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

Abstract

We established 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(L0 ⊕ L1 ⊕ L2), where Li are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.

An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons

“…Actually, thanks to our Corollary 2.11, we recover the result of Han Lie [22] without the need of special test configuration, see Corollary 3.13. • Finally, as we shall explain in the next sections, the converse of Theorem 3.2 was proven by the second author for weights corresponding to extremal Kähler metrics on semisimple principal toric fibrations [17].…”

“…Weighted cscK toric manifolds. The results from Section 2 are motivated by the study of the existence of weighted cscK metrics on toric manifolds, as studied in [17].…”

“…In the above formulas, by abuse of notation, ω P d denotes both the Kähler metric on P ℓ and its induced metric in 2πc 1 (O E (ℓ + 1)). The tuples (c a ) satisfying (17), parametrize the compatible Kähler classes. Furthermore, suppose that B is a product of Kähler-Einstein manifolds (B, ω B ) := k a=1 (B a , ω a ).…”

“…The interest for such fibrations was significantly renewed last year, when the second author proved in [17], using the breakthrough results of Chen and Cheng [7], that a uniform version of the Yau-Tian-Donaldson conjecture holds for semisimple principal toric fibrations, a very large class of toric fibrations. While it allows to translate the question of existence of extremal Kähler metrics on such manifolds into a question of convex geometry on their moment polytopes, it is not yet an explicitly checkable criterion, as the conditions to check still form an infinite dimensional space.…”

Extremal metrics on toric manifolds and homogeneous toric bundles

“…The appendix of [13] by Yuji Odaka showed that for non-singular spherical varieties, G-uniform K-stability is equivalent to existence of cscK metrics. Jubert (see [25]) showed the equivalence between the existence of extremal Kähler metrics on the total space of semi-simple principal toric fibrations and weighted uniform K-stability of the corresponding Delzant polytopes.…”