A Yau–Tian–Donaldson correspondence on a class of toric fibrations

Abstract

We establish a Yau-Tian-Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal Kähler metrics on a large class of toric fibrations, introduced by Apostolov-Calderbank-Gauduchon-Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature Kähler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal Kähler metrics on the total space and a suitable notion of weighted uniform K-stability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle P(L 0 ⊕ L 1 ⊕ L 2 ), where L i are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.Résumé. -Nous établissons une correspondance du type Yau-Tian-Donaldson, exprimée en terme d'un polytope de Delzant, concernant l'existence de métriques Kähler extrémales sur une large classe de fibrations toriques définie par Apostolov-Calderbank-Gauduchon-Tonnesen-Friedman et appelée semi-simple principal toric fibrations. Nous utilisons qu'une extrémale sur l'espace total correspond à une métrique à courbure scalaire constante pondérée (dans le sens de Lahdili) sur la fibre torique correspondante pour obtenir une équivalence entre l'existence des métriques extrémales sur l'espace total et une notion appropriée de K-stabilité uniforme pondéree du polytope de Delzant correspondant. En tant qu'application, nous montrons que le fibré en plan projectif P(L 0 ⊕ L 1 ⊕ L 2 ), où les L i sont des fibrés holomorphes au dessus d'une courbe elliptique, admet une métrique extrémale dans chaque classe de Kähler.

Weighted K–stability and coercivity with applications to extremal Kähler and Sasaki metrics

“…X /, where the weight function p. / is a polynomial depending on the fixed data .p a ; c a ; n a / of the construction. With this observation in mind, we show that (similarly to the case of semisimple rigid toric fibrations recently studied by Jubert [52]) the recent results Chen and Cheng [23] and He [47] can be used to obtain a converse of Theorem 1 in the case of semisimple principal fibrations.…”

“…We shall prove .iii/ D ) .ii/ and .i/ D ) .ii/. The arguments are very similar to the ones in the proof of [52,Theorem 1], where the case when .X; T / is toric is studied. The main idea is to show that, on a semisimple principal .X; T /-fibration, the continuity path used by Chen and Cheng [23] in the cscK case and its modification by He [47] to the extremal case can be adapted to bundle-compatible construction.…”

“…X / need not be extremal in general. We finally note that in the case of toric fibre, [52] provides a further equivalence with a certain weighted notion of uniform K-stability of the corresponding Delzant polytope. If , furthermore, the fibre .X; T / is a smooth toric Fano variety, then this equation is equivalent to the vanishing of the Futaki invariant (3) associated to the weights .pv; z w/ on X .…”

“…Proof This first two equalities are established in [6] (see the proof of Lemma 8) in the special case when .X; ! ; T X / is a toric variety, whereas the third identity is proved in [52] (also in the case when .X; T X / is toric). These computations extend to the general setting with no substantial additional difficulty (by using Lemma A.2 for the second identity), but we include them below for the sake of self-containedness.…”

“…Furthermore, the solution ' t .y/ is smooth as a function on OEt 0 ; 1/ Y . The main observation of [52] is that, with a suitable choice for Q , the path (66) can in fact be reduced to a continuity path on X . To see this we observe that, by [47, Proposition 3.1], one can take Q in (66) to be of the form Q D z !…”