Weighted extremal Kähler metrics and the Einstein-Maxwell geometry of projective bundles

Abstract

We study the existence of weighted extremal Kähler metrics in the sense of [4,32] on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined in [5], and they include the projective line bundles [29] and their blow-downs [31], thus providing a most general setting for extending the existence theory for extremal Kähler metrics pioneered by a seminal construction of Calabi [12]. We obtain a general existence result for weighted extremal metrics on admissible manifolds, which yields many new examples of conformally Kähler, Einstein-Maxwell metrics of complex dimension m > 2, thus extending the recent constructions of [30,38] to higher dimensions. For each admissible Kähler class on an admissible projective bundle, we associate an explicit function of one variable and show that if it is positive on the interval (−1, 1), then there exists a weighted extremal Kähler metric in the given class, whereas if it is strictly negative somewhere in (−1, 1), there is no Kähler metrics of constant weighted scalar curvature in that class. We also relate the positivity of the function to a notion of weighted K-stability, thus establishing a Yau-Tian-Donaldson type correspondence for the existence of Kähler metrics of constant weighted scalar curvature in the rational admissible Kähler classes on an admissible projective bundle. Weighted extremal orthotoric metrics are examined in an appendix.

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

“…Furthermore, assuming also that ω ∈ 2πc 1 (L) for a polarization L on M, we let K be the vector field on L defined by Ǩ and a positive Killing potential for Ǩ. In this setting, the search for K-extremal Kähler metrics in ζ given by the Calabi ansatz has been recently studied in [6]. The authors show there that (similarly to the extremal case studied in [4]), there exists a polynomial P (z) of degree ≤ m + 2 which is positive on the interval (−1, 1) if and only if (M, J) admits a K-extremal Kähler metric in ζ, and in this case, the extremal Kähler metric is given by the Calabi ansatz.…”

“…The authors show there that (similarly to the extremal case studied in [4]), there exists a polynomial P (z) of degree ≤ m + 2 which is positive on the interval (−1, 1) if and only if (M, J) admits a K-extremal Kähler metric in ζ, and in this case, the extremal Kähler metric is given by the Calabi ansatz. On the other hand, the computations from [6,40] also show that for each z 0 ∈ (−1, 1), P (z 0 ) computes (up to a positive multiple) the relative weighted Donaldson-Futaki invariant of a smooth Kähler test configuration (M , A z 0 ) associated to (M, J, ζ); furthermore, (M , A z 0 ) is a polarized test configuration precisely when z 0 ∈ (−1, 1) ∩ Q. Thus, the weighted K-stability established in Theorem 1 only yields that P (z) must be positive on (−1, 1) ∩ Q should a K-extremal metric exist, whereas Conjecture 5.8 mentioned above would imply the positivity of P (z) everywhere on (−1, 1), i.e., the existence of a K-extremal metric of Calabi type.…”

“…The induced S 1 -action on M corresponds to fibrewise rotations in the trivial factor O of O ⊕ L, and ( 38) is S 1 -invariant with m ω = z being the corresponding momentum map with image P = [−1, 1] ⊆ R. For any real constants b > |a|, f (z) = az + b is a positive Killing potential with respect to (38) for the Killing vector field a Ǩ where K is the generator of the S 1 -action, and the constant b is essentially the normalization constant κ > 0 of §3.2. The f -scalar curvature of the Kähler metric (38) is computed in [6] to be (39) Scal f (g…”

“…Up to a suitable renormalization, this is the setting of [4,6], except that we allowed p j = 0 (which introduces a reducible manifold (M, J) on which nontrivial solutions of the f -extremal equation can exist when f = const), and did not introduce here blow-downs. Indeed, when p j = 0, in order to get to the notation of [4], one simply needs to replace ω j the uniqueness result of [41] (see also Lemma 4.9 in this paper), we conclude as in the proof of Theorem 2 in [4] that the initial (a Ǩ, b)-extremal metric in ζ c must also admit a hamiltonian 2-form of order 1 associated to Ǩ, a contradiction.…”

Globally conformally Kähler Einstein metrics on certain holomorphic bundles

“…Non-Einstein cKEM examples are given by LeBrun [27,28] on CP 1 × CP 1 and the one-point-blowup of CP 2 , by Koca-Tønnesen-Friedman [23] on ruled surfaces of higher genus, and by Futaki-Ono [17] on CP 1 × M where M is a compact constant scalar curvature Kähler manifold of arbitrary dimension. For more research on cKEM metrics, please refer to Apostolov-Maschler [5], Apostolov-Maschler-Tonnesen-Friedman [6], Futaki-Ono [18] and Lahdili [24,25,26].…”