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.