The special features of toric differential geometry can be traced back to the fact that the group of Hamiltonian diffeomorphisms which commute with the action is abelian. In general, the action of a compact group $G$ on a symplectic manifold $(M,\omega)$ is called “multiplicity-free” if it has this property. This is equivalent to saying that the all the $G$-invariant functions Poisson-commute. The theory has been developed by a number of authors. The analogous notion in algebraic geometry is that of a spherical variety. The theory of extremal metrics and the Mabuchi functional in this setting has been studied by Alexeev and Katzarkov [2] and by Raza [27] and Podesta and Spiro [25] have extended the theorem of Wang and Zhu for Fano manifolds in this direction. There is also related work of Bielwaski [5]. We will now outline some of these ideas.

There is a general classification of multiplicity-free manifolds ([35], [20]), but rather than attempting to discuss the most general situation we focus on a simple class of examples. Pick a maximal torus $T$ in the compact connected Lie group $G$ and let $V$ be the dual of the Lie algebra of $T$. There is a weight lattice $\Lambda\subset V$. Pick a positive Weyl chamber in $V$ and consider an integral Delzant polytope $P$ whose closure is contained in the interior of this chamber. We construct a manifold from this data and as usual we can take either a symplectic or complex point of view.

Complex

The choice of a Weyl chamber defines a Borel subgroup $B$ of the complexified group $G^{c}$, containing the complexified torus $T_{c}$. For example if $G=U(m)$ the Borel subgroup is the group of complex matrices with zeros below the diagonal. Then we have a generalised flag manifold $Y=G^{c}/B$, which is a compact complex manifold. There is a homomorphism from $B$ to $T^{c}$ which is a left inverse to the inclusion. Now form the toric manifold $X$ associated to the polytope $P$. Then $T^{c}$ acts holomorphically on $X$ and so $B$ does also via the homorphism above. So we get a complex manifold

$$Z=G^{c}\times_{B}X,$$

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with a holomorphic fibration $\pi:Z\rightarrow Y$, having fibre $X$. The group $G^{c}$ acts on $Z$ and $\pi$ is a $K^{c}$-equivariant map. Further, we have a $T^{c}$-equivariant line bundle $L\rightarrow X$ so the same construction yields a $G^{c}$-equivariant line bundle ${\cal L}\rightarrow Z$ which restricts to $L$ on each fibre. We can identify $H^{0}(Z,{\cal L})$ with the sections of the vector bundle $\pi_{*}({\cal L})$ over $F$. Recall that there is a standard basis for $H^{0}(X,L)$ labelled by the lattice points $\nu$ in $\overline{P}$. This yields an isomorphism between $\pi_{*}({\cal L})$ and the direct sum of line bundles $\xi_{\nu}\rightarrow F$ associated to these weights. The Borel-Weil theorem asserts that the holomorphic sections of $\xi_{\nu}$ define the irreducible representation $W_{\nu}$ of $G^{c}$ with highest weight $\nu$. So we see that, as a representation of $G^{c}$,

$$H^{0}(Z,{\cal L})=\bigoplus_{\nu\in\overline{P}}W_{\nu}.$$

In particular the representation is “multiplicity-free”, in the sense that all irreducibles appear with multiplicity at most one. This is the same as saying that the algebra of $G^{c}$-equivariant endomorphisms of $H^{0}(Z,{\cal L})$ is commutative. The terminology “multiplicity free” in the symplectic setting is derived by analogy with this.

Notice that replacing $P$ by a multiple $kP$ yields the same complex manifold $Z$ but replaces ${\cal L}$ by ${\cal L}^{k}$. Translating $P$ by $\nu$ does not change $Z$ but changes the line bundle ${\cal L}$ to ${\cal L}\otimes\pi^{*}(\xi_{\nu})$. In none of the above do we use the fact that $\overline{P}$ lies in the interior of the positive Weyl chamber. This is exactly the condition which implies that ${\cal L}$ is an ample line bundle over $Z$.

Example Take $G=SU(2)$, so $V$ can be identified with ${\bf R}$ and the positive Weyl chamber with the positive reals. Let $P$ be the interval $(p_{1},p_{2})$. Then $X=Y={\bf C}{\bf P}^{1}$ and $Z$ is the blow-up of the complex projective plane atone point. As $p_{1},p_{2}$ vary we get all positive line bundles ${\cal L}$ over $Z$.

For the symplectic description we start by writing $Y=G/T$, and think of $G$ as a principal $T$-bundle over $Y$. As a manifold $Z$ is the associated bundle $G\times_{T}X$. Now $T$ has a Hamiltonian action on $X$. In general suppose a Lie group $K$ has a Hamiltonian action on a symplectic manifold $(M,\Omega)$ and we have a principal $K$-bundle $E\rightarrow U$. Then there is a canonical closed $2$-form $\tilde{\Omega}$ on the associated bundle $E\times_{K}M$ which restricts to $\Omega$ (in the obvious sense) on each fibre. Indeed this is true in the “universal” case when we take the group of all Hamiltonian diffeomorphisms of a symplectic manifold. This theory is explained in detail in [21], Sect. 6.1). It is easy to say explicitly how this works in the case at hand. Choose a basis of $V={\rm Lie}(T)^{*}$. The basis elements can be regarded as left-invariant $1$-forms $\alpha_{i}$ on $G$ and also as the components of a connection form on the $T$-bundle $G\rightarrow Y$. The moment map $\mu:X\rightarrow V$ has components, relative to this basis, which we denote by $x^{i}$, in line with our previous notation. Since the moment map is equivariant we can also regard $\mu$ as a map from $Z$ to $V$ and the components $x^{i}$ as functions on $Z$. Restrict to the open set $Z_{0}\subset Z$ corresponding to the open set $X_{0}\subset X$ where $T$ acts freely. This can be identified with the product $P\times G$, so we can also regard $\alpha_{i}$ as $1$-forms on $X_{0}$. Then we set

