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

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.