Gromov-Hausdorff limits and Holomorphic isometries

The aim of this paper is to study pointed Gromov-Hausdorff Convergence of sequences of K\"ahler submanifolds of a fixed K\"ahler ambient space. Our result shows that lower bounds on the scalar curvature imply convergence to a smooth K\"ahler manifold satisfying the same curvature bounds, and admitting a holomorphic isometry in the same ambient space. We then apply this convergence result to prove that there are no holomorphic isometries of a non-compact complete K\"ahler manifold with asymptotically non-negative ones into a finite dimensional complex projective space endowed with the Fubini-Study metric.