Here we consider a complex vector bundle $E$ over a complex manifold $X$. We want to study the interaction between two structures

• •

  A hermitian metric on $E$;

• •

  A holomorphic structure on $E$, which can be defined by a $\overline{\partial}$-operator

  $$\overline{\partial}:\Omega^{0}(E)\rightarrow\Omega^{0,1}(E).$$

A basic fact is that given both of these structures there is a unique compatible unitary connection, in the sense that the $\overline{\partial}$-operator is the $(0,1)$-component of the covariant derivative. Now there are two ways of setting up the theory. In the first—the traditional point of view in complex geometry, as is [14] for example—we fix a holomorphic structure and consider the various Hermitian metrics. Then we have, for example, the formula

for the curvature tensor in a local holomorphic trivialisation, where the metric is defined by a matrix-valued function $h$. In the second point of view—closer to what one does in general Yang-Mills theory—we fix the Hermitian metric and consider various $\overline{\partial}$-operators. We can identify the set of these operators with the space ${\cal A}$ of unitary connections on $E$. This point of view brings in two infinite dimensional groups. First, the group $U(E)$ of unitary automorphisms of $E$ and second the group $GL(E)$ of general linear automorphisms. Then $GL(E)$ acts on the space of $\overline{\partial}$-operators by conjugation, and hence on the set ${\cal A}$ of connections. The $\overline{\partial}$-operators which define equivalent holomorphic structures are exactly those which are in the same orbit of the $GL(E)$-action.

The advantage of this second point of view comes when studying the “jumping” of holomorphic structures. This arises from the fact that the $GL(E)$ orbits are not usually closed in ${\cal A}$. Fix a Kahler metric on the base space $X$ and use this to define the Yang-Mills functional: the $L^{2}$ norm of the curvature. When one seeks Yang-Mills connections compatible with a given holomorphic structure ${\cal E}$ one attempts to minimise this functional over a $GL(E)$ orbit in ${\cal A}$. But it may happen that there is no minimum, in the simplest case because the infimum is achieved at a point in ${\cal A}$ in the closure but not in the orbit itself. Then one finds a Yang-Mills connection not on the original holomorphic bundle ${\cal E}$, but on another one ${\cal E}^{\prime}$, such that there are arbitrarily small deformations of ${\cal E}^{\prime}$ which are isomorphic to ${\cal E}$. This lies at the root of the solution of the link between Yang-Mills theory and stability of holomorphic bundles expressed by the Kobayashi-Hitchin conjecture [3], [33], [7].

Now we pass on to Kahler geometry. We study the interaction between two structures on an underlying manifold $M$: a complex structure and a symplectic form. We require these to be algebraically compatible in the sense that the symplectic form is the imaginary part of a hermitian metric. As before there are two points of view we can take. In the first—the conventional point of view in complex differential geometry—we fix the complex structure and vary the Kahler form. If we choose a reference form $\omega_{0}$ and vary in the fixed cohomology class then (at least when $M$ is compact) any other form can be represented by a Kahler potential, in the shape

For the alternative point of view we fix a symplectic form $\omega$ and consider the space ${\cal J}$ of algebraically-compatible almost-complex structures on $M$. Then the group ${\rm SDiff}$ of symplectomorphisms of $(M,\omega)$ acts on ${\cal J}$, and this is the analogue of the unitary gauge group $U(E)$ in the previous case. We consider the subset ${\cal J}_{{\rm int}}$ of integrable almost complex structures, which is preserved by ${\rm SDiff}$. This is partitioned into equivalence classes under the relation $J_{1}\sim J_{2}$ if $(M,J_{1}),(M,J_{2})$ are isomorphic as complex manifolds. Although the group ${\rm SDiff}$ does not have a true complexification one can argue that the equivalence classes in ${\cal J}_{{\rm int}}$ are formally the orbits of such a (mythical) complexified group, in the sense that they behave that way at the level of tangent spaces and Lie algebras [8].

The focus of this article is on the existence question for four different kinds of special Kahler metrics, working within a fixed Kahler class on a compact manifold.

1. 1.

   Extremal Kahler metrics The definition is due to Calabi [6]. They are critical points (and in fact local minima) of the Calabi functional

   $$\int_{M}|{\rm Riem}(\omega)|^{2}d\mu_{\omega},$$

   where $\omega$ varies over the Kahler metrics in a fixed Kahler class and ${\rm Riem}$ is the Riemann curvature tensor. The Euler-Lagrange equation is

   $$\overline{\partial}({\rm grad}S_{\omega})=0,$$

   where ${\rm grad}$ is the gradient operator defined by $\omega$ and $S(\omega)$ is the scalar curvature. In other words, the vector field ${\rm grad}S_{\omega}$ should be a holomorphic vector field. On the face of it, this is a sixth order partial differential equation for the Kahler potential $\psi$.

2. 2.

   Constant scalar curvature Kahler metrics These are just those with $S_{\omega}$ constant. Certainly they are extremal metrics (since the gradient vanishes), and if it happens that $M$ has no non-trivial holomorphic vector fields then an extremal metric must have constant scalar curvature.

3. 3.

   Kahler-Einstein metrics By definition these are those where the Ricci tensor is a multiple $\lambda\omega$. We will only consider the case when $\lambda$ is positive (the zero and negative cases being completely understood through the results of Yau and Aubin). By rescaling there is no loss in supposing that $\lambda=1$. Solutions can only exist when $M$ is a “Fano” manifold and the class $[\omega]$ is $-c_{1}(M)$.

4. 4.

   Kahler-Ricci solitons These again occur only in the Fano case. They are metrics for which

   $${\rm Ric}-\omega=L_{v}\omega,$$

   where $L_{v}$ is the Lie derivative along a holomorphic vector field $v$.

Obviously a Kahler-Einstein metric has constant scalar curvature. There is no simple relation between the other two classes—extremal metrics and Kahler-Ricci solitons— but they can each be thought of as variants of the theory which take account of the possible holomorphic vector fields on the manifold. All this is elucidated by the theory of the Futaki invariant. We will not go in to this in detail here, since we will see later how the theory works in explicit examples. Suffice it to say that in either situation the relevant holomorphic vector field which can be determined a priori from standard topological data. More precisely, the vector field it determined once we fix a maximal compact connected subgroup of the group of holomorphic automorphisms. In either situation, an extremal metric or Kahler-Ricci soliton will necessarily be Einstein/constant scalar curvature if the Futaki invariant vanishes.

There is. of course, as yet no general existence theory for these structures but at the conjectural level one can see a detailed analogy with the Yang-Mills case. We do not want to go into this further here—partly because there is a comprehensive recent survey article [24]—but proceed with our study of special classes of manifolds.