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.)

We now take a different point of view, following Guillemin [15] and Abreu [1], and also the general scheme outlined in the previous section. Thus we consider an open set in ${\bf R}^{n}\times{\bf R}^{n}$ with linear coordinates $x^{a},\theta_{a}$. More invariantly, we should write the ambient space as $V\times V^{*}$ where $V={\bf R}^{n}$, with coordinates $x^{a}$. We assume the open set has the form $Q\times V^{*}$ where $Q\subset V$ is convex. On this open set we consider the standard symplectic form

$$\Omega=\frac{1}{2}\sum dx^{a}d\theta_{a}.$$

This is preserved by the translations in the $\theta$ variables. More precisely we have a Hamiltonian action of the group $G=V^{*}$ on the symplectic manifold $Q\times V^{*}$ and the moment map is just the projection to $Q$, with components the coordinates $x^{a}$. We consider $G$-invariant almost-complex structures on $Q\times V^{*}$, algebraically compatible with $\Omega$. Now at each point such a structure is specified by a subspace of the complexified cotangent bundle which has a unique basis of the form

$$\epsilon_{a}=d\theta_{a}+Z_{ab}dx^{b},$$

where $(Z_{ab})$ is a symmetric complex matrix with positive definite imaginary part. (This is just the standard description of the Siegel upper half-space $Sp(n,{\bf R})/U(n)$.) So our almost-complex structure is represented by a matrix-valued function $(Z_{ab})$ and $G$-invariance specifies that $Z$ is a function of the variables $x^{a}$. Following our general scheme we should now determine when such an almost-complex structure is integrable. By definition this means that the $2$-forms

$$d\epsilon_{a}=\frac{\partial Z_{ab}}{\partial x^{c}}dx^{c}dx^{b}$$

can be expressed as $\sum\alpha_{ab}\wedge\epsilon_{b}$ and this only happens when all the $d\epsilon_{a}$ are zero (since $d\epsilon_{a}$ does not contain any terms involving $d\theta_{i}$). So the integrability condition is

$$\frac{\partial Z_{ab}}{\partial x^{c}}=\frac{\partial Z_{ac}}{\partial x^{b}}.$$

(3)

Now consider the action of the infinite-dimensional symplectomorphism group. In this situation we need to consider the symplectic diffeomorphisms that commute with the $G$-action. More precisely we want to take the Hamiltonian diffeomorphisms generated by functions that Poisson-commute with the generators of the $G$-action; but these are just the functions of the $x^{i}$ variables. The corresponding group ${\cal G}$ of diffeomorphisms can be identified with smooth functions on $Q$, where a function $f$ acts by taking a point $(\underline{x},\underline{\theta})$ to $(\underline{x},\underline{\theta}+Df)$. This gives an action on the space of almost-complex structures which simply takes $Z_{ab}$ to $Z_{ab}+f_{ab}$, where $f_{ab}$ is the Hessian of $f$.

Now consider the action of ${\cal G}$ on the integrable structures. The condition (3) implies, by the elementary “criterion for an exact differential”, that there are complex-valued functions $it_{a}$ such that

$$Z_{ab}=i\frac{\partial t_{a}}{\partial x^{b}}.$$

The fact that $Z_{ab}$ is symmetric implies, by the same criterion, that there is a single complex valued function $F$ such that $t_{a}=\frac{\partial F}{\partial x^{a}}$, in other words

$$Z_{ab}=\frac{\partial^{2}F}{\partial x^{a}\partial x^{b}}.$$

If we let $f$ be minus the real part of $F$ then the action of $f\in{\cal G}$ takes the structure $(Z_{ab})$ to a new structure with zero real part. So ,taking account of this diffeomorphism group, we can reduce to considering $Z=iY$, with $Y$ real and positive definite. Now the functions $t_{a}$ are real and $\epsilon_{a}=d(t_{a}+i\theta_{a})$ so $t_{a}+i\theta_{a}$ are local complex co-ordinates. (Thus we confirm the Newlander-Nirenberg integrability theorem in this special case.). Write $u$ for the imaginary part of the function $F$ above, so

$$Y_{ab}=\frac{\partial^{2}u}{\partial x^{a}\partial x^{b}}=u_{ab}.$$

Some linear algebra shows that the metric defined by the almost complex structure and the fixed form $\Omega$ is

$$\frac{1}{2}\sum u_{ij}dx^{i}dx^{j}+u^{ij}d\theta_{i}d\theta_{j},$$

(4)

where $(u^{ij})$ is the matrix inverse of the Hessian $(u_{ij})$.

