assumptions. We also prove a sharp diameter bound and Hessian levels linear in $n$ under larger dimension-free curvature bounds.