Scalar curvature rigidity of almost Hermitian spin manifolds which are   asymptotically complex hyperbolic
  This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold $(M,g)$ of real dimension $4n+2$ which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. The fact that we do not assume $g$ to be K\"ahler reflects in the inequality for the scalar curvature. 
Mario Listing