Center of mass and Kähler structures

In this short note we introduce a new center of mass argument on the space of complex structures on any tangent space of a Riemannian manifold. We use this to show there is a universal positive constant, depending only on the dimension of the manifold, according to which an almost hermitian manifold is Kahler if and only if the holonomy action, induced on the space of complex structures at a point, is preserved up to error given by this constant.