It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler metrics (cscK metrics). I was wondering if:

1) Is it known if the blow-up of $\mathbb P^2$ in one or two points accepts a cscK metric? 2) Are there examples of complex projective manifolds (or varieties with klt singularities) that do not accept any cscK metric for all polarisations? I would expect these varieties, if they exist, to be Fano.

References are welcome.


1 Answer 1


In general, reductiveness of the lie algebra of holomorphic vector fields is an obstruction to the existence of cscK metrics (constant scalar curvature Kähler), not just Kähler-Einstein metrics. In particular, the blow up of $\mathbb{P}^2$ in $1$ or $2$ points cannot admit a cscK metric in any Kähler class. I'm not sure of a good reference for this, I can't say that I have read the original articles proving this fact. In the Kähler-Einstein case, which I'm sure you are aware of, this is proven in ‎Moroianu's notes [1]. The cscK case is stated as Theorem 4.5.2 in an article of Shu [2]. Shu cites the original papers of Lichnerowicz and Matsushima (which are in French). Shu also shows that this lie algebra is not reductive for $\mathbb{P}^2$ blown up at $1$ or $2$ points.

One can also answer your first question algebraically. One consequence of the existence of a cscK metric is K-semistability (this is due to Donaldson [3]), which implies slope semistability, developed by Ross-Thomas [4] in analogy with slope stability of vector bundles. One can then see that the blow-up of $\mathbb{P^2}$ in one point is slope unstable for all polarisations, it is destabilised by the exceptional divisor $E$, this is Example 5.27 [4]. Hence, the blow-up of $\mathbb{P^2}$ in one point cannot admit a cscK metric for any polarisation. This is a special case of a result they prove about projective bundles, Theorem 5.13 [4]. It is perhaps worth noting that $\mathbb{P}^2$ blown up at $2$ points is slope stable [5] polarised by the anti-canonical class, so this method does not work in that case.

By contrast, for $3 \leq m \leq 8$, the blow-up of $\mathbb{P}^2$ at $m$ points has both polarisations which do not admit cscK metrics and polarisations which do. The non-existence result follows by Example 5.29 of Ross-Thomas's paper [4]. On the other hand, it is well known that they admit Kähler-Einstein metrics [6]. Precisely which polarisations admit cscK metrics is not known, however.

It would be interesting to know if there are examples of varieties which admit no cscK metrics for any polarisations but whose lie algebras of holomorphic vector fields are reductive. I am not aware of any such examples.

It may also be worth mentioning that I am not sure there is yet a definition for a cscK metric on a klt variety. The definition of Kähler-Einstein metrics on klt varieties was introduced be Berman [7], as far as I am aware. His definition is stronger than just "Kähler-Einstein on the smooth locus", he also requires that the "algebraic" and "differential" volumes agree, in the sense that $(-K_X)^n$ equals the volume of the smooth locus (of course this is automatic in the smooth case).

[1] Moroianu. Lectures on Kähler geometry

[2] Shu. Compact Complex Surfaces and Constant Scalar Curvature Kähler Metrics arXiv:math/0612013

[3] Donaldson. Lower bounds on the Calabi functional

[4] Ross, Thomas. An obstruction to the existence of constant scalar curvature Kähler metrics

[5] Panov, Ross Slope Stability and Exceptional Divisors of High Genus

[6] Tian. On Calabi’s conjecture for complex surfaces with positive first Chern class

[7] Berman K-polystability of $\mathbb{Q}$-Fano varieties admitting Kahler-Einstein metrics

Apologies if I have given too many references!


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