The conclusion of this is that we have another description of the local differential geometry, defined by a convex function $u$ of the variables $x^{a}$. The relation between this picture and that in complex co-ordinates discussed above is just the Legendre transform for convex functions. That is, given a convex function $u$ on $Q\subset V$ we define a function $\phi$ on an open set $Q^{*}\subset V^{*}$ by decreeing that

$$\phi(\underline{t})=\sum x^{a}t_{a}-u(\underline{x}),$$

where the point $\underline{x}\in V$ is the unique point where $Du=\underline{t}$. As is well-known, this transform expresses a symmetric relation between $u$ and $\phi$, so $u$ is the Legendre transform of $\phi$. Further, the Hessian $\phi^{ab}=\frac{\partial^{2}\phi}{\partial t_{a}t_{b}}$ is the inverse of the Hessian $u_{ab}$ of $u$ at the corresponding point. It is easy to see using this that the Legendre transform does give a Kahler potential for the same metric expressed in the complex co-ordinates. Conversely if we start with the complex description and a convex function $\phi$ then the Legendre transform gives the symplectic picture. More invariantly, the map $\underline{x}$ is characterised as the moment map for the action of the group of translations.

Thus we have two natural coordinate systems to use when discussing this local differential geometry, and of course we can transform any formulae from one set-up to the other. Working in the symplectic picture we set

$$F_{ijkl}=u_{ia}u_{jb}\frac{\partial^{2}u^{ab}}{\partial x^{k}\partial x^{l}}.$$

Then one finds that the Riemann curvature tensor is

$$F_{ijkl}\eta^{i}\wedge\eta^{k}\otimes\eta{j}\wedge\eta^{l},$$

(5)

where $\eta^{a}=dx^{a}+iu^{ab}d\theta_{b}$. So the four-index tensor $F$ is essentially the same as the curvature tensor. For example the norm if the Riemann curvature tensor is the same as the natural norm of $F$ i.e.

$$|F|^{2}=\sum F_{ijkl}F_{abcd}u^{ia}u^{jb}u^{kc}u^{ld}.$$

The Ricci tensor is in the same fashion, equivalent to the tensor

$$G_{ij}=F_{ijkl}u^{kl},$$

which can also be expressed as

$$G_{ij}=\frac{\partial^{2}L}{\partial x^{i}\partial x^{j}}$$

where $L=\log\det(u_{ij})$. The scalar curvature is given by another contraction yielding Abreu’s formula

$$S=G_{ij}u^{ij}=\sum_{ij}\frac{\partial^{2}u^{ij}}{\partial x^{i}\partial x^{j}}.$$

(6)

We mentioned in the previous section that in the general case the group of symplectomorphisms does not have a complexification, and this limits the practicality of the symplectic approach to Kahler geometry. But in this special situation there is a complexification of ${\cal G}$: simply the complex valued functions on $Q$ under addition. Further, in it is nearly true that this complexified group ${\cal G}^{c}$ acts on the set of almost complex structures, represented as matrix-valued functions $Z_{ab}$. The “action” is simply to map $Z$ to $Z+\frac{\partial^{2}F}{\partial x^{a}\partial x^{b}}$. It is only a local action because the condition that the imaginary part of $X$ is positive definite could be violated. Our discussion above asserts that all the integrable structures are in a single orbit of this complexified action and the parametrisation by the function $u$ is the parametrisation by an open set in the quotient ${\cal G}^{c}/{\cal G}$. Further, it is easy to verify in this framework that the scalar curvature given by the formula (6) is a moment map for the action of ${\cal G}$ with respect to the natural symplectic structure on the space of almost-complex structures (which is derived from the invariant symplectic form on the Siegel upper half space), see [9].