A YauTianDonaldson correspondence on a class of toric fibrations
Abstract
We established a YauTianDonaldson 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 ApostolovCalderbankGauduchonTonnesenFriedman and called semisimple 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 Kstability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle $\mathbb{P}(\mathcal{L}_0\oplus\mathcal{L}_1 \oplus \mathcal{L}_2)$, where $\mathcal{L}_i$ are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.12297
 Bibcode:
 2021arXiv210812297J
 Keywords:

 Mathematics  Differential Geometry
 EPrint:
 Final version, corrected and updated references, a clarification of the proof of Theorem 7.12 and Lemma 6.3. To appear in Annales de l'Institut Fourier