Now we consider another generalisation of toric geometry. Let $G$ be a compact Lie group and $G^{c}$ its complexification. Suppose $G^{c}$ acts holomorphically on a compact complex manifold $V$ and that there is a point $x_{0}\in V$ whose $G^{c}$ orbit is dense. We also want to suppose that the stabiliser $\Gamma\subset G^{c}$ is finite. Then the orbit is a copy of $G^{c}/\Gamma$ in $V$ and the complement is an analytic subvariety (which must contain a divisor if $X$ is Kahler). Of course the case of a toric manifold fits into this picture, except that in that case we can assume $\Gamma$ is trivial (but see the further discussion below). In the next section we will study a particular example of this set-up: the Mukai-Umemura manifold.

Now there is no loss of generality in supposing that $\Gamma$ lies in the compact group $G$ and we can study $G$-invariant Kahler metrics on $V$. Over the dense orbit these can be represented by Kahler potentials $\Phi$ on $G^{c}$ which are invariant under the two groups $G$ (acting by left multiplication) and $\Gamma$ (acting by right multiplication). In other words, $\Phi$ can be regarded as a function on the symmetric space $M=G^{c}/G$ which is invariant under the action of the finite group $\Gamma$ on $M$. We will denote the corresponding function on $M$ by $\phi$.

A finite group $\Gamma$ can enter in the toric case in slightly different way, but leading to the same conclusion. Suppose $\Gamma$ is a finite subgroup of $GL(n,{\bf Z})$ which preserves the polytope $P$ of a toric manifold $X$. (For example if $X$ is ${\bf C}{\bf P}^{n}$, so $P$ is the standard simplex, we can take $\Gamma$ to be the permutations of the $n$ coordinates.) Then there is a group $\hat{T}$ which fits into a split exact sequence

$$1\rightarrow T\rightarrow\hat{T}\rightarrow\Gamma\rightarrow 1$$

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and which acts on $X$. As a toric manifold, we know that we can represent $T$-invariant Kahler metrics on $X$ by potentials $\phi$ on ${\bf R}^{n}$, but now we can further restrict to $\hat{T}$-invariant metrics and these correspond to $\Gamma$-invariant functions $\phi$, for the natural action of $\Gamma$ on ${\bf R}^{n}$ (of course, this copy of ${\bf R}^{n}$ is really the dual of that containing $P$).

We now develop the local Kahler differential geometry in this situation, working in terms of a function $\phi$ on the symmetric space $M$. This has a standard connection on its tangent bundle, which is the Levi-Civita connection for any $G^{c}$-invariant metric. Thus we have a Hessian operator $\nabla^{2}$ taking functions on $M$ to sections of $s^{2}(T^{*}M)$. The tangent space of $V$ at a point $gx_{0}$ can be identified with the complexification of the tangent space of $M$ at the point $Gg$. Thus we have an identification with the symmetric tensors $s^{2}(T^{*}M)$ at $Gg$ with a subspace of $\Lambda^{1,1}TG^{c}$ at $g$. This just corresponds to embedding the real symmetric matrices in the complex Hermitian matrices.

We can see this as follows. First note that in the toric case this is just what we have seen when we identify the Kahler metric with the Hessian $\phi^{ab}$. For the general case, there is no loss in working at the point $g=1$. To evaluate $\nabla^{2}\phi$ on a tangent vector $v$ we take the geodesic $\gamma(t)$ in $M$ starting with initial velocity $v$. Then

$$\nabla^{2}\phi(v)=\frac{d^{2}}{dt^{2}}\phi(\gamma),$$

evaluated at $0$. Now geodesics in $G^{c}/G$ through the identity coset correspond to $1$-parameter subgroups in $G^{c}$ so we have a homomorphism $\tilde{\gamma}:{\bf C}\rightarrow G^{c}$, such that $\gamma(t)=K\tilde{\gamma}(it)\in M$. Then we are essentially reduced to the toric case, restricting to this $1$-parameter subgroup.

Thus the local Kahler geometry in this situation reduces to the study of convex functions on $M$ which, by definition, are those functions $\phi$ with $\nabla^{2}\phi>0$ at each point. Equivalently, they are functions which are convex along geodesics in $M$. Of course this is a generalisation of the case when $M={\bf R}^{n}=T^{n}_{c}/T^{n}$. We can go on to write out the equations we want to solve explicitly in this framework. The Kahler-Einstein equation, in the Fano case, is

$$\det\nabla^{2}\phi=e^{-\phi}.$$

For the scalar curvature; given a convex function $\phi$, we define an operator

$$\Delta_{\phi}(f)=(\nabla^{2}\phi)^{-1}.\nabla^{2}f,$$

where $(\nabla^{2}\phi)^{-1}$ is the quadratic form on $T^{*}M$ induced by the nondegenerate quadratic form $\nabla^{2}\phi$ on $TM$, in the usual way, and the dot denotes the contraction between $s^{2}TM$ and $s^{2}T^{*}M$. Then the scalar curvature of the Kahler metric defined by $\Phi$ is

$$S=\Delta_{\phi}(\log\det\nabla^{2}\phi).$$

Notice that these local constructions make sense on any manifold equipped with a connection and volume form.

There are some important differences between this theory in the case of a semi-simple group $G$ and that in the abelian, toric, case.

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  When we go beyond the local differential geometry we need to consider a class of “admissible” functions $\phi$ which define metrics which extend smoothly to $V$. This imposes some asymptotic growth conditions on $\phi$ (as in the toric case) but these can be more complicated, since they encode the structure of the compactification.

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  In the toric case the local equations are affine invariant, but there is no substitute for the affine group in the semi-simple case. In the semi-simple case we have a preferred metric which changes the character of the theory.

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  The geometry of $M$ in the semi-simple case has negative curvature, reflecting the non-abelian nature of $G$. This makes a radical difference to arguments involving volumes of balls etc.

Again, there seems to the author to be a lot of scope for development of this theory. For example one could consider a function $w$ on a Riemannian manifold of negative curvature which satisfies a differential inequality

$$\det\nabla^{2}w\geq e^{-w},$$

and try to establish analogs of the results proved by Wang and Zhu in the toric case.