Here we work in the neighbourhood of a point in the free orbit $X_{0}$. We can use the group action to define local co-ordinates. So we have complex co-ordinates

$$\tau_{a}=\frac{1}{2}(t_{a}+i\theta_{a})$$

say. The factor $2$ here will simplify the formulae later. Locally the isometry group acts by translations in the $\theta_{a}$ directions. (Later, when we work globally, the $\theta_{a}$ will become “angular” co-ordinates, with period $4\pi$.) Locally, a Kahler metric is given by $i\partial\overline{\partial}\phi$ for a function $\phi$ of the complex variables $\tau_{a}$. If this function only depends on the real parts $t_{a}$ then the metric will obviously be invariant under translations in the $\theta_{a}$ directions and it is not hard to see that any metric of the kind we are considering arises in this way. Now if we write $\phi=\phi(t_{a})$ then the tensor $i\partial\overline{\partial}\phi$ is just

$$\sum_{ab}\frac{\partial\phi}{\partial t_{a}\partial t_{b}}d\tau_{a}d\overline{\tau}_{b},$$

and this defines a positive Hermitian form if and only if the Hessian matrix of $\phi$ is positive definite; or in other words $\phi$ is a convex function of the real variables $\tau_{a}$. Thus the theory of convex functions on Euclidean spaces is embedded, as this translationally invariant case, in the theory of Kahler geometry. We write $\nabla^{2}\phi$ for the Hessian of $\phi$ and also use index notation $\nabla^{2}\phi=(\phi^{ab})$. The placing of the indices is unconventional but will be convenient later. We write $(\phi_{ab})$ for the inverse matrix. Explicitly the symplectic form $\omega$ is

$$\frac{1}{2}\sum\phi^{ab}dt_{a}\wedge d\theta_{b},$$

and the Riemannian metric is

$$\frac{1}{2}\left(\sum\phi^{ab}dt^{a}dt^{b}+\sum\phi^{ab}d\theta^{a}d\theta^{b}\right).$$

We regard the curvature tensor of this metric as an element of $\Lambda^{2}\otimes\Lambda^{2}$. Then the curvature tensor is

$$\sum R^{abcd}d\tau_{a}d\overline{\tau}_{b}\otimes d\tau_{c}d\overline{\tau}_{d},$$

where

$$R^{abcd}=\phi^{abcd}-\phi^{ac\lambda}\phi^{bd\mu}\phi_{\lambda\mu}.$$

(2)

(Here we use the summation convention over the repeated indices. The third and fourth order derivatives of $\phi$ are written as $\phi^{abc},\phi^{abcd}$ in the obvious way.) This formula for the curvature tensor is just the formula (1), expressed in our current notation.)