Kähler eigenvalue pinching implies Gromov–Hausdorff rigidity to CP^n

Establish that for every ε > 0 there exists a dimension-dependent δ(n, ε) > 0 such that any compact Kähler manifold (M, ω) of complex dimension n satisfying Ric(ω) > ω and λ_{n+3}(Δ_ω) < 1 + δ is biholomorphic to CP^n and the Gromov–Hausdorff distance d_{GH}((M, ω), (CP^n, ω_{CP^n})) is less than ε.

Background

The paper develops eigenvalue comparison and rigidity results for Kähler manifolds with positive Ricci lower bound, showing sharp rigidity when the first eigenvalue multiplicity reaches the maximum associated with (CP^n, ω{CP^n}). In the Riemannian setting, Petersen and Aubry proved an almost rigidity result: under positive Ricci curvature and near-maximal first eigenvalues, manifolds are close in Gromov–Hausdorff distance to the sphere. Motivated by this, the authors propose a Kähler analogue stating that pinching of the (n+3)-rd eigenvalue under Ric(ω) > ω should force biholomorphism to CP^n and Gromov–Hausdorff closeness to (CP^n, ω{CP^n}).

The authors prove this almost rigidity under the Kähler–Einstein assumption (Ric(ω) = ω) and an “almost Kähler–Einstein” framework, but leave the general Ric(ω) > ω case as a conjecture, indicating that new ideas may be needed. This conjecture is central to extending Riemannian eigenvalue pinching phenomena to the Kähler category.