The only K\"ahler manifold with degree of mobility $\ge 3$ is $(CP(n), g_{Fubini-Study})$

TL;DR

This paper proves that the only closed connected K"ahler manifold with a high degree of h-projective symmetry is the complex projective space with the Fubini-Study metric, confirming a special case of a classical conjecture.

Contribution

It establishes that a closed K"ahler manifold with degree of mobility at least 3 must be the Fubini-Study space or have affine equivalence, confirming a key case of the Obata-Yano conjecture.

Findings

Only the Fubini-Study metric has degree of mobility ≥ 3 on closed manifolds.

A closed manifold with an essential group of h-projective transformations is isometric to complex projective space.

Generalizes Tanno's 1978 result to pseudo-Riemannian K"ahler manifolds.

Abstract

The degree of mobility of a (pseudo-Riemannian) K\"ahler metric is the dimension of the space of metrics h-projectively equivalent to it. We prove that a metric on a closed connected manifold can not have the degree of mobility $\ge 3$ unless it is essentially the Fubini-Study metric, or the h-projective equivalence is actually the affine equivalence. As the main application we prove an important special case of the classical conjecture attributed to Obata and Yano, stating that a closed manifold admitting an essential group of h-projective transformations is $(CP(n), g_{Fubini-Study})$ (up to a multiplication of the metric by a constant). An additional result is the generalization of a certain result of Tanno 1978 for the pseudo-Riemannian situation.