Gromov-Hausdorff limits and holomorphic isometries

The aim of this paper is to study pointed Gromov-Hausdorff Convergence of sequences of Kähler submanifolds of a fixed Kähler ambient space. Our result shows that lower bounds on the scalar curvature imply convergence to a smooth Kähler 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 Kähler manifold with asymptotically non-negative curvature into a finite dimensional complex projective space endowed with the Fubini-Study metric.