Examples of constant scalar curvature kähler metric that is not kahler einstiein

Question

It is well known that if the first Chern class is proportional to the kähler class given, then every cscK in that class has to be kähler Einstein. So there are two directions to generate examples as far as I can think of:

1). It is a result if we have a cscK in $[w]$ then for any kähler class sufficiently close to $[w]$ the existence of cscK is guaranteed. So maybe we can try perturb our class on manifolds where $C_1(M) < 0$.

2). May be we can consider manifolds where $C_1(M)$ is semiample.

I have no idea. Are there any standard examples which are relatively simple to calculate?

Answer 1

Here is a non-compact example.

Consider the half-space $ \mathbb{C} \times \mathbb{H}$ as a subset of $ \mathbb{C}^2$. We use the Kahler potential $$\Psi = \frac{x_1^2}{x_2} - \log(x_2).$$ Here, $z_1 =x_1+ \sqrt{-1} y_1$ and $z_2 =x_2+ \sqrt{-1} y_2$.

This metric has constant scalar curvature but the Ricci potential is $3 \log(x_2)$, so is not Kahler-Einstein.

This Kahler manifold is known as the Siegel-Jacobi space and has been studied in quite a bit of detail. To give two examples, Mathieu Molitor wrote a paper "Gaussian distributions, Jacobi group and Siegel-Jacobi space" describing the geometry in considerable detail. It was also studied by Jae-Hyun Yang in "Geometry and Arithmetic on the Siegel-Jacobi Space."

Answer 2

$\newcommand{\Proj}{\mathbf{P}}$Examples of compact constant scalar curvature Kähler manifolds that are not Einstein are constructed by solving explicit ODEs in On existence of Kähler metrics with constant scalar curvature, Osaka J. Math., 31 (1994), 561–595. If $M$ is a compact almost-homogeneous space with real hypersurface orbits and two ends, and if the connected automorphism group is reductive, then the set of Kähler classes containing a Kähler metric of constant scalar curvature is a real-algebraic hypersurface that separates the Kähler cone.

The simpest non-product example is $\Proj^{3}$ blown up along two skew lines. We may view this Fano threefold as the completion of the line bundle $\mathcal{O}_{\Proj^{1}}(-1) \otimes \mathcal{O}_{\Proj^{1}}(1) \to \Proj^{1} \times \Proj^{1}$ (with the line bundles pulled back to $\Proj^{1} \times \Proj^{1}$ by projection to the respective factors). The Kähler cone has three parameters, which may be viewed loosely as the sizes $a_{1}$ and $a_{2}$ of the $\Proj^{1}$ factors and the size $2b$ of the $\Proj^{1}$ fibre. [See note below] If we scale to make $b = 1$, the resulting slice of the Kähler cone is the open quadrant $a_{1}$, $a_{2} > 1$.

The point $(a_{1}, a_{2}) = (2, 2)$ is proportional to the anticanonical class, which contains an Einstein-Kähler metric by work of Koiso and Sakane (Non-homogeneous Kähler-Einstein metrics on compact complex manifolds, in Springer Lecture Notes 1201, 1986, 165-179).

The set of classes containing a constant scalar curvature representative consists of the diagonal $a_{1} = a_{2}$ and a branch of the right hyperbola $(a_{1} - 1)(a_{2} - 1) = 1$.

Note: More literally, let $\omega$ denote the unit-area Fubini-Study form on $\Proj^{1}$. In the Kähler class $(a_{1}, a_{2}, b)$, the zero section has Kähler form cohomologous to $(a_{1} - b)\omega \times (a_{2} + b)\omega$ and the infinity section has Kähler form cohomologous to $(a_{1} + b)\omega \times (a_{2} - b)\omega$.