New extremal Kähler metrics on projective bundles

@misc{jubert2026extremal,
  title = {{New extremal Kähler metrics on projective bundles}},
  author = {Simon Jubert},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01094},
  eprint = {2609.01094},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01094},
}

TY  - UNPB
TI  - New extremal Kähler metrics on projective bundles
AU  - Simon Jubert
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01094
UR  - https://arxiv.org/abs/2609.01094
AB  - Consider a holomorphic vector bundle E over a compact complex curve C which decomposes as a sum of stable vector bundles. For the projectivization ℙ(E), we prove that the existence of a compatible extremal almost Kähler (aK) metric of involutive type in the sense of Lejmi is equivalent to the existence of a Calabi extremal Kähler metric. This result rests on the Yau--Tian--Donaldson correspondence in terms of the moment polytope Δ for ℙ(E), proved by the author and Yin in a previous work. The main advantage is that compatible extremal aK metrics of involutive type are solutions to a second-order linear PDE, rather than a fourth-order nonlinear PDE for Calabi's extremal Kähler metrics. As an application, we prove that when E has rank $4$ and C is an elliptic curve or the projective line, ℙ(E) is a Calabi dream manifold, i.e. admits an extremal Kähler metric in every Kähler class.
ER  -