$$\tilde{\Omega}=d(\sum x^{i}\alpha_{i})$$

on $Z_{0}$. On each fibre the $1$-forms $\alpha_{i}$ can be identified with the $d\theta_{i}$ and we recover the form $\sum dx^{i}d\theta_{i}$. The point is that, although the $1$-forms $\alpha_{i}$ do not extend over $Z$, the closed $2$-form $\tilde{\Omega}$ does. This is fairly clear from the corresponding discussion on the fibres. The condition that $\overline{P}$ lies inside an open Weyl chamber is exactly the condition that the form $\tilde{\Omega}$ is symplectic. The $G$-invariant functions on $Z$ are just the composite of $\mu$ with functions on $\overline{P}$ and these all Poisson-commute.

An important object in this theory is the “Duistermaat-Heckmann”function $W$ on $V={\rm Lie}(T)^{*}$. It is a polynomial function which, on the open Weyl chamber, gives the symplectic volume of the corresponding coadjoint orbit. Algebraically it is the product of the positive roots, where the roots are viewed as linear functions on $V$. The push-forward $\mu_{*}(\tilde{\Omega}^{N})$of the symplectic measure on $Z$ is the restriction to $\overline{P}$ of $(2\pi)^{n}W$ times the Lebesgue measure on $V$. Thus if we identify functions on $\overline{P}$ with $G$-invariant functions on $Z$ the operation of integration over $Z$ corresponds to the weighted integral

$$\int_{\overline{P}}fWd\underline{x}.$$

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Raza extended the symplectic point of view on toric differential geometry, as outlined (2.1.2) above, to this setting [27]. The orthogonal complement with respect to $\tilde{\Omega}$ defines a field of horizontal subspaces in $Z$, transverse to the fibres. Any $G$-invariant almost-complex structure on $Z$, compatible with $\tilde{\Omega}$, must respect this decomposition and agree with the standard complex structure, induced from $Y$, in the horizontal subspace. So such almost-complex structures correspond to the same $T$-invariant almost-complex structures on $X$ which we studied before, and the integrable structures are determined by an admissible symplectic potential $u$ on $\overline{P}$, as before. The whole difference in the theory resides in the weight function $W$. Raza shows that the scalar curvature of the metric on $Z$ defined by a symplectic potential $u$ is

$$\frac{1}{W}\ \frac{\partial^{2}Wu^{ij}}{\partial x^{i}\partial x^{j}}+f_{G},$$

where $f_{G}$ is function determined by the group $G$. In fact if we let $\sigma\in{\rm Lie}(T)^{*}$ be the sum of the positive roots of $G$ then

$$f_{G}=W^{-1}(W_{i}\sigma^{i}):$$

the derivative of $\log W$ in the direction $\sigma$. This extends Abreu’s formula in the toric case, and also a formula of Calabi, for the case when $K=SU(2)$ ([6], [18]). There there seems to be considerable scope for extending the analytical theory developed in the toric case to this more general setting, similar to the work of Szekelyhidi in [28].

Now we consider the Fano case, where the line bundle ${\cal L}$ is $K_{Z}^{-1}$. This requires, first, that the fibre $X$ be Fano. Recall that there is a preferred centre $\nu_{0}$ in $P$ (the centre of mass of the boundary). The second requirement, to identify ${\cal L}$ with $K_{Z}^{-1}$, is that $\nu_{0}$ is equal to $\sigma$, the sum of the positive roots. (To see this, observe that the line bundle over $Y$ associated to the weight $\sigma$ is the $K_{Y}^{-1}$.) In Section 3 we took this centre to be the origin, but here that would conflict with the Weyl chamber structure. So, given a polytope $P$ satisfying these two conditions above, and an admissible symplectic potential $u$, we define

$$h=(x^{i}-\sigma^{i})u_{i}-u.$$

Then $L-h$ is smooth on $\overline{P}$. The Ricci soliton condition is

$$L-h=G+\sum c_{i}x^{i},$$

for suitable constants $c_{i}$. This falls into the class of equations we considered in 3.2, and the existence theorem of Podesta and Spiro is another illustration of our result there.

What we have discussed is the simplest class of multiplicity-free manifolds. One gets other examples in at least two ways.

• •

  One can allow the boundary of $\overline{P}$ to touch the boundary of the Weyl chamber.

• •

  One can consider polytopes contained in proper affine subspaces of ${\rm Lie}(T)^{*}$.

There seems to be considerable scope for developing this theory, both in the Fano case and for extremal metrics. In the latter case one could hope to extend the results proved for toric varieties, along the lines of the work of Szekelyhidi [28] in the case when $G=SU(2)$.

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.

• •

  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.

• •

  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.

• •

  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.