# On the groups of c-projective transformations of complete K\"ahler manifolds

## TL;DR

This paper proves that for complete connected K"ahler manifolds, the group of c-projective transformations is almost contained within the complex affine transformations, with only two possible exceptions, confirming a stronger form of the Yano-Obata conjecture.

## Contribution

It establishes a bound on the index of the c-projective transformation group, showing it is at most two unless the manifold is complex projective space with the Fubini-Study metric.

## Key findings

- The index of the group of complex affine transformations in the c-projective group is at most two.
- Equality occurs only for complex projective space with Fubini-Study metric.
- This result confirms a stronger version of the Yano-Obata conjecture for complete K"ahler manifolds.

## Abstract

We show that for any complete connected K\"ahler manifold the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the K\"ahler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini-Study metric. This establishes a stronger version of the recently proved Yano-Obata conjecture for complete K\"ahler manifolds.