Gromov-Hausdorff limits and Holomorphic isometries

TL;DR

This paper investigates the convergence behavior of sequences of Kähler submanifolds under Gromov-Hausdorff limits, showing that scalar curvature bounds lead to smooth limits with holomorphic isometries, and applies this to non-existence results for certain holomorphic isometries.

Contribution

It establishes new convergence results for Kähler submanifolds with scalar curvature bounds and proves non-existence of specific holomorphic isometries into projective space.

Findings

Lower scalar curvature bounds imply convergence to smooth Kähler manifolds with the same bounds.

Such limits admit holomorphic isometries in the ambient space.

No holomorphic isometries exist from non-compact complete Kähler manifolds with asymptotically non-negative curvature into projective space.

Abstract

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.