Scalar curvature and rigidity of odd-dimensional complex hyperbolic spaces (Q1298111)

We consider rigidity properties of complex hyperbolic spaces of odd complex dimension among Kähler manifolds whose geometry and complex structure are strongly asymptotic to that of the model space. The main result of the paper can be stated as follows: any complete Kähler spin manifold close to the model at infinity whose scalar curvature is no less than the one of the model must be isometric to it. The proof first parallels well-known arguments of Riemannian geometry (known as positive mass arguments) to produce a full set of Kählerian Killing spinors on any of these manifolds. We then show that this fact implies rigidity.