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

We establish a Yau–Tian–Donaldson type correspondence, expressed in terms of single Delzant polytope, concerning the existence extremal Kähler metrics on large class toric fibrations, introduced by Apostolov–Calderbank–Gauduchon–Tonnesen-Friedman and called semi-simple principal fibrations. use that an metric total space corresponds to weighted constant scalar curvature (in sense Lahdili) corresponding fiber order obtain equivalence between suitable notion uniform K-stability polytope. As application, we show projective plane bundle ℙ(ℒ 0 ⊕ℒ 1 2 ), where ℒ i are holomorphic line bundles over elliptic curve, admits every class.