There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) exploited further.
The trick applies to certain Kähler classes $\mathfrak{k}$ of a complex manifold $M$, and allows one to give upper and lower bounds for the scalar curvature of an extremal Kähler metric in $\mathfrak{k}$, if such a metric exists. This is important later in the paper -- in dimension 2 it allows one to control the Sobolev constant of such a metric, and thus to control the degenerations of sequences of such metrics.
The trick appears at the end of Section 3. I will describe it in a more general setting than Chen-LeBrun-Weber's (hopefully not making any mistakes in this generalization). Let $M$ be a complex manifold, fix a maximal connected compact subgroup $K$ of the group of automorphisms of $M$, and let $\mathfrak{k}$ be a Kähler class of $M$. According to Futaki-Mabuchi there is a canonically determined holomorphic vector field $\Xi_\mathfrak{k}$ which is the extremal Kähler vector field of a $K$-invariant extremal Kähler metric $\omega\in\mathfrak{k}$, if any exists. Suppose that $\mathfrak{k}$ satisifes
Important Property. The holomorphic vector field $\Xi_\mathfrak{k}$ generates a closed 1-parameter subgroup ($\cong S^1$) of automorphisms of $M$.
(Remark: in Chen-LeBrun-Weber's setting, all bilaterally symmetric Kähler classes of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$ have this property.)
Then the range (max minus min) of the scalar curvature of $\omega$ is equal to $\lambda \ \mathfrak{k}\cdot [F]/4\pi$, where $F$ is a rational curve in $M$ which is the closure of a ``generic'' orbit of the group ($\cong\mathbb{C}^*$) of automorphisms generated by $\Xi_\mathfrak{k}$ and its complexification, and where the real number $\lambda$ is such that $\lambda^{-1}\Xi_\mathfrak{k}$ is the standard generator of the $S^1$.
Since the average of the scalar curvature of $\omega$ is cohomologically determined by $\mathfrak{k}$, this then gives upper and lower bounds on the scalar curvature of $\omega$.
Here are my questions (assuming that my slight generalization of Chen-LeBrun-Weber's statement is correct).
- First, is there any general way to determine which Kähler classes $\mathfrak{k}$ satisfy the Important Property?
- Second, if not, what are the known examples of Kähler classes with the Important Property? (The examples I can think of are (1) Chen-LeBrun-Weber's; (2) all Kähler classes on a manifold for which $K=S^1$.)
- And thirdly, in any known examples (other than Chen-LeBrun-Weber's) with the Important Property, has it been computed/can someone compute the explicit bounds on scalar curvature that are obtained?
