Curvature and the c-projective mobility of Kaehler metrics with hamiltonian 2-forms

TL;DR

This paper explores the relationship between curvature, hamiltonian 2-forms, and c-projective mobility in Kähler metrics, providing explicit classifications and necessary conditions for higher mobility.

Contribution

It establishes necessary and sufficient conditions for Kähler metrics to have high c-projective mobility, linking curvature nullity to the existence of hamiltonian 2-forms and classifying such metrics.

Findings

Kähler metrics with high mobility admit hamiltonian 2-forms

Curvature nullity is necessary for mobility ≥ 3

Explicit classification of metrics with mobility ≥ 3

Abstract

The mobility of a Kaehler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kaehler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kaehler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kaehler metrics with hamiltonian 2-forms, we describe explicitly the Kaehler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an Appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.