The author has written at length on this topic in other papers, so we shall be rather brief here. Expressed in terms of a symplectic potential $u$ the condition for an extremal metric is that the scalar curvature

$$S(u)=-u^{ij}_{ij},$$

is an affine-linear function on $P$. More generally, it is natural in this context to consider the prescribed scalar curvature equation $S(u)=A$ for some given function $A$ on $P$. This can be expressed as a variational problem. Recall that our polytope $P$ comes with preferred defining inequalities $\lambda_{r}(\underline{x})\geq c_{r}$. These linear functions $\lambda_{r}$ define a measure $d\sigma$ on the boundary of $P$ (just a multiple of standard Lebesgue measure on each codimension-$1$ face). Then, given a function $A$ on $P$ we define a linear functional

$$L_{A}(f)=\int_{\partial P}fd\sigma-\int_{P}Afd\underline{x}.$$

Now define a nonlinear functional by

$${\cal F}_{A}(u)=L_{A}(u)-\int_{P}\log\det\nabla^{2}u\ d\underline{x}.$$

Then an admissible symplectic potential $u$ which satisfies the equation $u^{ij}_{ij}=-A$ is an absolute minimiser of the functional ${\cal F}_{A}$.

The functional ${\cal F}_{A}$ is a variant of the Mabuchi functional, which is defined in the general Kahler context. It is a convex functional on the space of convex functions on the polytope $P$. The equation $u^{ij}_{ij}=-A$, together with the Guillemin boundary conditions asserts that the functional $L_{A}$ is represented by the inverse of the Hessian of $u$ in the sense that

$$L_{A}(f)=\int_{P}u^{ij}f_{ij},$$

(10)

for all test functions $f$. We see immediately from this that if a solution $u$ is to exist then $L_{A}$ must vanish on the affine linear functions $f$. This is set of $n+1$ linear constraints on the function $A$. If we take $A$ to be the constant

$$\frac{{\rm Vol}(\partial P,d\sigma)}{{\rm Vol}(P,d\underline{x})},$$

then $L_{A}$ vanishes on the constant functions $f$. The restriction of this functional $L_{A}$ to the linear functions $f$ is the Futaki invariant, in this special setting. Otherwise said, this is essentially the difference between the centre of mass of $(\partial P,d\sigma)$ in ${\bf R}^{n}$ and the centre of mass of $(P,d\underline{x})$. If this Futaki invariant does not vanish then we cannot have a constant scalar curvature metric, but there is a unique affine-linear function $A$ satisfying the constraint above, and we seek an extremal metric with this prescribed scalar curvature.

It is not true that any toric variety admits an extremal metric. To see this observe that if a solution exists then the weak formulation (10) implies that $L_{A}(f)\geq 0$ for convex functions $f$ (with strict inequality if $f$ is, say, smooth and not affine linear). But one can construct examples of toric surfaces where $L_{A}$ does not satisfy this condition, for the affine-linear $A$ above. To fit this in with the discussion of Section 1, imagine following a minimising sequence $u^{(\alpha)}$ for the functional ${\cal F}_{A}$, in the case when no solution exists (there would be a similar discussion for the Calabi functional). Then the typical phenomenon (which one can see explicitly in some simple examples, and probably holds in general) is that $u^{(\alpha)}$ behaves like

$$u^{(\alpha)}\sim C_{\alpha}v,$$

where $C_{\alpha}$ are real, $C_{\alpha}\rightarrow\infty$ and $v$ is a piecewise-linear convex function on $\overline{P}$. Differential geometrically this corresponds to the collapsing of some directions in the torus fibration over the parts of $P$ where the derivative of $v$ is discontinuous. Algebro-geometrically, the data $v$ describes a toric degeneration of $X$ into a singular toric variety $X_{0}$ (at least, this is the case if $v$ is defined by “rational data”). In other words we have a picture much like that sketched in 1.1, except that rather than “jumping” to a different complex structure on the same underlying smooth manifold we have to allow singularities. (In fact a similar thing happens in the Yang-Mills case in higher dimensions, where the limiting structures may be sheaves rather than holomorphic bundles.)

In this way, one has a good understanding of one mechanism by which existence can fail. The more formidable problem is to see if this is the only way. More precisely, it is natural to make the

We refer to [9], [10], [11] for more information about this, particularly in the case when $n=2$.